0.1$; Figure 7 and see an example in Figure 9) restorations tend to occur when the 21 cm signals possess significant large-scale ($k\\lesssim 1{h ~\\rm Mpc^{-1}}$) components along line of sight, which contribute a non-negligible part to the simulated 21 cm spectra (Figure 10). The best restoration of the 21 cm signals can be obtained with the three-narrow-segment fitting technique (Figures 7 and 9; Table 4), and in single-wide-segment fittings the scatter of power restoration on $\\gtrsim 6h^{-1}$ Mpc scales is larger when compared with our three-narrow-segment fittings (Table 4). We also show the result of the two-narrow-segment approach, which is similar to three-narrow-segment approach but has one less segment. In general the results of two-narrow-segment approach are better than those of the single-narrow-segment separations, but the goodness of separation is more or less sensitive to the choice of $\\Delta z\\_{\\rm gap}$ or $\\Delta \\nu\\_{\\rm gap}$ (Figure 7).\nWe also have applied above separation approaches to higher redshifts of *z*0 = 9, 10, 12, and 14, which correspond to reference frequencies of *ν*0 = 142.0 MHz, 129.1 MHz, 109.3 MHz, and 94.7 MHz for redshifted 21 cm signals, respectively (Table 5). At these redshifts, where the cosmic reionization model adopted in this work (Santos et al. 2003, 2005; §2.1) predicts that the HII bubbles are growing in their earlier stage and less extended, the separation quality of our three-narrow-segment approach is still the best, although statistically the separation quality of the single-wide-segment approach is almost equally good.\nConstraints on EoR Physics\n--------------------------\nIn fiducial reionization models such as the one adopted in this work (Eq. (1)), the power of the redshifted 21 cm signals is mainly determined by the mean halo bias *b* and the average ionization fraction *x**e* at a given redshift (Fig. 11), whereas nowadays both *b* and *x**e* are poorly constrained and suffer large uncertainties among different theoretical reionization models (Santos et al. 2003). In future observations, well restored 21 cm powers will greatly help derive *b* and *x**e*, and then deduce HII bubble size and growth rate. Since powers integrated in different scale ranges follow different contour lines on the *x**e* − *b* diagram (e.g., $w|\\_{k<1 h/{\\rm Mpc}}$ and $w|\\_{k<5 h/{\\rm Mpc}}$ on Fig. 11), it is feasible to constrain *b* and *x**e* simultaneously using the overlapping area of any two of these contours (Fig. 12), the accuracy of which depends on how accurate we can restore the corresponding 21 cm powers. In Figure 12 and Table 3 we clearly see that, for an ideal noise condition (1 mK), the tightest constraints (*b* = 3.50− 0.10+ 0.25 and *x**e* = 0.84− 0.07+ 0.05) are obtained using the powers restored with our three-narrow-segment separation approach, when compared with the input model values (*b* = 3.50 and *x**e* = 0.84; Santos et al. 2003). When using the single-wide-segment approach, the obtained *b* = 3.50− 0.20+ 10.5 and *x**e* = 0.83− 0.65+ 0.13 are also unbiased but their errors are larger by about one order of magnitude compared with three-narrow-segment approach. When using the single-narrow-segment approach both of the constraints (2.85 ≤ *b* ≤ 14 and 0.14 ≤ *x**e* ≤ 0.98) are very poor. For a realistic noise condition (60 mK), the three-narrow-segment separation approach still works well (*b* = 3.55− 0.15+ 0.75 and *x**e* = 0.83− 0.19+ 0.07; Fig. 13 and Table 3) whereas the single-wide-segment approach introduces larger errors in the result (*b* = 3.75− 0.35+ 1.65 and *x**e* = 0.76− 0.26+ 0.12). In such a sense we conclude that the three-narrow-segment quadratic polynomial fitting technique is the best approach to restore the 21 cm signals.\nDetection of BAO signals at Intermediate Redshifts with Three-Segment Separation Approach\n-----------------------------------------------------------------------------------------\nWe have also studied whether or not our three-segment separation technique can be applied to intermediate frequency bands to detect the characteristic BAO signals (Ansari et al. 2012), which can be used as a sensitive probe to investigate the nature of dark matter and to constrain cosmological models (e.g., Bharadwaj et al. 2009 for a review; Visbal et al. 2009; Bagla et al. 2010). We apply the single-narrow-segment, single-wide-segment and three-narrow-segment approaches to the redshift range of 0.35 − 0.65, which corresponds to 861 MHz − 1052 MHz in detection. The power spectrum of the brightness temperature of the 21 cm signals is assumed to follow that of the matter density field\n \n$$P\\_{\\rm 3D,21cm}(k,z)=\\bar{T}^2\\_{\\rm 21cm}(z)b^2(z)P\\_{\\rm 3D,matter}(k,z)$$ \n(Seo et al. 2010), where $\\bar{T}\\_{\\rm 21cm}(z)$ is the brightness temperature of the redshifted 21 cm signals averaged at *z*, and *b*(*z*) is the mean bias. According to Barkana & Loeb (2007), Pritchard & Loeb (2008), and Chang et al. (2008), *b*(*z*) = 1 after the cosmic reionization, and $\\bar{T}\\_{\\rm 21cm}(z)$ is estimated to be\n \n$$\\bar{T}\\_{\\rm 21cm}(z)=\\frac {188x\\_{\\rm HI}(z)\\Omega\\_{\\rm H,0}h(1+z)^2} {\\sqrt{\\Omega\\_{\\rm M}(1+z)^3+1-\\Omega\\_{\\rm M}}}~{\\rm mK},$$ \nwhere $x\\_{\\rm HI}(z)$ is the neutral hydrogen fraction at *z*, and $\\Omega\\_{\\rm H,0}$ is the ratio of the total hydrogen mass density to the critical density at *z* = 0. In literature (e.g., Zwaan et al. 2005; Prochaska et al. 2005; Rao et al. 2006) a conservative estimation $x\\_{\\rm HI}(z)\\Omega\\_{\\rm H,0}= 0.00037$ is often assumed for the neutral hydrogen fraction at intermediate redshifts.\nAs a simple test, we carry out both single-segment and three-segment separations ($\\Delta z\\_{\\rm gap}=0.15,~ 0.20,~ 0.25$ or $\\Delta \\nu\\_{\\rm gap}=100,~ 150,~ 200$ MHz) at *z*0 = 0.5 with a channel width *d**ν* = 0.1 MHz, a channel number *N* = 500 (i.e., a segment width $\\Delta\\nu\\_{\\rm segment}=Nd\\nu=50$ MHz), and $\\sigma\\_{\\rm noise}|\\_{z=0.5}$=20 mK. As shown in Figure 14 and Table 6, the three-segment separation approach (e.g., $\\bar{r}\\_{\\rm fit}=0.016\\pm0.004$ for $z\\_{\\rm segment}=\\{0.5,~0.7,~0.3\\}$) yields better results than the single-segment separation approach ($\\bar{r}\\_{\\rm fit}=0.036\\pm0.015$). The power loss in the reconstruction for the $\\gtrsim 30h^{-1}$ Mpc scale structures is $\\bar{w}\\_{\\rm loss}|\\_{k<0.2h/{\\rm Mpc}}=1-\\bar{w}|\\_{k<0.2h/{\\rm Mpc}}\\leq 0.02$ with three-segment separation approach while $\\bar{w}\\_{\\rm loss}|\\_{k<0.2h/{\\rm Mpc}}=0.19\\pm0.16$ with single-segment separation approach. Therefore we conclude that the three-segment separation approach can successfully restore the 21 cm signals from intermediate redshifts better than the single-segment approach, showing its potential to be employed in detecting the BAO signals that typically possess characteristic ≃ 150 comoving Mpc scales. More details on the simulation of the BAO signals and its separation from the foreground will be presented in the next paper since it is beyond the scope of this work.\nSUMMARY\n=======\nIn this work we construct a complex foreground model and compare between different separation approaches by carrying out statistical analysis of the restored 21 cm spectra and corresponding power spectra, as well as their constraints on mean halo bias *b* and average ionization fraction *x**e*. At *z* = 8 the best restoration of 21 cm signals and the tightest determination of *b* and *x**e* can be obtained with the three-narrow-segment quadratic polynomial fitting technique proposed in this work. We illustrate that the new technique also works well at redshifts higher than 8, and is a potentially useful tool to probe the BAO at intermediate redshifts.\nWe sincerely thank the referee for his/her careful review and valuable comments on our work, which help polish and improve the manuscript. This work was supported by the Ministry of Science and Technology of the People’s Republic of China (973 Program; Grant Nos. 2009CB824900 and 2009CB824904), the National Science Foundation of China (Grant Nos. 10878001, 10973010, 11125313, and 11203041), the Shanghai Science and Technology Commission (Grant No. 11DZ2260700), and Shanghai Jiao Tong University Innovation Fund for Postgraduates.\nAnsari, R., Campagne, J. E., Colom, P., et al. 2012,, 540, A129\nBaek, S., Semelin, B., Di Matteo, P., Revaz, Y., & Combes, F. 2010,, 523, A4\nBagla, J. S., Khandai, N., & Datta, K. K. 2010,, 407, 567\nBarkana, R., & Loeb, A. 2007, Reports on Progress in Physics, 70, 627\nBharadwaj, S., Sethi, S. K., & Saini, T. D. 2009,, 79, 083538\nBowman, J. D., Morales, M. F., & Hewitt, J. N. 2009,, 695, 183\nChang, T.-C., Pen, U.-L., Peterson, J. B., & McDonald, P. 2008, Physical Review Letters, 100, 091303\nChapman, E., Abdalla, F. 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L. 2005,, 359, L30\n[]\n[Tab:publ-works]\n| | | |\n| --- | --- | --- |\n| Component | Mean Value | rms |\n| | (K) | (K) |\n| Galaxy synchrotron emission | 237 | 24 |\n| Galaxy free-free emission | 0.877 | 0.205 |\n| Galaxy clusters | 0.084 | 2.06 |\n| Extragalactic discrete sources | 88 | 7937 |\n| Extragalactic discrete sources (under $S\\_{\\rm cut}$) | 23 | 29 |\n| Total foreground | 326 | 7937 |\n| Total foreground (with discrete sources under $S\\_{\\rm cut}$) | 261 | 38 |\n[]\n[Tab:publ-works]\nc|cc Parameter & Value \nField of view & 10∘ × 10∘ \nTotal effective area $nA\\_{\\rm eff}$ & 17658 m2 \nNumber of baselines *n*(*n* − 1)/2& (81 × 80)/2 \nFrequency coverage & 50 − 200 MHz \nFrequency resolution *d**ν*& 40 kHz \nImage spatial resolution Ω& 1 × 10− 6 sr \nObserving time *τ*& 1 yr \n$\\sigma\\_{\\rm noise}$ at 157.8 MHz & 60 mK \n[]\n[Tab:publ-works]\nc|c|cc|cc\n&&Mean Halo Bias&Average Ionization Fraction \n&$\\bar{w}|\\_{k<1h/{\\rm Mpc}}$&$\\bar{w}|\\_{k<5h/{\\rm Mpc}}$&*b*&*x**e* \n&1.00&1.00&3.50&0.84 \n[.0ex][c]$\\sigma\\_{\\rm noise}|\\_{z=8}$ \n=60 mK &[.0ex][c]Single-narrow-segment \n($\\Delta\\nu\\_{\\rm segment}=2$ MHz) &[0ex][c]0.25 ± 0.04&[0ex][c] 0.66 ± 0.02 &[0ex][c] – &[0ex][c] – \n&&& \n &[0ex][c]Single-wide-segment \n($\\Delta\\nu\\_{\\rm segment}=2$ MHz) &[0ex][c]0.97 ± 0.05&[0ex][c] 1.01 ± 0.02 &[0ex][c] 3.75− 0.35+ 1.65 &[0ex][c] 0.76− 0.26+ 0.12 \n&&& \n &[0ex][c]Three-narrow-segment \n($\\Delta\\nu\\_{\\rm segment}=2$ MHz) & [0ex][c]1.00 ± 0.05& [0ex][c]1.01 ± 0.02 &[0ex][c] 3.55− 0.15+ 0.75 &[0ex][c] 0.83− 0.19+ 0.07 \n&&& \n[.0ex][c]$\\sigma\\_{\\rm noise}|\\_{z=8}$ \n=1 mK &[.0ex][c]Single-narrow-segment \n($\\Delta\\nu\\_{\\rm segment}=2$ MHz) &[0ex][c]0.64 ± 0.24&[0ex][c] 0.83 ± 0.13 &[0ex][c][2.85, 14] &[0ex][c] [0.14, 0.98] \n&&& \n &[0ex][c]Single-wide-segment \n($\\Delta\\nu\\_{\\rm segment}=20$ MHz) & [0ex][c]0.98 ± 0.12& [0ex][c]0.99 ± 0.05 &[0ex][c] 3.50− 0.20+ 10.5 &[0ex][c] 0.83− 0.65+ 0.13 \n&&& \n &[0ex][c]Three-narrow-segment \n($\\Delta\\nu\\_{\\rm segment}=2$ MHz) & [0ex][c]1.00 ± 0.02& [0ex][c]1.00 ± 0.02 &[0ex][c] 3.50− 0.10+ 0.25 & [0ex][c]0.84− 0.07+ 0.05 \n&&& \n Note. *a* Calculations are carried out using the fiducial reionization model of Santos et al. (2003, 2005). \n[]\n[Tab:publ-works]\nc|crcc\n&&$\\bar{r}\\_{\\rm fit}$&$\\bar{w}|\\_{k<1h/{\\rm Mpc}}$ \n[0.0ex][c]Single-narrow-segment \n($\\Delta\\nu\\_{\\rm segment}= 2$ MHz)&&[0.0ex][c]0.153 ± 0.070&[0.0ex][c]0.64 ± 0.24 \n&&& \n[0.0ex][c] Single-wide-segment \n($\\Delta\\nu\\_{\\rm segment}=20$ MHz)&&[0.0ex][c]0.055 ± 0.034&[0.0ex][c]0.98 ± 0.12 \n&&& \n[0.0ex][c] Two-narrow-segment \n($\\Delta\\nu\\_{\\rm segment}= 2$ MHz) &\n[0.0ex][c] $\\Delta z\\_{\\rm gap}$& − 1.5 &0.116 ± 0.083&1.09 ± 0.51 \n& & − 1.0 &0.099 ± 0.072&1.03 ± 0.35 \n& & − 0.5 &0.082 ± 0.054&0.97 ± 0.23 \n& &0.5 &0.062 ± 0.037&0.92 ± 0.16 \n& &1.0 &0.056 ± 0.034&0.91 ± 0.14 \n& &1.5 &0.050 ± 0.028&0.92 ± 0.13 \n & [0.0ex][c] $\\Delta \\nu\\_{\\rm gap}$ \n(MHz)& − 30 &0.048 ± 0.026&0.91 ± 0.12 \n& & − 20 &0.052 ± 0.030&0.91 ± 0.13 \n& & − 10 &0.062 ± 0.037&0.91 ± 0.16 \n& &10 &0.084 ± 0.059&0.96 ± 0.22 \n& &20 &0.098 ± 0.069&1.02 ± 0.31 \n& &30 &0.116 ± 0.086&1.13 ± 0.84 \n[0.0ex][c] Three-narrow-segment \n($\\Delta\\nu\\_{\\rm segment}= 2$ MHz) & [0.0ex][c] $\\Delta z\\_{\\rm gap}$&0.5 &0.028 ± 0.010&1.00 ± 0.02 \n& &1.0 &0.024 ± 0.006&1.00 ± 0.02 \n& &1.5 &0.023 ± 0.005&1.00 ± 0.02 \n & [0.0ex][c] $\\Delta \\nu\\_{\\rm gap}$ \n(MHz)&10 &0.027 ± 0.009&1.00 ± 0.02 \n& &20 &0.023 ± 0.006&1.00 ± 0.02 \n& &30 &0.022 ± 0.005&1.00 ± 0.02 \nNote. *a* We run 1000 rounds of simulation and separation for each separation approach. $\\bar{r}\\_{\\rm fit}$ is the mean relative root-mean-square error for the restoration of the 21 cm signals, and $\\bar{w}|\\_{k<1h/{\\rm Mpc}}$ is the mean power restoration on $\\gtrsim 6h^{-1}$ Mpc scales. Errors indicate the 68% confidence limits.\n[]\n[Tab:publ-works]\nc|c|ccccc *z*0&Parameter&Single-Narrow-Segment&Single-Wide-Segment&Three-Narrow-Segment&Three-Narrow-Segment \n&&($\\Delta\\nu\\_{\\rm segment}=2$ MHz)&($\\Delta\\nu\\_{\\rm segment}=20$ MHz)&($\\Delta z\\_{\\rm gap}=1$)&($\\Delta \\nu\\_{\\rm gap}=10$ MHz) \n 9&$\\bar{r}\\_{\\rm fit}$&0.145 ± 0.067&0.053 ± 0.028&0.032 ± 0.007&0.033 ± 0.009 \n&$\\bar{w}|\\_{k<1 h/{\\rm Mpc}}$&$0.66\\hspace{0.5em}\\pm0.23\\hspace{0.5em}$&$0.99\\hspace{0.5em}\\pm0.10\\hspace{0.5em}$&$1.00\\hspace{0.5em}\\pm0.03\\hspace{0.5em}$&$1.00\\hspace{0.5em}\\pm0.03\\hspace{0.5em}$ \n 10&$\\bar{r}\\_{\\rm fit}$&0.141 ± 0.062&0.058 ± 0.025&0.042 ± 0.010&0.043 ± 0.009 \n&$\\bar{w}|\\_{k<1 h/{\\rm Mpc}}$&$0.67\\hspace{0.5em}\\pm0.23\\hspace{0.5em}$&$0.99\\hspace{0.5em}\\pm0.11\\hspace{0.5em}$&$1.01\\hspace{0.5em}\\pm0.05\\hspace{0.5em}$&$1.01\\hspace{0.5em}\\pm0.05\\hspace{0.5em}$ \n 12&$\\bar{r}\\_{\\rm fit}$&0.146 ± 0.055&0.075 ± 0.022&0.067 ± 0.014&0.066 ± 0.013 \n&$\\bar{w}|\\_{k<1 h/{\\rm Mpc}}$&$0.73\\hspace{0.5em}\\pm0.21\\hspace{0.5em}$&$1.01\\hspace{0.5em}\\pm0.11\\hspace{0.5em}$&$1.01\\hspace{0.5em}\\pm0.08\\hspace{0.5em}$&$1.01\\hspace{0.5em}\\pm0.07\\hspace{0.5em}$ \n 14&$\\bar{r}\\_{\\rm fit}$&0.157 ± 0.050&0.099 ± 0.024&0.094 ± 0.020&0.093 ± 0.020 \n&$\\bar{w}|\\_{k<1 h/{\\rm Mpc}}$&$0.75\\hspace{0.5em}\\pm0.22\\hspace{0.5em}$&$1.02\\hspace{0.5em}\\pm0.13\\hspace{0.5em}$&$1.02\\hspace{0.5em}\\pm0.11\\hspace{0.5em}$&$1.02\\hspace{0.5em}\\pm0.11\\hspace{0.5em}$ \nNote. *a* Parameters are defined in the same way as in Table 4.\n[]\n[Tab:publ-works]\nc|crcc\n&&$\\bar{r}\\_{\\rm fit}$&$\\bar{w}|\\_{k<0.2h/{\\rm Mpc}}$ \n[0.0ex][c]Single-segment \n($\\Delta\\nu\\_{\\rm segment}= 50$ MHz) &&[0.0ex][c]0.036 ± 0.015&[0.0ex][c]0.81 ± 0.16 \n& \n[0.0ex][c] Three-segment \n($\\Delta\\nu\\_{\\rm segment}= 50$ MHz) & [0.0ex][c] $\\Delta z\\_{\\rm gap}$&0.15 &0.016 ± 0.004&1.00 ± 0.02 \n& &0.20 &0.016 ± 0.004&1.00 ± 0.02 \n& &0.25 &0.016 ± 0.003&1.00 ± 0.02 \n & [0.0ex][c] $\\Delta \\nu\\_{\\rm gap}$ \n(MHz)&100 &0.016 ± 0.004&1.00 ± 0.02 \n& &150 &0.015 ± 0.003&1.00 ± 0.02 \n& &200 &0.015 ± 0.003&1.00 ± 0.02 \nNote. *a* Parameters are defined in the same way as in Table 4, whereas the wave number range is confined as $k<0.2h~{\\rm Mpc}^{-1}$ to calculate relative power restoration.\n1D power spectra of 21 cm signals calculated at some typical redshifts by adopting the fiducial reionization model (§2.1; Santos et al. 2003, 2005).\nTelescope arrangement of the 21CMA array. Each telescope is an assembly of 127 logarithmic periodic antennas.\nDensity of visibility measurements in the 21CMA uv plane at 150 MHz with an integration time of 24 hours (§2.2.2). \n Difference between restored ($T\\_{\\rm fore}^{\\,\\prime}$) and input ($T\\_{\\rm fore}$) foregrounds calculated with single-segment separation approach by using different polynomial approximations (§3.1). The calculations are repeated for 1000 times for each given $\\Delta\\nu\\_{\\rm segment} = 2i$ MHz (*i* = 1, 2, ⋯, 40) to determine the mean value of $|T\\_{\\rm fore}^{\\,\\prime}-T\\_{\\rm fore}|$. \n Mean restored power spectra for the 21 cm signals from 10000 sky pixels with $\\sigma\\_{\\rm noise}|\\_{z=8}$=60 mK, as compared with the theoretical 21 cm power spectrum at *z*0 = 8 (black). (a)-(c) are the condition with *d**ν* = 40 kHz, and (d)-(f) with *d**ν* = 100 kHz. Residuals obtained in the *χ*2 test between the restored and theoretical 21 cm power spectra in each case are also plotted in the lower panels.\n A comparison between three separation techniques discussed in §3 and §4. Upper panel: Sketch diagram for the spatial distribution of ionized bubbles along line of sight at *z*0 = 8. Lower panel: Corresponding spectrum of the simulated 21 cm signals (black solid), along with the best-fits (red solid: single-narrow-segment; green solid: single-wide-segment; blue star: three-narrow-segment) of $T\\_{\\rm fore}^{\\prime}-T\\_{\\rm fore}$, which presents the Mpc-scale components of 21 cm signals that have been entangled with the foreground. Clearly the best-fit spectrum of single-narrow-segment is significantly biased by the Mpc-scale components of the bubble, and best-fit spectrum of single-wide-segment is biased by the tens of Mpc-scale components of the giant bubbles. The black dotted line marks $T\\_{\\rm fore}^{\\prime}-T\\_{\\rm fore}=0$, which is the most ideal result for the restoration of the 21 cm signals.\n(a) Fractional distributions of the relative root-mean-square error $r\\_{\\rm fit}$ for single-narrow-segment (black solid), single-wide-segment (red square), two-narrow-segment with *z* gap (magenta solid; upper left to lower right are $\\Delta z\\_{\\rm gap}=1.5,~1.0,~\\cdots,~-1.5$, respectively), two-narrow-segment with *ν* gap (blue dash; upper left to lower right are $\\Delta\\nu\\_{\\rm gap}=-30,~-20,~\\cdots,~30$ MHz, respectively), three-narrow-segment with *z* gap (red solid; upper left to lower right is $\\Delta z\\_{\\rm gap}=1.5,~1.0,~0.5$, respectively), and three-narrow-segment with *ν* gap (green dash; upper left to lower right are $\\Delta\\nu\\_{\\rm gap}=30,~20,~10$ MHz, respectively) separation approaches, each of which is calculated based on 1000 rounds of simulation and separation with $\\sigma\\_{\\rm noise}|\\_{z=8}$=1 mK (Table 4; §4.1). (b) Average power spectra of input and restored 21 cm signals and their 68% dispersion limits. The restorations are performed with single-narrow-segment (blue) and three-narrow-segment (magenta) quadratic polynomial fitting techniques based on 1000 rounds of simulation and separation for each.\nOne case for successful separation with the single-narrow-segment quadratic polynomial fitting technique, for which the segment width is chosen to be 2 MHz, $\\sigma\\_{\\rm noise}|\\_{z=8}$=1 mK, and our complex foreground model is applied (§4.1). (a) Spectra of input 21 cm signals $T\\_{\\rm 21cm}$, restored 21 cm signals plus noise $T\\_{\\rm 21cm}^{\\rm raw}$, noise $T\\_{\\rm noise}$, and the difference between $T\\_{\\rm 21cm}^{\\rm raw}$ and $T\\_{\\rm 21cm}$ are shown in black, blue, green and red, respectively. (b) Corresponding power spectra for (a). Power spectrum of the complex foreground (cyan) is also plotted for comparison. \nSame as Figure 8 but for one case of failed separation at *z*0 = 8 with the single-narrow-segment approach (§4.1). Spectra are marked in the same colors as in Figure 8. \n Relative root-mean-square error $r\\_{\\rm fit}$ for the single-narrow-segment separation at *z*0 = 8 and the power of Mpc-scale components of the simulated 21 cm signals (§4.1). When the fits of single-narrow-segment separation is not acceptable ($r\\_{\\rm fit}>0.1$), there exist a clear correlation between $r\\_{\\rm fit}$ and $P\\_{\\rm 21cm}|\\_{k<1h/{\\rm Mpc}}$, i.e., $\\log P\\_{\\rm 21cm}|\\_{k<1h/{\\rm Mpc}}=0.89 \\log r\\_{\\rm fit} -1.95$ (solid) as can be derived with maximum likelihood method. \nRelative power restorations *w* for (a) $\\gtrsim 6h^{-1}$ Mpc and (b) $\\gtrsim 1h^{-1}$ Mpc scales, each of which is obtained at *z*0 = 8 ($\\sigma\\_{\\rm noise}|\\_{z=8}$=1 mK) with different separation approaches and averaged over 1000 times (Table 3), and their dependence on mean halo bias *b* and average ionization fraction *x**e*. Dashed lines show the 68% errors (§4.2).\n*x**e* and *b* determined by overlapping the $w|\\_{k<1 h/{\\rm Mpc}}$ and $w|\\_{k<5 h/{\\rm Mpc}}$ contours shown in Figures 11(a) and (b) on the *x**e* − *b* map. The input model parameters (*x**e* = 0.84, *b* = 3.50) are shown as a black diamond, meanwhile the best restored parameters are shown with a red cross. Allowed parameter ranges (68% level) are marked in yellow (Table 3). \nSame as Figure 12 but for $\\sigma\\_{\\rm noise}|\\_{z=8}$=60 mK. \nSame as Figure 7 but for *z*0 = 0.5 (Table 6; §4.3).\n"},"data_id":{"kind":"string","value":"arxiv_0183855"}}},{"rowIdx":45992,"cells":{"content":{"kind":"string","value":") – (2.943013355541281,0.5748759344428223) – cycle; (3.2438189126237993,0.6280566799665583) – (3.1418629537083134,0.6280566799665585) – (3.1418629537083134,0.5261007210510723) – (3.2438189126237993,0.5261007210510723) – cycle; (3.1418629537083134,0.6280566799665585) – (3.0460861438180094,0.6280566799665586) – (3.0460861438180094,0.5322798700762547) – (3.1418629537083134,0.5322798700762545) – cycle; (3.0460861438180094,0.6280566799665586) – (2.943013355541281,0.6280566799665588) – (2.943013355541281,0.5249838916898304) – (3.0460861438180094,0.5249838916898303) – cycle; (2.943013355541281,0.5748759344428223) – (2.8859463279440996,0.574875934442822) – (2.8859463279440996,0.5178089068456407) – (2.943013355541281,0.5178089068456407) – cycle; 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(1.4013396432940843,0.6892102033421779)– (1.4497514024230376,0.5618692262717914); (1.4648801457878167,0.92850344203525)– (1.5527565138874866,0.7135460689481155); (1.5527565138874866,0.7135460689481155)– (1.5527565138874866,0.5864650639606508); (1.7326495478173052,0.9635677811925659)– (1.7326495478173052,0.7308627183203932); (1.7326495478173052,0.7308627183203932)– (1.6724233927019152,0.5583838105636105); (1.7326495478173052,0.7308627183203932)– (1.7887759241380015,0.554284031768917); (2.235417273239036,1.4625868852677506)– (2.0309701129296442,0.8942036576058552); (2.235417273239036,1.4625868852677506)– (2.4173853069152877,0.871724530972716); (2.0309701129296442,0.8942036576058552)– (1.9421935623364832,0.6190441408465115); (2.0309701129296442,0.8942036576058552)– (2.124161596012736,0.6234590733364421); (2.4173853069152877,0.871724530972716)– (2.2818860108224763,0.5983295064467444); (2.4173853069152877,0.871724530972716)– (2.4204266524062668,0.5976845932961146); (2.4173853069152877,0.871724530972716)– (2.5559259484990777,0.6013708519377239); (3.0819023883268373,1.3889321581512355)– (2.7234251942819374,0.8272695099918191); (3.0819023883268373,1.3889321581512355)– (2.8840156384402382,0.869864519948035); (3.0819023883268373,1.3889321581512355)– (3.09341613408254,0.778459458507818); (3.0819023883268373,1.3889321581512355)– (3.3928956110263973,0.7797855386464796); (2.7234251942819374,0.8272695099918191)– (2.723425194281938,0.6240840558773009); (2.8840156384402382,0.869864519948035)– (2.884015638440238,0.7518690857459509); (2.884015638440238,0.7518690857459509)– (2.8840156384402382,0.6338736515438649); (2.8840156384402382,0.6338736515438649)– (2.8554821246416475,0.5444117311403699); (2.8840156384402382,0.6338736515438649)– (2.9144798417426903,0.5463424206442314); (3.09341613408254,0.778459458507818)– (2.994549749679645,0.5765202858281945); (3.09341613408254,0.778459458507818)– (3.0939745487631614,0.5801682750214066); (3.09341613408254,0.778459458507818)– (3.1928409331660563,0.5770787005088154); (3.3928956110263973,0.7797855386464796)– (3.2922178147107495,0.5823099381569325); (3.3928956110263973,0.7797855386464796)– (3.3898461138927654,0.5814794431488177); (3.3928956110263973,0.7797855386464796)– (3.4905239102084127,0.5792604410233004); (3.770986154714497,1.1619605433968896)– (3.6435650364862546,0.8313539710541298); (3.770986154714497,1.1619605433968896)– (3.872578881771756,0.8055255798831467); (3.6435650364862546,0.8313539710541298)– (3.592315258018243,0.6794182954076224); (3.6435650364862546,0.8313539710541298)– (3.694629066607704,0.6792325470610602); (3.872578881771756,0.8055255798831467)– (3.804662075105622,0.6186001500927967); (3.872578881771756,0.8055255798831467)– (3.932083193333863,0.6101876549887721); (3.932083193333863,0.6101876549887721)– (3.9772515510532336,0.5195223993758777); (3.932083193333863,0.6101876549887721)– (3.931759003805608,0.5195267500217758); (3.932083193333863,0.6101876549887721)– (3.886590646086238,0.519846588904133); (3.804662075105622,0.6186001500927967)– (3.837462996196433,0.5323924480593911); (3.804662075105622,0.6186001500927967)– (3.7779586846343247,0.5262949174398779); (3.694629066607704,0.6792325470610602)– (3.7195064146155903,0.6030525011973251); (3.7195064146155903,0.6030525011973251)– (3.7195064146155907,0.5517498033414762); (3.694629066607704,0.6792325470610602)– (3.6689777176797795,0.603826502117364); (3.6689777176797795,0.603826502117364)– (3.6689777176797795,0.5540718061015921); (3.592315258018243,0.6794182954076224)– (3.5656599094657593,0.603225697076132); (3.592315258018243,0.6794182954076224)– (3.616722744696082,0.6024199982658909); (3.616722744696082,0.6024199982658909)– (3.6160004682727536,0.5512713508664012); (3.5656599094657593,0.603225697076132)– (3.5648542106555174,0.5521628618458094); (2.,8.)– (2.,6.);\n(4.,8.)– (0.,8.); (0.,8.)– (0.004368420828023068,0.5076994075353054); (0.004368420828023068,0.5076994075353054)– (4.007847529220136,0.501168446510196); (4.007847529220136,0.501168446510196)– (4.,8.); (-0.002668709472009345,8.64) node[anchor=north west] *I*\\**μ*(*ω*) = *μ**A*(∂*T*) = *c*2(*A*); (2,7.25) node[anchor=north west] *ω*; (-0.5050465391305167,4.349537230555931) node[anchor=north west] 1;\n(-0.03481734532263382,3.85) node[anchor=north west] *Q**α*; (1.02,5) node[anchor=north west] *α*;\n(-0.03481734532263382,2) node[anchor=north west] *Q**β*; (0.3,2.7331243768413462) node[anchor=north west] *β*;\n[main 2] (i) Let *T* = (*V*, *E*) be a rooted tree and *μ**A* be the equilibrium measure (for *p* = 2) for a set *A* ⊆ ∂*T*.\nThen, there exists a square tiling {*Q**α*}*α* ∈ *E* of the rectangle *R* = [0, *c*2(*A*)] × [0, 1], where the combinatorics of the tiling are prescribed by *T* and the square *Q**α* has side of length ℓ(*α*) = *μ**A*(∂*T**α*).\n(ii) Conversely, suppose a rectangle *R* = [0, *c*] × [0, 1], *c* ≤ 1, is square-tiled by {*Q**α*}*α* with combinatorics given by a rooted tree *T*.\nThen there exists an *F**σ* subset *A* of ∂*T* such that the measure *μ*(∂*T**α*) = ℓ(*α*), is the equilibrium measure of *A*, and then *c*2(*A*) = *c*.\nA consequence of this theorem is that the different tilings having the same tree-combinatorics can be parametrized by a family of *F**σ* subsets of ∂*T*.\nTheorem [main 2] is very much related to results in,,. Benjamini and Schramm proved that the equilibrium measure of a planar graph’s boundary is associated to a tiling of a cylinder by squares, and on trees, analytically, this is the content of. The converse statement is not true: there are tilings of cylinders whose combinatorics are prescribed by a planar graph, in which the sizes of the tiles do not reflect the equilibrium measure of the graph’s boundary; trees provide plenty of counterexamples. Theorem [main 2] provides, in the special case of trees, the correct bijection between tilings and discrete potential theoretic objects. It would be very interesting having a generalization of this statement to the whole class of planar graphs.\nAgain in the linear case, Theorem [main 1] is related to some beautiful theorems of Kai-Lai Chung (see, ), interpreting equilibrium measures in terms of last exit times for stochastic processes. The results of Chung, and of Benjamini and Schramm, have a probabilistic statement, or proof. As we work in the nonlinear case 1 < *p* < ∞, we do not expect probabilistic methods to apply here. Even in the linear case, however, it would be interesting having a converse to Benjamini and Schramm’s theorem on planar graphs, and this might have a probabilistic proof. See the monographs, and for thorough introductions to the stochastic processes which are here relevant.\nThe paper is organized as follows. Section [SecPotential] is devoted to present some preliminary results of Potential Analysis on trees. Section [SecRescaling] contains the proof of (i) in Theorem [main 1], which follows quite easily from some rescaling properties we present. In Section [SecFlows] we provide some useful reinterpretation of equation in terms of edge functions and of measures. The proof of part (ii) of our characterization is more subtle and is given in Section [SecSufficient]. In Section [SecTiling] we discuss some new results on square tilings of rectangles which follow from Theorem [main 1] with *p* = 2. Finally, in Section [SecBranching] we show how capacities can be expressed by a recursive formula involving branched continued fractions, and we exploit this fact to give another reformulation of Theorem [main 1].\nIt is a pleasure to acknowledge useful discussions with Davide Cordella and Nikolaos Chalmoukis.\nPotential Theory on the tree\n============================\nLet *T* be a tree with root vertex *o* and root edge *ω*, as described in the Introduction. A vertex is said to be a *leaf* if it is the endpoint of one edge only. Being mainly interested in infinite trees, we will assume that *T* has no leaves except *o*; this is a simplification but not a restriction. Given *α* ∈ *E*, we write *b*(*α*) and *e*(*α*) for its beginning and ending vertex, respectively, i.e., they are connected by *α* and *b*(*α*) < *e*(*α*). We denote by ∣*α*∣ the *level* of an edge *α*, which is the number of edges preceding *α* in the geodesic to the root *ω*. With this definition we have ∣*ω*∣ = 0. The level of a vertex is the level of the subsequent edge, ∣*b*(*α*)∣ = ∣*α*∣. We define the *sons* of an edge *α* as the elements of the set *s*(*α*) = {*β* ∈ *E* : *β* ∼ *α*, ∣*β*∣ = ∣*α*∣ + 1}. Given an edge *α*, we write *T**α* for the subtree rooted at *α* having as edge set {*β* ∈ *E* : *β* ≥ *α*}, and we call it the *α*-*tent*. The boundary ∂*T**α* of the *α*-tent is exactly the set of labels corresponding to rays passing through *α*. We put on ∂*T* the topology generated by the boundary of tents, namely the one having as a basis {∂*T**α*}*α* ∈ *E*.\nWe say that *S* is a subtree of *T* if it is a tree having the same root as *T* and having as vertex set a subset of *V*. Observe that subtrees are exactly those subsets that can be obtained subtracting from *T* a countable union of tents.\nIt is clear that the standard counting metric # does not extend properly to the boundary of an infinite tree. We enodow $\\overline{V}$ with a different metric: given two points $\\xi,\\eta\\in \\overline{V}$, we define their *confluent* to be the vertex *ξ* ∧ *η* = max{*x* ∈ *V* : *x* ≤ *ξ*, *x* ≤ *η*}. Then, we define the *visual metric*\n \n*d*(*ξ*, *η*) = *e*− ∣*ξ* ∧ *η*∣. \nThe reader familiar with Gromov’s theory of hyperbolic spaces can observe that for *ξ*, *η* ∈ *V*, expression ([gromovmetric]) coincides with the Gromov product on (*V*, #), given by\n \n$$(x,y)\\_o=\\frac{1}{2}\\left(\\#(x,o)+\\#(y,o)-\\#(x,y)\\right), \\quad x,y\\in V.$$ \nIn fact, one can extend such a product to points in ∂*T* by setting (*ξ*, *η*)*o* = lim(*x*, *y*)*o*, where the limit is taken for *x* → *ξ*, *y* → *η* along the rays labeled by *ξ* and *η*. With this notation we have *d*(*ξ*, *η*) = (*ξ*, *η*)*o*. It is not hard to see that ([gromovmetric]) defines a distance on $\\overline{V}$, which in fact is an ultrametric, and that $\\left(\\overline{V},d\\right)$ is a compact metric space (see, for example, ). Moreover, the topology induced by this metric on ∂*T* is the tent topology introduced before.\nThere is a one-to-one correspondence between compact sets in ∂*T* and boundaries of subtrees, in the following sense.\n[prop:tree with prescribed compact boundary] A set *K* ⊆ ∂*T* is compact if and only if there exists a subtree *T**K* ⊆ *T* such that *K* = ∂*T**K*.\nThe fact that the boundary of a subtree is compact follows directly from the definition of subtree. Conversely, if *K* is compact, consider the subtree *S* ⊆ *T* having as edge set *E*(*S*) = *P*(*K*) : = {*α* ∈ *E* : *α* < *ξ*, for some *ξ* ∈ ∂*K*}. Clearly *K* ⊆ ∂*S*. On the other hand, if *ξ* ∈ ∂*S*, by definition of boundary of a tree, *P*({*ξ*}) ⊆ *E*(*S*). Now, suppose by contraddiction that $\\xi\\notin K$. Then, by compactness, there exists and an edge *α* ∈ *E* such that ∂*T**α*⋂*K* = ∅ and *ξ* ∈ ∂*T**α*. Hence, $\\beta\\notin P(K)$ for *β* ∈ *P*({*ξ*}) ⊆ *E*(*S*) with ∣*β*∣ ≥ ∣*α*∣, leading to a contraddiction.\nWe present now a Nonlinear Potential Theory on the tree *T* which falls within the axiomatics developed in §2.3-2.5 of the treatise of Adams-Hedberg. We consider the compact metric space $(\\overline{V},d)$ and we make *E* into a measure space by endowing it with the counting measure. We introduce the kernel $k:\\overline{V}\\times E\\to \\mathbb{R}$, given by the indicator function *k*(*ξ*, *α*) = I({(*ξ*, *α*) : *α* < *ξ*}). Observe that *k*( ⋅ , *α*) is continuous on ∂*T*, since ∂*T**α* is open.\nGiven a function *f* : *E* → R and *p*, *p*ʹ ∈ (1, + ∞) such that 1/*p* + 1/*p*ʹ = 1, we define the **p* − potential* of *f*, $I\\_pf: \\overline{V} \\rightarrow \\mathbb{R}\\cup\\{\\pm \\infty \\} $, by\n \n*I**p**f*(*ξ*) = ∑*α* ∈ *E**k*(*ξ*, *α*)*f*(*α*)∣*f*(*α*)∣*p*ʹ − 2 = ∑*E* ∋ *α* < *ξ**f*(*α*)∣*f*(*α*)∣*p*ʹ − 2. \nWe fix the convention *I**p**f*(*o*) = 0 and we ease the notation by simply writing *I**f* in place of *I*2*f*.\nThe *co-potential* of a charge *μ* ∈ M(∂*T*) is defined as the edge function\n \n$$I^\\ast\\mu(\\alpha)=\\int\\_{\\overline{V}}k(\\xi,\\alpha)d\\mu(\\xi)=\\mu\\left(\\partial T\\_\\alpha\\right), \\quad \\alpha\\in E.$$ \nObserve that ⟨*I**f*, *μ*⟩*L*2(∂*T*) = ⟨*f*, *I*\\**μ*⟩ℓ2(*E*).\nTo the charge *μ* one can then associate the *nonlinear potential* *V**p**μ*(*ξ*) = *I**p**I*\\**μ*(*x*), $\\xi\\in \\overline{V}$. It is easily seen that for *p* = 2 one recovers the logarithmic potential of *μ* on the metric space $(\\overline{V},d)$,\n \n$$V\\_2\\mu(\\xi)=\\int\\_{\\partial T}\\log\\frac{1}{d(\\xi,\\nu)}d\\mu(\\nu), \\quad \\xi\\in \\overline{V}.$$ \nThe **p* − energy* of the charge *μ* is given by\n \nE*p*(*μ*) = ∫∂*T**V**p**μ*(*ξ*)*d**μ*(*ξ*) = ‖*I*\\**μ*‖*p*ʹ*p*ʹ. \nSetting Ω*A* = {*f* ∈ ℓ*p*(*E*) : *I**f* ≥ 1 on *A*}, we define the *p*-*capacity* of a set *A* ⊆ ∂*T* as\n \n*c**p*(*A*) = inf*f* ∈ Ω*A*‖*f*‖*p**p*. \nA property holds *c**p*-a.e. on *A* if it holds everywhere on *A* but for a subset of zero *p*-capacity. A function *f* is *admissible* for *A* ⊆ ∂*T* if $f\\in\\overline{\\Omega\\_A}^{\\ell^p}$. It can be proved that $\\overline{\\Omega\\_A}^{\\ell^p}=\\lbrace f\\in\\ell\\_+^p: \\ If\\geq 1 \\ c\\_p-a.e. \\ \\mbox{on} \\ A \\rbrace$, see, and that there exists a unique function $f^A\\in \\overline{\\Omega\\_A}^{\\ell^p}$, called *p*-*equilibrium function* for *A*, such that\n \n$$c\\_p(A)=\\min\\_{f \\in \\overline{\\Omega\\_A}^{\\ell^p}}\\Vert f\\Vert\\_p^p=\\Vert f^A\\Vert\\_p^p.$$ \nMoreover, for such a function it holds *I**f**A* = 1 *c**p*-a.e. on *A*.\nWe call a point *ξ* ∈ *A* *irregular* for *A* if *I**f**A*(*ξ*) ≠ 1. Clearly the set of irregular points for any set *A* has zero *p* − capacity.\nAs a set function, *p*-capacity is monotone, countably subadditive and regular from inside and outside, i.e.,\n* $\\displaystyle c\\_p\\big(\\bigcup\\_n A\\_n\\big)=\\lim\\_n c\\_p(A\\_n)$, for any increasing sequence (*A**n*) of arbitrary subsets of ∂*T*.\n* $\\displaystyle c\\_p\\big(\\bigcap\\_n(K\\_n)\\big)=\\lim\\_n c\\_p(K\\_n)$, for any decreasing sequence (*K**n*) of compact subsets of ∂*T*.\nA set *A* ⊆ ∂*T* is *capacitable* if *c**p*(*A*) = sup{*c**p*(*K*) : *K* ⊆ *A*, *K* compact} = inf{*c**p*(*G*) : *F* ⊇ *A*, *G* open}. From (in) and (out) follows that all Suslin sets, and in particular all Borel sets, are capacitable, see.\nObserve that without losing generality, in the definition of *p* − capacity the infimum can be taken over functions supported on the *predecessor set* of $\\overline{A}$, $P(\\overline{A}):=\\lbrace \\alpha\\in E: \\ \\alpha<\\xi, \\text{ for some } \\xi\\in \\overline{A}\\rbrace$. Namely, the capacity of a compact set *K* only depends on the combinatorics of *P*(*K*) and not on the rest of the tree, and if *T**K* ⊆ *T* is the subtree having *K* as a boundary (see Proposition [prop:tree with prescribed compact boundary]), we have *c**p*(*K*) = *c**p*(∂*T**K*), where the right handside is intended as the capacity when *T**K* is taken as the ambient space.\nIt is possible to give a dual definition of capacity in terms of measures on ∂*T* rather than of functions on *E*. The proof of the following theorem is a straightforward adaptation of an equivalent result for capacities in R*n* arising by regular enough kernels.\nSuppose that *A* ⊆ ∂*T* is a Suslin set. Then\n \n*c**p*(*A*) = sup{*μ*(*A*)*p* : *μ* ≥ 0, supp(*μ*) ⊆ *A*, E*p*(*μ*) ≤ 1}. \nMoreover, there exists a unique positive Borel measure *μ**A* supported in $ \\overline{A} $, called the *p* − equilibrium measure of *A*, such that\n \n$$\\mu^A(\\overline{A})=c\\_p(A)=\\mathcal{E}\\_p(\\mu^A),$$ \nand *I*\\**μ**A*(*α*)*p*ʹ − 1 = *f**A*(*α*), *α* ∈ *E*.\nWith this definition of capacity, it is clear that if a property (*P*) holds *c**p* − a.e. on *A* ⊆ ∂*T*, then it holds *μ* − a.e. on *A* for every measure *μ* with E*p*(*μ*) < ∞. To see this, let *B* : = {*ξ* ∈ *A* : ¬(*P*)}, so that *c**p*(*B*) = 0. The measure $\\displaystyle\\nu:=\\frac{\\mu|\\_B}{\\mathcal{E}\\_p(\\mu)^{1/p^\\prime}}$ satisfies E*p*(*ν*) ≤ 1. Then, *ν*(*B*) ≤ *c**p*(*B*) = 0, from which it follows *μ*(*B*) = 0.\nWe conclude the section by showing that in ∂*T* there exist compact subsets with arbitrary *p*-capacity.\n[every capacity] Let *T**n* be a homogeneus tree of degree *n*. For each real number *t* ∈ [0, *c**p*(∂*T**n*)] there exists a compact subset *K**t* of the boundary ∂*T**n* such that *c**p*(*K**t*) = *t*.\nEach edge *α* of *T**n*, except the root, can be given an index *i*(*α*) ∈ {0, …, *n* − 1} which distinguishes it from the other *n* − 1 edges *β* such that *b*(*β*) = *b*(*α*). We can define a map Λ : ∂*T**n* → [0, 1] associating to each point *ξ* = {*α**j*}*j* = 1∞ ∈ ∂*T**n* the number having expansion in base *n* given by\n \nΛ(*ξ*) = ∑*j* = 1∞*i*(*α**j*)*n*− *j*. \nThe map Λ is clearly onto but it fails to be injective because of the multiple representations of the rational numbers. Still, Λ− 1(*t*) has at most two points. Moreover, Λ is continuous, since\n \n∣Λ(*ξ*) − Λ(*η*)∣ ≤ (*n* − 1)∑*j* = ∣*ξ* ∧ *η*∣ + 1∞*n*− *j* ≈ *n*− ∣*ξ* ∧ *η*∣ → 0, as *ρ*(*ξ*, *η*) → 0. \nNow, consider the function *φ* : [0, 1] → R given by *φ*(*t*) = *c**p*(Λ− 1[0, *t*]). This is an increasing map, and we know that *φ*(0) = 0 and *φ*(1) = *c**p*(∂*T**n*). By the subadditivity of *c**p*, the continuity of Λ and the regularity of capacity, we have\n \n*φ*(*t* + *ɛ*) − *φ*(*t*) ≤ *c**p*(Λ− 1[*t*, *t* + *ɛ*]) → *c**p*(Λ− 1{*t*}). \nThe right handside equals zero, since the preimage of a single point under Λ is finite. By similar reasoning we estimate *φ*(*t*) − *φ*(*t* − *ε*). It follows that *φ* is continuous and *φ*([0, 1]) = [0, *c**p*(∂*T**n*)]. The result is obtained picking *K**t* = Λ− 1[0, *φ*− 1(*t*)].\nPreliminaries\n=============\nA tree is a connected graph with no cycles. Given a tree *T*, we denote by *E* the set of its edges and by *V* the set of its vertices. We always assume *T* to be *locally finite*, in the sense that the number deg(*x*) of edges departing from any vertex *x* must be finite. We do not ask the family {deg(*x*)}*x* ∈ *V* to be uniformly bounded. Given two vertices *x*, *y*, we write *x* ∼ *y* if they are connected by one (and only one) edge. Similarly, for *α*, *β* ∈ *E*, we write *α* ∼ *β* if they have a common vertex. Observe that given any two vertices, there is only one path for travelling from one to the other without ever passing from the same edge twice. We call such a path a *geodesic*. We can indistinctly think of a geodesic as a sequence of connected edges or as a sequence of connected vertices. The tree *T* will be always assumed to be *rooted*, i.e., there exists a distinguished root vertex *o*, connected to one and only one other vertex through the root edge *ω*.\nIn particular, we decide to order with respect to a time parametrization of geodesics emanating from the root, and we write *x* ≥ *y* if and only if *y* ∈ [*o*, *x*]. The same order relation extends to edges. If *x* ∼ *y* and *y* > *x*, the edge connecting them is denoted by [*x*, *y*]. Conversely, given *α* ∈ *E*, we write *b*(*α*) and *e*(*α*) for its beginning and ending vertex, respectively, i.e., they are connected by *α* and *b*(*α*) < *e*(*α*).\nThe root induces a partial order on *V* (and similarly on *E*). We decide to order with respect to a time parametrization of geodesics emanating from the root, and we write *x* > *y* if and only if *y* ∈ [*o*, *x*]. Thus, edges automatically get an orientation and we can identify *E* with the subset of *V* × *V* made of points (*x*, *y*) such that *x* ∼ *y* and *y* > *x*. In such a case we write [*x*, *y*] for the edge connecting *x* and *y*, while the geodesic segment [*y*, *x*] is not considered to be an edge.\nWe write M(∂*T*) for the family of charges (signed Borel measures) on the boundary of the tree, and M+(∂*T*) for the sub-family of (nonnegative) measures. Let *f* : *E* → R, *g* : *V* → R and *μ* ∈ M(∂*T*). We define\nWe can think of *V* and *E* as measure spaces endowed with the counting measure. Hence, write ℓ*p*(*V*) and ℓ*p*(*E*) for the spaces of *p* − summable functions on *V* and *E* respectively.\nWe define the *gradient* of a function *g* : *V* → R at (*x*, *y*) ∈ *V* × *V* to be the anti-symmetric difference operator\n \n∇*g*(*x*, *y*) = I({(*x*, *y*) : *x* ∼ *y*}*d**g*(*x*, *y*). \nIf *α* = [*x*, *y*] is an edge, with the natural orientation induced by the root, with some abuse of notation we also write ∇*g*(*α*) for ∇*g*(*x*, *y*). The *divergence* of a function *φ* : *V* × *V* → R at *x* ∈ *V* is given by div*φ*(*x*) = ∑*y* ∼ *x**φ*(*x*, *y*). For *p* ∈ (1, + ∞), we define the *p-Laplacian* of *g* as $\\Delta\\_p g =\\operatorname{div}\\big(\\nabla g|\\nabla g|^{p-2}\\big)$. Explicitely, we have\n \n$$\\Delta\\_p g (x)=\\sum\\_{y\\sim x}\\big(g(y)-g(x)\\big)|g(y)-g(x)|^{p-2} \\quad x\\in V.$$ \nWe say that *g* is *p* − *harmonic* if Δ*p**g* ≡ 0 on *V*. Observe that *g* is 2 − harmonic if and only if it holds the mean value property *g*(*x*) = ∑*y* ∼ *x**g*(*y*)/deg(*x*).\nWe also introduce the Sobolev space *W*1, *p*(*T*) = {*g* : *V* → R : ∇*g* ∈ ℓ*p*}. The **p* − energy* of *g* on a set *V*ʹ ⊆ *V* is the associated seminorm\n \n∣*g*∣1, *p*, *V*ʹ = ∑*x* ∈ *V*ʹ∣∇*g*(*x*)∣*p*. \nRescaling of capacities\n=======================\nIn this section we show that equilibrium measures rescale under changes of the root in a tree, in a sense that will be more clear soon. This is a point where trees behave much more simply than general planar graphs, and constitutes the key property to prove (*i*) in Theorem [main 1]. We introduce a subscript notation to indicate which is the root of the tree we are referring to. For example, give the rooted tree *T* and some edge *α* ∈ *E*, we write *I**p*, *α* for the *p* − potential operator acting on functions defined on the edges of the *α* − tent *T**α*, $I\\_{p,\\alpha}f=I\\_p(\\bigchi\\_{T\\_{\\alpha}}f)$. Accordingly, *V**p*, *α**μ* = *I**p*, *α**I*\\**μ*, E*p*, *α*(*μ*) = ∑*β* ≥ *α*∣*I*\\**μ*(*β*)∣*p*ʹ and *c**p*, *α* denotes the capacity when we consider *T**α* as ambient space. In the same fashion, if *A* is a subset of ∂*T*, *A**α* = *A* ∩ ∂*T**α*.\nA question arises: if *A* is a subset of ∂*T* and *μ* its *p* − equilibrium measure, which is the *p* − equilibrium measure *μ**α* for *A**α* in the tent *T**α*? It is natural to expect that it is a rescaling of the measure *μ*, i.e., *μ**α* = *k**α**μ*∣*A**α* for some positive constant *k**α*. In such a case, for *c**p* − a.e. *ξ* in *A**α*, we would have\n \n$$1=V\\_{p,\\alpha}\\mu\\_\\alpha(\\xi)=k\\_{\\alpha}^{p^\\prime-1}V\\_{p,\\alpha}\\mu(\\xi)=k\\_{\\alpha}^{p^\\prime-1}\\Big(V\\_p\\mu(\\xi)-V\\_p\\mu(b(\\alpha))\\Big)=k\\_{\\alpha}^{p^\\prime-1}\\Big(1-V\\_p\\mu(b(\\alpha))\\Big).$$ \nIt follows that the only possible candidate rescaling constant is\n \n$$\\label{k}\nk\\_{\\alpha}=\\Big(1-V\\_p\\mu(b(\\alpha))\\Big)^{1-p}.$$ \nWe now prove that our ansatz is correct. This was already observed in.\n[prop:rescaling] Let *μ* be the *p* − equilibrium measure for a set *A* ⊆ ∂*T*. Then,\n \n$$\\mu\\_{\\alpha}:=k\\_{\\alpha}\\mu|\\_{A\\_{\\alpha}}=\\frac{\\mu |\\_{\\partial T\\_\\alpha}}{\\Big(1-V\\_p\\mu(b(\\alpha))\\Big)^{p-1}},$$ \nis the *p* − equilibrium measure for *A**α* ⊆ ∂*T**α*.\nLet *φ**α* be the (unique) *p* − equilibrium function of *A**α* in *T**α*. We want to prove that *φ**α* = *I*\\**μ**α**p*ʹ − 1. Define the edge function,\n \n$$f(\\beta)= \\begin{cases}\n\\Big(1-V\\_p\\mu(b(\\alpha))\\Big)\\varphi\\_{\\alpha}(\\beta) & \\text{if } \\beta\\geq\\alpha\\\\\nI^\\ast\\mu(\\beta)^{p'-1} & \\text{otherwise }.\n\\end{cases}$$ \nThen *f* is admissible for *A* ⊆ ∂*T*, since for *c**p* − a.e. *ξ* ∈ *A* \\ *A**α*, *I**f*(*ξ*) = *V**p**μ*(*ξ*) = 1, while for *c**p* − a.e. *ξ* ∈ *A**α* it holds\n \n$$If(\\xi)= I\\_{\\alpha}f(\\xi)+If(b(\\alpha))=\\Big(1-V\\_p\\mu(b(\\alpha))\\Big)I\\_{\\alpha}\\varphi\\_{\\alpha}(\\xi)+V\\_p\\mu(b(\\alpha))=1.$$ \nClearly *I*\\**μ**p*ʹ − 1 is an admissible function for *A**α* ⊆ ∂*T**α*, since *I**α**I*\\**μ**p*ʹ − 1 ≤ *I**I*\\**μ**p*ʹ − 1 = 1, *c*.*p* − a.e. on *A*. Hence, it must be ‖*φ**α*‖ℓ*p*(*T**α*)*p* ≤ ‖*I*\\**μ**p*ʹ − 1‖ℓ*p*(*T**α*)*p* = ‖*I*\\**μ*‖ℓ*p*ʹ(*T**α*)*p*ʹ. It follows\n \n$$\\begin{split}\n\\Vert f\\Vert\\_p^p &=\\Big(1-V\\_p\\mu(b(\\alpha))\\Big)^p\\Vert \\varphi\\_{\\alpha}\\Vert\\_{\\ell^p(T\\_\\alpha)}^p+\\Vert I^\\ast\\mu^{p'-1}\\Vert\\_{\\ell^p(T\\setminus T\\_\\alpha)}^p\\\\\n&\\leq\\Vert I^\\ast\\mu\\Vert\\_{\\ell^{p'}(T\\_\\alpha)}^{p'}+\\Vert I^\\ast\\mu\\Vert\\_{\\ell^{p'}(T\\setminus T\\_\\alpha)}^{p'}=\\mathcal{E}\\_p(\\mu)=c\\_p(A).\n\\end{split}$$ \nThen, *f* must be the equilibrium function for *A*, and by the uniqueness of the equilibrium function *f* = *I*\\**μ**p*ʹ − 1. This implies *φ**α* = *I*\\**μ**α**p*ʹ − 1.\nAs an immediate consequence we have the following.\n[cor] Let *A* ⊆ ∂*T* be a set of positive capacity and *α* ∈ *E*, *α* not the root edge of *T*, such that *A* ⊆ ∂*T**α*. Then *c**p*, *α*(*A*) > *c**p*(*A*).\nSince *k**α* > 1, we have *c**p*, *α*(*A*) = *μ**α*(*A*) = *k**α**μ*(*A*) > *μ*(*A*) = *c**p*(*A*).\nObserve by passing that the above corollary is supported by *visual intuition*, since we expect the capacity of a set to be larger if we look at the set from a closer point of view.\nWe can now give a necessary condition for a measure to be of equilibrium, thus proving (*i*) in Theorem [main 1].\n[Proof of (*i*) in Theorem [main 1]] We claim that for every *α* ∈ *E*, *μ* solves the following equation:\n \n$$\\label{eq:formula}\nI^\\ast\\mu(\\alpha)\\Big(1-V\\_p\\mu(b(\\alpha))\\Big)=\\mathcal{E}\\_{p,\\alpha}(\\mu).$$ \nSuppose the claim holds and let *g* = *V**p**μ*. It is easily seen that ∇*g*[*x*, *y*] = *I*\\**μ*[*x*, *y*]*p*ʹ − 1 for every edge [*x*, *y*] ∈ *E*. It follows that *g* solves the equilibrium equation.\nTo prove the claim, let *α* ∈ *E*. If *A**α* = ∅, then by the topology of the tree follows that also $\\overline{A}\\cap\\partial T\\_\\alpha=\\emptyset$. Since $\\operatorname{supp}(\\mu)\\subseteq \\overline{A}$, *I*\\**μ*(*α*) = 0, and hence *I*\\**μ*(*β*) = 0 for all *β* ≥ *α*, and trivially holds. Otherwise, on one hand we have\n \n$$c\\_{p,\\alpha}(A\\_{\\alpha})=\\mathcal{E}\\_{p,\\alpha}(\\mu\\_{\\alpha})=\\frac{\\mathcal{E}\\_{p,\\alpha}(\\mu)}{\\Big(1-V\\_p\\mu(b(\\alpha))\\Big)^p},$$ \nand on the other,\n \n$$c\\_{p,\\alpha}(A\\_{\\alpha})=\\mu\\_{\\alpha}(A\\_{\\alpha})=k\\_{\\alpha}\\mu(A\\_{\\alpha})=\n\\frac{I^\\ast\\mu(\\alpha)}{\\Big(1-V\\_p\\mu(b(\\alpha))\\Big)^{p-1}}.$$ \nMatching the two expressions we get the claim.\nFlows on edges and *p* − harmonic functions\n===========================================\nObserve that equation can be regarded as a special instance of an equation for edge functions:\n \n$$\\label{eq:edges}\nf(\\alpha)\\Big(1-I\\_pf(b(\\alpha))\\Big)=\\sum\\_{\\beta\\geq\\alpha}|f(\\beta)|^{p'}, \\quad f:E\\to\\mathbb{R}.$$ \nIn the particular case when the function *f* is the co-potential of a charge *μ* on the ∂*T*, the above equation reduces to. It is clear that a necessary condition for an edge function *f* to be the co-potential of a charge is to satisfy the following additivity condition\n \n*f*(*α*) = ∑*β* ∈ *s*(*α*)*f*(*β*), for all *α* ∈ *E*. \nA function *f* : *E* → R fulfilling is said to be a *flow* on *T*.\nIn a full characterization for edge functions which are co-potential of charges is given. The nonnegative case, however, is easy to rule out: an edge function *f* is a nonnegative flow if and only if *f* = *I*\\**μ* for some nonnegative measure *μ* on ∂*T*.\nIt turns out that solutions of with bounded *p* − potential are always flows, and hence they have a measure representation.\n[prop:main per funzioni] Let *f* : *E* → R be a solution of such that *I**p**f* < 1 on *V*. Then *f* = *I*\\**μ* for some nonnegative measure *μ* on ∂*T*.\nFirst of all observe that for a function *f* solving, the condition *I**p**f* < 1 automatically implies that *f* ≥ 0 on *E*. We want to show that *f* satisfies for every *α* ∈ *E*. If *f*(*α*) = 0 for some edge *α*, then the right handside of equation ([eq:edges]) says that *f*(*β*) = 0 on all edges *β* ≥ *α* and then clearly holds in *α*. Now, consider edges *α* such that *f*(*α*) ≠ 0. We have\n \n$$\\label{eq:main per funzioni}\n\\begin{split}\n \\sum\\_{\\beta\\ge\\alpha}f(\\beta)^{p'}-f(\\alpha)^{p'}&=\\sum\\_{\\beta\\in s(\\alpha)}\\sum\\_{\\gamma\\ge\\beta}f(\\beta)^{p'}=\\sum\\_{\\beta\\in s(\\alpha)}f(\\beta)\\Big(1-I\\_p f(b(\\beta))\\Big)\\\\\n &=\\Big(1-I\\_p f(e(\\alpha))\\Big)\\sum\\_{\\beta\\in s(\\alpha)}f(\\beta)\\\\\n &=\\Big(1-I\\_p f(e(\\alpha))\\Big)\\Big(-f(\\alpha)+\\sum\\_{\\beta\\in s(\\alpha)}f(\\beta)\\Big)+\\sum\\_{\\beta\\ge\\alpha}f(\\beta)^{p'}-f(\\alpha)^{p'}.\n \\end{split}$$ \nSince 1 − *I**p**f*(*e*(*α*)) > 0, for every *α*, then *f* is a flow, and the result follows.\nA similar calculation as in the proof of Proposition [prop:main per funzioni] can be used to show that, if *f* is a (not necessarily positive) flow and holds for ∣*α*∣ large, then it holds everywhere. We give here the explicit details.\nLet *α* ∈ *E* and suppose that *f* is a flow solving for *β* ∈ *s*(*α*). Then *f* solves also in *α*.\nFollowing the line of and using the flow property of *f* we easily obtain\n \n$$\\begin{split}\n\\sum\\_{\\beta\\ge\\alpha}|f(\\beta)|^{p'}&=|f(\\alpha)|^{p'}+\\Big(1-I\\_p f(e(\\alpha))\\Big)\\sum\\_{\\beta\\in s(\\alpha)}f(\\beta)\\\\\n&=f(\\alpha)\\Big( f(\\alpha)|f(\\alpha)|^{p'-2}+1-I\\_p f(e(\\alpha))\\Big)=f(\\alpha)\\Big(1-I\\_p f(b(\\alpha))\\Big).\\qedhere\n\\end{split}$$ \nWe end the section by pointing out that the flow condition for edge functions has a neat counterpart for vertex functions. In fact, it is easy to observe that *f* is a flow if and only if *I**p**f* is *p* − harmonic on *V* \\ {*o*}. In this context, the *p*-*Laplacian* of a function *g* : *V* → R at a vertex *x* is given by\n \n$$\\Delta\\_p g (x):=\\sum\\_{y\\sim x}\\Big(g(y)-g(x)\\Big)\\Big|g(y)-g(x)\\Big|^{p-2},$$ \nand *g* is *p*-*harmonic* if Δ*p**g* ≡ 0 on *V* \\ {*o*}. From Proposition [prop:main per funzioni] then immediately follows\nLet *g* : *V* → R be a solution of with ‖*g*‖∞ < 1. Then *g* is *p* − harmonic.\nProof of Theorem [main 1]\n=========================\nIn this section, we prove (*i**i*) in Theorem [main 1], namely that for a measure with bounded nonlinear potential it is sufficient to solve equation to be an equilibrium measure.\n[Proof of (ii) in Theorem [main 1]] Let *g* : *V* → R, with ‖*g*‖∞ < 1, be a solution of. Without loss of generality, normalize assuming *g*(*o*) = 0 (the function *g̃* = (*g* − *g*(*o*))/(1 − *g*(*o*)) solves, is bounded by 1 and vanishes at *o*). Set *f*(*α*) = ∇*g*(*α*)*p* − 1, *α* ∈ *E*. Then *f* solves and *I**p**f* = *g* is bounded by 1. It follows by Proposition [prop:main per funzioni] that there exists a nonnegative Borel measure *μ* on ∂*T* such that *f* = *I*\\**μ* and, consequently, *g* = *V**p**μ*.\nObserve that *μ* has finite *p* − energy: since it solves at every edge *α*, by evaluating at *α* = *ω* we get E*p*(*μ*) = *μ*(∂*T*) = *V**p*(*μ*)(*e*(*ω*))*p* − 1 < 1. Moreover, *V**p**μ*(*ξ*) = lim*x* → *ξ**V**p**μ*(*x*) ≤ 1. We show that indeed *V**p**μ* = 1, *μ* − a.e. on ∂*T*.\nLet *S**N* = {*α* ∈ *E* : ∣*α*∣ = *N*}. To each *N* ∈ N we associate a piecewise-constant function Φ*N* on the boundary, Φ*N*(*ξ*) = (1 − *V**p**μ*(*b*(*α*))) for *ξ* ∈ ∂*T**α*, *α* ∈ *S**N*. Then we have\n \n$$\\begin{split}\n0&\\leq\\int\\_{\\partial T} \\Phi\\_N(\\xi)d\\mu(\\xi)=\\sum\\_{\\alpha\\in S\\_N}\\int\\_{\\partial T\\_{\\alpha}} \\Phi\\_N(\\xi)d\\mu(\\xi)=\\sum\\_{\\alpha\\in S\\_N}(1-V\\_p\\mu(b(\\alpha)))I^\\ast\\mu(\\alpha)\\\\\n&=\\sum\\_{\\alpha\\in S\\_N}\\mathcal{E}\\_{p,\\alpha}(\\mu)=\\sum\\_{|\\beta|\\geq N}I^\\ast\\mu(\\beta)^{p'}=\\mathcal{E}\\_p(\\mu)-\\sum\\_{|\\beta|< N}I^\\ast\\mu(\\beta)^{p'}.\n\\end{split}$$ \nSince E*p*(*μ*) < ∞, it follows that\n \n∫∂*T*Φ*N*(*ξ*)*d**μ*(*ξ*) → 0, as *N* → + ∞. \nAlso, $\\Phi\\_N(\\xi)\\searrow\\Phi(\\xi):=1-V\\_p\\mu(\\xi)\\geq 0$ as *N* → + ∞. Hence, by monotone convergence theorem, we obtain\n \n∫∂*T*Φ(*ξ*)*d**μ*(*ξ*) = ∫∂*T*(1 − *V**p**μ*(*ξ*))*d**μ*(*ξ*) = 0, \nwhich gives *V**p**μ*(*ξ*) = 1, *μ* − a.e. on ∂*T*.\nConsider now the *irregular points* for *μ*, i.e., with the *μ* − measure zero set\n \nI(*μ*) = {*ξ* ∈ supp(*μ*) : *V**p**μ*(*ξ*) < 1}. \nLet *B**n* = {*ξ* ∈ supp(*μ*) : *V**p**μ*(*ξ*) ≤ 1 − 1/2*n*}. Clearly *B**n* ⊆ *B**n* + 1 and I(*μ*) = ⋃*n**B**n*. Fix *ɛ* > 0, and choose a collection of edges (*α**j**n*)*n* ∈ N, *j* ∈ J*n*, such that {∂*T**j*, *n*}*j* ∈ J*n* is an open cover of *B**n*, where *T**j*, *n* is the *α**j**n* − tent. Without loss of generally, we can assume that (*α**j**n*)*n* ∈ N, *j* ∈ J*n* satisfies the following:\n* *B**n* ⊆ ⋃*j* ∈ J*n*∂*T**j*, *n*\n* *T**j*, *n*⋂*T**i*, *l* = ∅, for (*j*, *n*) ≠ (*i*, *l*)\n* ∣J*n*∣ = *m**n* ∈ N\n* ∑*j* ∈ J*n**I*\\**μ*(*α**j**n*) < *ɛ*/2*n*.\nIn fact, if the intersection in (ii) was not empty, one of the two would be contained in the other and can be replaced by it in the covering family. Moreover, all the sublevel sets *B**n* are compact, since *V**p**μ* is continuous. Hence, for each *n*, we can extract a finite subcover so that *B**n* ⊆ ⋃*j* ∈ J*n*∂*T**j*, *n* with ∣J*n*∣ = *m**n* ∈ N, which is the finiteness condition on the index set in (iii). Finally, condition (iv) comes from the outer regularity of the measure *μ*: since 0 = *μ*(*B**n*) = inf{*μ*(∂*T**α*) : *α* ∈ *E*, ∂*T**α* ⊇ *B**n*}, there exist sequences (*α**j**n*)*j* such that *μ*(∂*T**j*, *n*) → 0 and we can assume to properly extract each subcover from one of those.\nWrite ∂*T* = *F**ɛ*⋃*G**ɛ*, where *G**ɛ* : = ⋃*j*, *n*∂*T**j*, *n* and *F**ɛ* = ∂*T* \\ *G**ɛ*. Observe that,\n \n*μ*(∂*T*) ≥ *μ*(*F**ɛ*) = *μ*(∂*T*) − *μ*(*G**ɛ*) = *μ*(∂*T*) − ∑*j*, *n**I*\\**μ*(*α**j**n*) ≥ *μ*(∂*T*) − ∑*n**ɛ*/2*n* = *μ*(∂*T*) − *ɛ*. \nHence we have,\n \n*μ*(∂*T*) = lim*ɛ* → 0*μ*(*F**ɛ*). \nNow, *V**p**μ* ≡ 1 on *A**ɛ* : = supp(*μ*) ∩ *F**ɛ*, so that *V**p**μ* is a *p*-admissible function for *A**ɛ*. Then, by definition of capacity we have\n \n*c**p*(*A**ɛ*) ≤ E*p*(*μ*∣*F**ɛ*) ≤ E*p*(*μ*) = *μ*(∂*T*). \nOn the other hand, the measure $\\displaystyle\\nu^{\\varepsilon}:=\\frac{\\mu|\\_{ F\\_{\\varepsilon}}}{\\mathcal{E}\\_p(\\mu|\\_{F\\_{\\varepsilon}})^{1/p^\\prime}}$ is admissible for *A**ɛ*, since E*p*(*ν**ɛ*) = 1. By the dual definition of capacity it follows that\n \n$$\\label{dual definition}\nc\\_p(A\\_\\varepsilon)\\geq\\nu^{\\varepsilon}(A\\_\\varepsilon)^p\\geq\\Big(\\frac{\\mu(F\\_{\\varepsilon})}{\\mu(\\partial T)^{1/p^\\prime}}\\Big)^p.$$ \nWe are now ready to build up the candidate *F**σ* set. Let {*ɛ*(*k*)}*k* ∈ N be a sequence of positive numbers such that $\\varepsilon(k)\\searrow 0$ as *k* → + ∞. Define *A* : = ⋃*k**A**ɛ*(*k*), which is clearly an *F**σ* set. Observe that we can assume that the covers related to each choice of *ɛ* are taken so that *G**ɛ*(*k* + 1) ⊆ *G**ɛ*(*k*). Therefore, $A\\_{\\varepsilon(k)}\\nearrow$ as *k* → + ∞. It follows that\n \n$$\\mu(A)=\\lim\\_{k\\to\\infty}\\mu(A\\_{\\varepsilon(k)})=\\mu(\\partial T)-\\lim\\_{k\\to\\infty}\\mu(G\\_{\\varepsilon(k)})\\geq\\mu(\\partial T)-\\lim\\_{k\\to\\infty}\\sum\\_n\\frac{\\varepsilon(k)}{2^n}=\\mu(\\partial T),$$ \nwhile the reverse inequality is trivially true. Using this together with ([definition]) and ([dual definition]) and the regularity of the *p* − capacity, we obtain\n \n$$c\\_p(A)=\\lim\\_{k\\to +\\infty}c\\_p(A\\_{\\varepsilon(k)})=\\mu(\\partial T)=\\mu(A).\\qedhere$$ \nIt is clear from the proof that the situation is much easier for measures with no irregular points.\nSuppose that *μ* ∈ M+(∂*T*) solves and *V**p**μ* ≡ 1 on supp(*μ*). Then, *μ* is the *p*-equilibrium measure of supp(*μ*).\nInfinite square tilings\n=======================\nIn Brooks, Smith, Stone and Tutte considered the problem of tiling a rectangle with a finite number of squares and proved that to any finite connected planar graph *G* can be associated such a tiling. The same graph can produce different tilings. Chosen any two vertices in *G*, they show how the associated tiling can be built in such a way to reflect this choice. In Benjamini and Schramm extended this result to the infinite case, showing that infinite graphs can produce infinite tilings. Theorem [main 1] can be reformulated, for *p* = 2, in terms of square tilings of a rectangle: part (*i*) of the theorem is essentially equivalent to the infinite tiling theorem by Benjamini and Schramm, in the special case when *G* is a rooted tree *T*, hence providing a new and different proof of it. More interestingly, part (*i**i*) provides a converse result, in a sense that will be more clear once introduced the proper terminology.\nGiven a rectangle *R*, i.e, a closed planar region whose boundary is a rectangle, we say that a family of squares *Q* = {*Q**j*}*j* is a *square tiling* of *R* if int(*Q**i*)⋂int(*Q**j*) = ∅, for *i* ≠ *j*, and $R=\\overline{\\bigcup\\_j Q\\_j}$. By rotation invariance of the problem we always think rectangles and squares to have sides parallel to the coordinates axes of R2, and we talk about upper (lower) and left (right) sides, as well as horizontal and vertical sides, in the obvious way. We write *B*(*j*) and *E*(*j*) for the upper and lower side of *Q**j*, respectively.\nWe say that *the combinatorics of a family *Q* of squares in the plane are prescribed by a tree *T* if the followings are true.* \n* The squares in the family are indexed by the edges of the tree, *Q* = {*Q**α* : *α* ∈ *E*}.\n* *B*(*α*) ⊆ *E*(*β*) whenever *b*(*α*) = *e*(*β*).\nTheorem [main 1] can be reformulated, for *p* = 2, in terms of square tilings, leading to Theorem [main 2]. We detail out the proof here.\n[Proof of Theorem [main 2]] (i) Given a tree *T* with root edge *ω* and a set *A* ⊆ ∂*T*, let {*Q**α*}*α* ∈ *E* be a family of squares such that *Q**α* has side of length ℓ(*α*) = *μ*(∂*T**α*), being *μ* = *μ**A* the equilibrium measure of *A*. By the additivity of *μ* we can place the squares on the plane in such a way that,\n \n*E*(*β*) = ⋃*β* ∈ *s*(*α*)*B*(*α*). \nWith this choice, the combinatorics is prescribed by *T*. Moreover, it is clear that the interiors of the squares in the family are pairwise disjoint and that ⋃*α**Q**α* is both vertically and horizontally convex (its intersection with any vertical and horizontal line is either empty, or a point, or a line segment). Now, let *R* be the rectangle having vertical sides of length 1 and upper side coinciding with the upper side of *Q*(*ω*), so being of length $\\mu(\\partial T)=\\mu(\\overline{A})=c\\_2(A)$. Denote by ∣ ⋅ ∣ the area measure. Then,\n \n$$|R|=c\\_2(A)=I^\\*\\mu(\\omega)=\\mathcal{E}\\_2(\\mu)=\\sum\\_{\\alpha\\in E}\\mu(\\partial T\\_\\alpha)^2=\\sum\\_{\\alpha\\in E}|Q\\_\\alpha|=\\big|\\bigcup\\_{\\alpha} Q\\_\\alpha\\big|.$$ \nIt is then enough to show that the family of squares is contained in *R* to prove that it is a tiling. It is clear that all the family {*Q**α*} lies in between the two vertical sides of *R*, and that the horizontal room is fully filled, by additivity of the measure. Moreover, as already observed in the proof of part (ii) of Theorem [main 1], for every *ξ* ∈ ∂*T* it must be\n \n1 ≥ *V*2*μ*(*ξ*) = ∑*α* < *ξ**μ*(∂*T**β*) = ∑*α* < *ξ*ℓ(*α*). \nIt follows that ⋃*α**Q**α* ⊆ *R* and {*Q**α*}*α* is a tiling. \n(ii) Let the rectangle *R* be tiled according to the combinatorics of a tree *T*, as described above. Then for each *α* ∈ *E*, we have:\n \n$$\\sum\\_{\\beta \\geq\\alpha}|Q(\\beta)|=\\ell(\\alpha)\\bigg(1-\\sum\\_{\\beta<\\alpha}\\ell(\\beta)\\bigg).$$ \nHence, it is immediate that if we define a measure on ∂*T* by *μ*(∂*T**α*) = ℓ(*α*), it solves equation. Then the harmonic function *g* = *V*2*μ* is bounded by 1 on *V* and solves. By Theorem [main 1], *μ* must be the equilibrium measure of some *F**σ* subset *A* of ∂*T*.\nIt might be interesting to informally discuss some features of the tiling, and its relation to the set *A*. The example below can provide a useful illustration of what we are here saying in general terms.\nFor each *ξ* in ∂*T*, let *α**n* be the only edge of level *n* along the geodesic labelled by *ξ*, and choose a point *x**n* + 1 in *Q**α**n*. Then, it is immediate that lim*n* → ∞*x**n* = : *λ*(*ξ*) exists in *R*, and that it does not depend on the precise location of the *x**n*’s in the corresponding squares. Let *π*(*ξ*) be the orthogonal projection of *λ*(*ξ*) onto the lower side of *R*, identified with [0, *c*2(*A*)].\n2 Let *A* ⊆ ∂*T* and *μ**A* its equilibrium measure. The following facts are easy to check:\n* if we replace *T* by its subtree obtained keeping only the edges *α* with *μ**A*(∂*T**α*) > 0, then tiling does not have degenerate squares, and *c*2(*A* ∩ ∂*T**α*) > 0 for all edges *α*.\n* let *A*′ be the set of regular points for *μ**A*, i.e. *A*′ = {*ξ* ∈ ∂*T* : *V*2*μ**A*(*ξ*) = 1}: then *μ**A*′ = *μ**A*, hence they induce the same tiling.\n* *π* is injective but possibly at countably many points and surjective from ∂*T* onto [0, *c*2(*A*)];\n* *μ**A*(*π*− 1(*A*)) = ℓ(*A*) for all measurable sets *A* ⊆ [0, *c*2(*E*)] (where ℓ denotes length measure on [0, *c*2(*A*)]);\n* let Ex(*A*) : = {*ξ* ∈ ∂*T* : *π*(*ξ*) ≠ *λ*(*ξ*)}: then, Ex(*A*) = ∂*T* \\ *A*′;\nFor the boundary point *ξ* identified by the geodesic {*x**j*}*j*, *λ*(*ξ*) does not lie on the bottom side of the *R*.\nWe can assume from now that the tree has been prune as in (i) and that *A* = *A*′. Still, we see below that the combinatorics of the tree are not, by themselves, enough to determine a rectangle *R* and a square tiling of it. They are, if we assume that the set *A* in the Theorem [main 2] is *closed, but they are not in general. This is in striking contrast with the case of finite trees, or more generally graphs. However, if *c*2(*A*) < *c*2(∂*T*), then a price has to be paid. In fact, in that case* \n \n*c*2(Ex(*A*)) ≥ *c*2(∂*T*) − *c*2(*A*) > 0, \ni.e., the exceptional set Ex(*A*) is rather large, although, clearly, 0 = *μ**A*(Ex(*A*)) = ℓ(*π*(Ex(*A*))). To enlighten this phenomenon, we provide an example of a set which is densly spread out on the boundary having exceptional set of full capacity.\n[A regular set of dyadic combinatorics and arbitrarily small capacity with positive capacity in every subtree] Let *ɛ* > 0 be any small number, and *T* = *T*2 a dyadic tree with edge root *ω*. Let *n* = *n*(*ω*) be the number of steps one has to move to the left, starting from the root, before finding an edge *α**n**ω* such that *c**p*(∂*T**n*, *ω*) ≤ *ɛ*/2, where *T**n*, *ω* denotes *α**n**ω* − tent. Let {*α*1*ω*, …, *α**n**ω*} be the geodesic from the root to *α**n**ω*, and *β**j* be the right brother of *α**j**ω*, i.e. the only edge with *b*(*β**j*) = *b*(*α**j**ω*). In each subtree *T**β**j*, *j* = 1, …, *n*, starting from the root *β**j*, move to the left, say *n*(*β**j*) steps, until you find an edge *α**n*(*β**j*)*β**j* such that *c**p*(∂*T**n*(*β**j*), *β**j*) ≤ *ɛ*/(22*n*). Then we iterate the process: {*α*1*β**j*, …, *α**n*(*β**j*)*β**j*} is the geodesic from *β**j* to *α**n*(*β**j*)*β**j* and *γ**i* = *γ**i*(*j*) the right brother of *α**i**β**j*. In each subtree *T**γ**i* we individuate as before, always moving to the left, subtrees with *c**p*(∂*T**n*(*γ**i*), *γ**i*) ≤ *ɛ*/(23*n*(*β**j*)), and so on. Let\n \n*A*1 = ∂*T**α**n**ω*, *A*2 = ⋃*j* = 1*n*∂*T**n*(*β**j*), *β**j*, *A*3 = ⋃*j* = 1*n*⋃*i* = 1*n*(*β**j*)∂*T**n*(*γ**i*)*γ**i*, …, and set *A* = ⋃*k**A**k*. \nBy construction, for every *α* ∈ *E* the tree *T**α* contains a tent with boundary in *A*. Since tents have positive capacity (for example by the rescaling property of Proposition [prop:rescaling]), it follows that *c**p*(*A* ∩ ∂*T**α*) > 0 for every *α* ∈ *E*. On the other hand,\n \n$$c\\_p(A)\\leq\\sum\\_k c\\_p(A\\_k)\\leq \\sum\\_k \\frac{\\varepsilon}{2^k}=\\varepsilon.$$ \nFor the regularity, observe that by construction for every point *ξ* ∈ *A*, there exists some edge *α* such that *ξ* ∈ ∂*T**α* ⊆ *A*. Therefore, if *μ*, *μ**α* are the equilibrium measures for *A* and ∂*T**α* respectively, we have\n \n$$V\\_p\\mu(\\xi)=V\\_{p,\\alpha} \\mu(\\xi)+V\\_p \\mu(b(\\alpha))=\\big(1-V\\_p\\mu(b(\\alpha)) \\big)V\\_{p,\\alpha} \\mu^\\alpha(\\xi)+V\\_p\\mu(b(\\alpha))=1,$$ \nby the regularity of homogeneous trees.\nMoreover, given any real number *c* ∈ (0, 1), it is possible to build a tree with *c*2(∂*T*) = *c*. It follows that we can perform square tilings of rectangles with any ratio of sides.\nPRIMA AVEVAMO UNA PROP PER DIMOSTRARE L’AFFERMAZIONE SOPRA, L’HO TOLTA E TOGLIEREI ANCHE L’AFFERMAZIONE PER NON APPESANTIRE L’ARTICOLO....POSSO RECUPERARLA QUANDO SCRIVERÒ L’ARTICOLETTO SULLA REGOLARITÀ DEI BORDI\nBranched continued fractions\n============================\nTheorem [main 1] can be reformulated in terms of *branched continued fractions*. Besides adding further interesting structure to the class of equilibrium measures, this provides a recursive formula for concretely calculating capacity of sets. An accessible survey on branched continued fractions is in.\n[cont frac] Let *f* : *E* → R^+ be any non-negative function such that *I**p**f* < 1 on *V*, and consider the associated rescaled function defined by\n \n$$\\label{sequence condition}\nc(\\alpha)=\\frac{f(\\alpha)}{\\Big(1-I\\_pf(b(\\alpha))\\Big)^{p-1}}.$$ \nThen, *f* is the potential of a measure *μ* on ∂*T* if and only if *c* is defined, for each edge *α* which is not a leaf, by the following recursive formula\n \n$$\\label{eq:precursive}\nc(\\alpha)=\\frac{\\displaystyle\\sum\\_{\\beta\\in s(\\alpha)}c(\\beta)}{\\left(1+\\left(\\displaystyle\\sum\\_{\\beta\\in s(\\alpha)}c(\\beta)\\right)^{p^\\prime-1}\\right)^{p-1}}.$$ \nBy, *f*(*ω*) = *c*(*ω*). Denote by *α*− the parent of *α*, i.e. the only edge *α*− such that *α* ∈ *s*(*α*−). For every *α* ≠ *ω*, we have\n \n$$\\begin{split}\nf(\\alpha)^{p'-1}&=c(\\alpha)^{p^\\prime-1}\\Big(1-I\\_pf(b(\\alpha))\\Big)=c(\\alpha)^{p^\\prime-1}\\left(1-I\\_pf(b(\\alpha^-))-f(\\alpha^-)^{p'-1}\\right)\\\\\n&=c(\\alpha)^{p^\\prime-1}\\left(\\frac{f(\\alpha^-)^{p'-1}}{c(\\alpha^-)^{p^\\prime-1}} -f(\\alpha^-)^{p'-1}\\right)=c(\\alpha)^{p^\\prime-1}f(\\alpha^-)^{p'-1}\\frac{1-c(\\alpha^-)^{p^\\prime-1}}{c(\\alpha^-)^{p^\\prime-1}}.\n\\end{split}$$ \nIterating we obtain,\n \n*f*(*α*) = *c*(*α*)∏*γ* < *α*(1 − *c*(*γ*)*p*′ − 1)*p* − 1. \nHence, for any chosen edge *α* which is not a leaf, it holds\n \n∑*β* ∈ *s*(*α*)*f*(*β*) = ∏*γ* ≤ *α*(1 − *c*(*γ*)*p*′ − 1)*p* − 1∑*β* ∈ *s*(*α*)*c*(*β*). \nNow, *f* is the co-potential of a measure if and only if the flow condition holds. Namely, using, holds if and only if\n \n*c*(*α*) = (1 − *c*(*α*)*p*′ − 1)*p* − 1∑*β* ∈ *s*(*α*)*c*(*β*), \nwhich is equivalent to, as can be seen solving with respect to *c*(*α*).\nBy the rescaling properties of equilibrium measure (Proposition [prop:rescaling]), we have that if *μ* is the *p* − equilibrium measure for a set *A* ⊆ ∂*T*, then *c*(*α*) = *c**p*, *α*(*A**α*). This gives us an algorithm to calculate the capacity of a set in ∂*T* in terms of successive tents capacities. Moreover, by relation we deduce that capacities can be expressed by means of branched continued fractions. For example, by we obtain the expression\n \n$$c\\_2(\\partial T)=\\frac{1}{1+\\displaystyle\\frac{1}{\\displaystyle\\sum\\_{\\beta\\in s(\\omega)}\\displaystyle\\frac{1}{1+\\displaystyle\\sum\\_{\\gamma\\in s(\\beta)}\\frac{1}{1+\\displaystyle\\dots}}}} \\quad.$$ \nIn the same structure was observed for the *total resistence* *R* of an infinite tree without edges of degree 1. In particular, for such a class of trees, we obtain the relation\n \n$$c\\_2(\\partial T)=\\frac{1}{1+R}.$$ \nTo end the section, we give a reformulation of Theorem [main 1] which provides a characterization of equilibrium measures by means of an equation for capacities.\n(i) Let *μ* the equilibrium measure for a set *A* ⊆ ∂*T* and *f* = *I*\\**μ*. Then the function *c* = *c**f* given by solves\n \n*c*(*α*)(1 − *c*(*α*)*p*ʹ − 1) = ∑*β* > *α**c*(*β*)*p*ʹ∏*α* ≤ *γ* < *β*(1 − *c*(*γ*)*p*ʹ − 1)*p*, *α* ∈ *E*. \n(ii) Let *μ* be a measure on ∂*T* such that *V**p*(*μ*) < 1 on *V*. Set *f* = *I*\\**μ*. If the function *c* = *c**f* solves, then there exists an F*σ* set *A* ⊆ ∂*T* such that *μ* is its *p*-equilibrium measure.\nGiven any measure *μ*, setting *f* = *I*\\**μ* and *c* = *c**f*, by ([sequence condition]) we have\n \n$$f(\\alpha)\\big(1-I\\_pf(b(\\alpha))\\big)=\\frac{f(\\alpha)^{p'}}{c(\\alpha)^{p'-1}}, \\quad \\alpha\\in E.$$ \nIf *μ* is an equilibrium measure then holds, and conversely if *μ* is such that *V**p*(*μ*) < 1 and solves it is the equilibrium measure of some F*σ* set. But holds if and only if\n \n*f*(*α*)*p*ʹ(1 − *c*(*α*)*p*ʹ − 1) = *c*(*α*)*p*ʹ − 1∑*β* > *α**f*(*β*)*p*ʹ. \nBut by Proposition [cont frac], we know that *c*(*α*) solves, or equivalently, *f*(*α*) is defined by. Hence, sobstituting above we get\n \n*c*(*α*)*p*ʹ∏*γ* < *α*(1 − *c*(*γ*)*p*′ − 1)*p*(1 − *c*(*α*)*p*ʹ − 1) = *c*(*α*)*p*ʹ − 1∑*β* > *α**c*(*β*)*p*ʹ∏*γ* < *β*(1 − *c*(*γ*)*p*′ − 1)*p*, \nwhich is ([formula for capacities]).\n"},"data_id":{"kind":"string","value":"arxiv_0183859"}}},{"rowIdx":45993,"cells":{"content":{"kind":"string","value":"{aligned}$$ \nSo we have\n \n$$\\begin{aligned}\n\\frac{1}{N}R(\\hat{\\kappa}\\_1,\\ldots,\\hat{\\kappa}\\_K)=\\frac{1}{N}R(\\kappa^\\*\\_1,\\ldots,\\kappa\\_K^\\*)+O(N^{-\\frac{q-3-2q\\alpha}{2q}})-\\frac{1}{N}R(\\kappa\\_1^0,\\ldots,\\kappa\\_K^0)<0\\end{aligned}$$ \nas *N* → ∞. This is a contradiction because *κ̂**k* are the maximizers of *R*(*κ*1, …, *κ**K*). Since\n$\\displaystyle P\\left(\\bigcap\\limits\\_{k,l}A\\_{kl}\\right)=1-P\\left(\\bigcup\\limits\\_{k,l}A^c\\_{kl}\\right)\\geq 1-\\sum\\limits\\_{l-k\\geq ml}^NP\\left(A\\_{kl}^c\\right)=1-\\frac{B\\_1}{\\epsilon^{2q}}$.\nso $\\hat{\\kappa}\\_j\\overset{p}{\\rightarrow} \\kappa\\_j^0$.\n**Proofs of Theorem [BICconsist3]:**\nIf *L* < *K*, then there should be a change point *λ**j*0 that can not estimated consistently. Then following the proof in Theorem [esticonsis3],\n \n$$\\begin{aligned}\n&&-\\frac{1}{N}R(\\hat{\\kappa}\\_1,\\ldots,\\hat{\\kappa}\\_L)+LC\\_N/N+\\frac{1}{N}R(\\hat{\\kappa}\\_1,\\ldots,\\hat{\\kappa}\\_K)-KC\\_N/N\\\\\n&<&-c+O\\_p(N^{-\\frac{q-3-2q\\alpha}{2q}})-(K-L)C\\_N/N<0\\end{aligned}$$ \nas *N* → ∞.\nIf *K̂* > *K*, then still every change point *λ**i* should be estimated consistently. So, there should be a change point *κ**j* − 1 < *κ**k*\\* < *κ**j*. Then by Lemma [ch3l2] and [ch3l3], $\\displaystyle R(\\kappa\\_{j-1},\\kappa\\_k^\\*,\\kappa\\_j)-R(\\kappa\\_{j-1},\\kappa\\_{j})=O\\_p(N^{-\\frac{q-3-2q\\alpha}{2q}})$. So $BIC\\_L-BIC\\_K=O\\_p(N^{-\\frac{q-3-2q\\alpha}{2q}})+(K-L)C\\_N/N<0$, as *N* → ∞. Here we complete the proofs of Theorem [BICconsist3].\n99 Brockwell, P. J., and Davis, R. A. (1991), *Time Series: Theory and Methods (2nd Edition)*, Springer-Verlag. Davis, R. A., Lee, T. C. M., and Rodriguez-Yam, G. A. (2006), “Structural Break Estimation for Nonstationary Time Series Models,” *Journal of the American Statistical Association*, 101(473), 223-239. Fan, J. and Yao, Q., (2003), *Nonlinear Time Series*, Springer. Goldberger, A. L., Amaral, L. A. N., Glass, L., Hausdorff, J. M., Ivanov, P. Ch., Mark, R. G., Mietus, J. E., Moody, G. B., Peng, C-K., Stanley, H. E. (2000), “PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals”, *Circulation* 101(23), e215-e220. 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(2001), “Automatic Statistical Analysis of Bivariate Nonstationary Time Series,” *Journal of the American Statistical Association*, 96(454), 543-560. Parzen, E. (1982), “Maximum Entropy Interpretation of Autoregressive Spectral Densities,” *Statistics and Probability*, 1, 2-6. Parzen, E. (1983), “Time Series ARMA Model Identification by Estimating Information,” Technical Report, Texas A&M Research Foundation Project No. 4226T, N-37. Preuss, P., Puchstein, R., and Dette, H. (2015), “Detection of Multiple Structural Breaks in Multivariate Time Series,” *Journal of the American Statistical Association*, 110(510), 654-668. Priestley, M. B. (1981), *Spectral Analysis and Time Series*, Academic Press. Rao, T. S. (1993), *Developments in Time Series Analysis*, Chapman and Hall/CRC. Shore, J. E. (1981), “Minimum Cross-Entropy Spectral Analysis,” *IEEE Transsactions on Acoustics, Speech, and Signal Processing*, ASSP-29(2), 230-237. Woodroofe, M. B., and Van Ness, J. W. (1967), “The Maximum Deviation of Sample Spectral Densities,” *The Annals of Mathematical Statistics*, 38(5), 1558-1569. Yao, Y.-C. (1988), “Estimating the Number of Change-Points Via Schwarz’ Criterion,” *Statistics & Probability Letters*, 6(3), 181-189. Yao, Y.-C., and Au, S. T. (1989), “Least-Squares Estimation of a Step Function,” *Sankhyā*, 51(3), 370-381. Zou, C., Yin, G., Feng, L., and Wang, Z. (2014), Nonparametric Maximum Likelihood Approach to Multiple Change-Point Problems. *The Annals of Statistics*, 42(3), 970-1002.\n\nNonparametric Multiple Change Point Detectionfor Non-Stationary Time Series\n===========================================================================\n**Author’s Footnote:**\nZixiang Guan is PhD, Department of Mathematics and Statistics, University of Calgary (Email: `zixiang.guan@ucalgary.ca`), Gemai Chen is Professor (Email: `gchen@math.ucalgary.ca`), Department of Mathematics and Statistics, University of Calgary, Alberta, Canada.\n**Abstract**\nThis article considers a nonparametric method for detecting change points in non-stationary time series. The proposed method will divide the time series into several segments so that between two adjacent segments, the normalized spectral density functions are different. The theory is based on the assumption that within each segment, time series is a linear process, which means that our method works not only for classic time series models, e.g., causal and invertible ARMA process, but also preserves good performance for non-invertible moving average process. We show that our estimations for change points are consistent. Also, a Bayesian information criterion is applied to estimate the member of change points consistently. Simulation results as well as empirical results will be presented.\nKeywords: Changepoint; Dynamic Programming; Spectrums; BIC; Kullback-Leibler Divergence\nIntroduction\n============\nTime series analysis is a well-developed branch of statistics with a wide range of applications in engineering, economics, biology and so on. Generally speaking, when investigating the theoretical properties as well as analyzing real data both in time domain and frequency domain, it is often assumed that time series is stationary. However in application, stationarity may be violated. How to analyze non-stationary time series is challenging. In ARIMA model ( Brockwell and Davis 1991), data is differenced finite times so that it reduces to ARMA process. In another way, we may assume that non-stationary process consists of several stationary ones. Our goal is to segment time series properly.\nConsider a sequence of data {*X*1, …, *X**N*} and let *τ*00, *τ*10, *τ*20, …, *τ**K*0, *τ**K* + 10 be nonnegative integers satisfying 0 = *τ*00 < *τ*10 < *τ*20 < … < *τ**K*0 < *τ**K* + 10 = *N*. We assume that within the *j*th segment of data, i.e., *τ**j* − 10 + 1 ≤ *t* ≤ *τ**j*0, 1 ≤ *j* ≤ *K* + 1, *X**t* is stationary. *τ**j*0 are called structural breaks, or change points, which are unknown. *K* is the number of change points. From the perspective of time domain, we can assume that each stationary segment can be modeled by appropriate statistical models while its structure varies across different segments. Davis, Lee and Rodriguez-Yam (2006) divided non-stationary time series into several different autoregressive processes. Kitagawa and Akaike (1978) detected change points by AIC criterion. In frequency domain, Ombao, Raz, Von Sachs and Malow (2001) used a family of orthogonal wavelet called SLEX to partition non-stationary process into stationary ones. Lavielle and Ludeña (2000) estimated change points using Whittle log-likelihood when time series was parametric. In Korkas and Fryzlewicz (2017), Locally Stationary Wavelets was applied to estimate the second-order structure, then Wild Binary Segmentation was imposed to divide the time series into several segments based on CUSUM statistics.\nWhen autocovariance function is absolutely summable, it is well known that stationary process has spectral density function, which is the Fourier transformation applied to autocovariance function (Brockwell and Davis 1991). Spectral density function preserves a good property which is that after being properly normalized, it becomes a well-defined probability density function (Priestley 1981). In probability theory, there are some existing functions to measure the difference between probability density functions. Kullback-Leibler divergence (K-L divergence) is one of them.\nIn Kullback and Leibler (1951), a divergence function was introduced to measure the discrimination information between two distribution functions. The original definition is as follows. Suppose there are two probability distributions, *Z*1(*x*) and *Z*2(*x*). *z*1(*x*) and *z*2(*x*) denote the probability density functions of *Z*1 and *Z*2, respectively. Then Kullback-Leibler (K-L) divergence from *Z*2 to *Z*1 is\n$\\displaystyle D\\_{\\mathrm{KL}}(Z\\_1\\|Z\\_2)=\\int z\\_1(x)\\log\\left(\\frac{z\\_1(x)}{z\\_2(x)}\\right)dx$.\nAs we can see, K-L divergence is defined for probability density function, which is non-negative. Since spectral density function is also non-negative, we will generalize K-L divergence so that it is applicable to non-negative functions, which is still called Kullback-Leibler divergence in this article and will be applied later in the forthcoming sections. The definition is as follows. Suppose we have two non-negative functions, *f*1 and *f*2, defined on a common support. Then the Kullback-Leibler divergence of *f*1(*x*) with respect to *f*2(*x*) is\n$\\displaystyle D\\_{\\mathrm{KL}}(f\\_1\\|f\\_2)=\\int f\\_1(x)\\log\\frac{st(f\\_1)}{st(f\\_2)}dx$,\nwhere $st(f\\_1)=\\frac{f\\_1(x)}{F\\_1}$, $st(f\\_2)=\\frac{f\\_2(x)}{F\\_2}$, and *F*1 = ∫*f*1(*x*)*d**x*, *F*2 = ∫*f*2(*x*)*d**x*. By Gibbs’s inequality, this new K-L divergence is still non-negative and is equal to 0 if and only if *f*1 = *c**f*2 almost everywhere, where *c* is an appropriate constant. In this article, we apply K-L divergence to normalized spectral density functions to define our objective function. Then we estimate change points by maximizing the objective function. That is, we find the locations where discrepancy between different spectral density functions reaches its maximum. When estimating spectral density function, we adopt the classical method. That is, we first calculate periodogram then smooth it by choosing appropriate spectral window. Although we assume that each stationary segment is a linear process, which is a class of stationary time series more general than autoregressive process and ARMA model, we calculate spectral density function without estimating any parameters. So our change point detection method is nonparametric, which we call it nonparametric spectral change-point detection (NSCD). In application, we do not know the number of change points, so a BIC criterion,which is similar to Yao (1988) and Zou, Yin, Feng, and Wang (2014), is proposed. The consistency of both change point estimation and BIC criterion can be guaranteed. Dynamic programming algorithm (Hawkins 2001) is used, but due to its computational complexity, we adopt the screening algorithm (Zou *et al*. 2014). Also, Pruned Exact Linear Time (PELT) by Killick, Fearnhead and Eckley (2012) can also boost the speed of algorithm when estimating the number and locations of change points simultaneously.\nThe rest of this article is organized as follows. In Section 2, we describe our objective functions. Asymptotic properties are presented in Section 3. Implementation and further details of our algorithm will be given in Section 4. In Section 5, numerical simulations as well as comparison with several methodologies are shown. The analysis of EEG data is presented in Section 6.\nModel and Methodology [ch3s1]\n=============================\nConsider non-stationary time series $X\\_t=\\sum\\limits\\_{j=-\\infty}^{+\\infty} a\\_j(k)\\xi\\_{t-j}$, $\\xi\\_j\\overset{i.i.d}{\\sim}(0,\\sigma^2)$, *τ**k* − 10 < *t* ≤ *τ**k*0, *k* = 1, …, *K* + 1 with *τ*00 = 0, *τ**K* + 10 = *N*, *τ**k*0 − *τ**k* − 10 = *N**k*. Here we assume that all *ξ**t* are independent and identically distributed. {*a**j*(*k*)} satisfy $\\sum\\limits\\_{j}|a\\_j(k)|<+\\infty$, ∀*k*, so that when *τ**k* − 10 < *t* ≤ *τ**k*0, spectral density function exists, denoted by *f**k* (Brockwell and Davis 1991). If *E**X**t* ≠ 0, we can subtract mean from observations by *X**t* − *μ*. Otherwise, we assume that *E**X**t* = 0 for all *t*. Our goal is to detect *τ**k* when *f**k* and *f**k* + 1 are different. If we could find a spectral density function *f* that it is different from all *f**k*, then a discriminant function can be applied so that it reaches its maximum at *τ**k*. Here we adopt Kullback-Leibler divergence and define our objective function as below:\n$\\displaystyle R(\\tau\\_1,\\ldots,\\tau\\_{K+1})=\\sum\\limits\\_{k=1}^{K+1}(\\tau\\_k-\\tau\\_{k-1})\\int\\_{-\\pi}^\\pi \\hat{f}\\_k(u)\\log\\frac{st(\\hat{f}\\_k)}{st(f)}du$.\nHere *F̂**k* = ∫− *π**π**f̂**k*(*u*)*d**u*, *F* = ∫− *π**π**f*(*u*)*d**u*, $st(\\hat{f}\\_k) = \\frac{\\hat{f}\\_k}{\\hat{F}\\_k}$, and $st(f)=\\frac{f}{F}$ so that *s**t*(*f̂**k*) and *s**t*(*f*) become probability density functions, and 0 < *τ*1 < *τ*2 < ⋯ < *τ**K* < *N* is a possible partition of the time series. There are two ways to find *f*. We can give an estimation of the spectral density function *f̂*0, based on the whole time series. Although the whole time series is non-stationary, the estimated spectral density function still converges to a well-defined function, which is the weighted sum of *f**k* (See Lemma [ch3l2] for details). If we know that data is not white noise, then $st(f)=\\frac{1}{2\\pi}$.\nThere are literatures concerning the application of K-L divergence in time series. Parzen (1982) used it to estimate the parameters of autoregressive process. Parzen (1983) extended it to estimate ARMA process. Shore (1981) calculated the minimum K-L divergence to estimate spectrum given its priori spectrum has an exponential form. See Rao (1993) for more reviews. From information point of view, K-L divergence measures the information gain when a new probability density function is used instead of the old one. So our objective function will detect the locations where we maximize the information gain when a new spectral density function *f* comes in.\nTo estimate the spectral density function, the classical methodology is adopted here. First, periodogram will be calculated as follows:\n$\\displaystyle I\\_{\\tau\\_{k-1}+1,\\tau\\_k}(\\lambda)=\\frac{1}{\\tau\\_k-\\tau\\_{k-1}}\\left|\\sum\\limits\\_{j=\\tau\\_{k-1}+1}^{\\tau\\_k}X\\_je^{-ij\\lambda}\\right|^2$\nSince periodogram is not consistent, we choose a spectral window, *W*(*u*), to smooth periodogram, then\n*f̂**k*(*λ*) = *m*∫− *π**π**W*(*m*(*λ* − *u*))*I**τ**k* − 1 + 1, *τ**k*(*u*)*d**u*, *λ* ∈ [ − *π*, *π*].\nThere are several choices for *W*(*u*) (Priestley 1981). Since we want the normalized *f̂**k* to be a probability density function, i.e., *f̂**k* ≥ 0, Bartlett kernel is chosen, which is *m**W*(*m**u*) = sin2(*m**u*/2)/(2*π**m*sin2(*u*/2)), where *m* is the bandwidth. One usually applies Fourier transformation to achieve an estimator on a grid of frequencies, denoted by Λ = {*λ*1, …, *λ**N**λ*}, where *N**λ* is the cardinality of Λ, which could tend to infinity as *N* goes to infinity. Here we still use integral to denote the summation of estimators. We may set *N**λ* = *N* so *N**λ* could tend to infinity, or Λ could be Nyquist frequency. Based on this grid of frequency, K-L divergence applied to normalized spectral density function is still non-negative, and equals zero if and only if two normalized spectral density functions are equal on [ − *π*, *π*].\nWhen estimating change points, we have no idea about the number of change points in the data, so *K* should be estimated. Yao and Au (1989), Zou *et al*. (2014) gave BIC criterion and showed its consistency. Following their work, we also propose a BIC criterion as follows:\n*B**I**C**L* = − max*τ*1′, …, *τ**K*′ + *L**C**N*.\nHere *C**N* is an appropriate constant which will be illustrated later. So our estimator *K̂* is chosen by minimizing the criterion above.\nAsymptotic Theory [ch3s2]\n=========================\nIn Section [ch3s1], we can see that *K* is a constant, and *τ**i*0, *i* = 1, 2, …, *K*, change with *N*. So there is a sequence of constants, 0 < *κ*10 < *κ*20 < ⋯ < *κ**K*0 < 1, such that {*X**t*, *t* = 1, 2, …, 1} is a realization of non-stationary time series with *τ**k*0 = [*κ**k*0*N*], *k* = 1, 2, …, *K*, where [*x*] denotes the largest integer which is not greater than *x*. Their estimators are denoted by *κ̂**k*, *k* = 1, …, *K*. We can estimate *τ**k*0 first, denoted by *τ̂**k*, by maximizing the objective function *R*(*τ*1, …, *τ**K*), then *κ̂**k* = *τ̂**k*/*N*. In literature, Yao and Au (1989) achieved a consistent estimation of *O**p*(1) for *τ̂**k*, which means that the difference between estimators and true change points is no bigger than a constant. Zou *et al*. (2014) also drew the same conclusion when the number of change points was constant. In Davis *et al*. (2006), the consistency was attained in the sense that ∣*κ̂**k* − *κ**k*∣ < *ε* in probability 1, where *ε* was some constant. The reason that estimators cannot converge as fast as those in Yao and Au (1989) and Zou *et al*. (2014) is that estimating AR process needs a sufficient number of samples. To guarantee estimate accuracy, we always find the next change point which is *m**l* away from the previous one. For example, after giving *τ̂**j*, we look for the next change point starting from *τ̂**j* + *m**l*. Our results are similar to Davis *et al*. (2006) and based on a set of discrete frequencies Λ. The following assumptions are needed to obtain the consistency:\n* *E**ξ**t*8*q* < ∞ where *q* is some integer satisfying *q* ≥ 3, {*a**j*(*k*)} converge absolutely, ∀*k*.\n* *W*(*v*) is a non-negative, even, bounded, integrable function, ∫− *π**π**W*(*v*)*d**v* = 1, \n$\\int\\_{-\\pi}^\\pi \\left(W(v)\\right)^{1-\\frac{1}{2q}}dv<\\infty$.\n* *N*min/*N* → *c*min > 0, as *N* → ∞, and *N*min > *m*, where *N*min = min1 ≤ *k* ≤ *K*(*τ**k*0 − *τ**k* − 10).\n* ∀*k*, *f**k* is everywhere positive and satisfies uniform Lipschitz condition:\n∣*f**k*(*u*1) − *f**k*(*u*2)∣ ≤ *B**f**k*∣*u*1 − *u*2∣\n∀*u*1 ∈ [ − *π*, *π*], *u*2 ∈ [ − *π*, *π*], where *B**f**k* is a constant.\n* *m* = *O*(*N**α*), where $\\frac{1}{4}\\leq\\alpha<\\alpha+\\frac{3}{2q}<\\frac{1}{2}$.\n* *w*(0) = 1, and *w*(*v*) has continuous derivatives to the order of 2*q*.\n* {*f**k*(*u*)} are linearly independent for *u* ∈ [ − *π*, *π*].\n* *m**l* = *c*ml*N*, where *c*ml > 0. *m**l* < *N*min\nIn A8, *m**l* is the minimal length of time series when estimating spectral density functions, since a sufficient number of observations is always necessary, especially when the convergence rate of estimators is slow. Also, *m**l* < *N**m**i**n* so that all change points are distinguishable. Assumption 7 is given to guarantee that any linear combination is not equal to *f**j*, ∀*j* = 1, …, *K*. Assumption 1-6 are similar to those in Woodroofe and Van Ness (1967) so that the $\\max\\limits\\_{\\lambda\\_j\\in\\Lambda}\\frac{\\hat{f}\\_k(\\lambda\\_j)}{f\\_k(\\lambda\\_j)}$ can be bounded in probability. Assumption 4 can be stronger so that in Assumption 5, *α* can be less than $\\frac{1}{4}$ (see Woodroofe and Van Ness (1967) for further details). Theorem [esticonsis3] gives the consistency of change point estimation.\n[esticonsis3] When *K* is known, $\\hat{\\kappa}\\_j\\overset{p}{\\rightarrow}\\kappa\\_j$, ∀*j* = 1, …, *K*.\nTo estimate the number of change points, a pre-specified upper bound *K*max satisfying *K* < *K*max will be given. Then BIC values for each 1 ≤ *L* ≤ *K*max are calculated and *K̂* will be the number where BIC reaches its minimum. The following theorem establishes the consistency of estimation of BIC criterion.\n[BICconsist3] If *C**N*/*N* → 0, $C\\_N/N^{\\frac{q+3+2q\\alpha}{2q}}\\rightarrow\\infty$, we have *P*(*K̂* = *K*) → 1.\nAlgorithm\n=========\nSimilar to Zou *et al*. (2014), our objective function is separable. For change point detection, a commonly adopted algorithm, Dynamic Programming (Hawkins 2001), can be applied. The main idea of Dynamic Programming is that estimation of *τ̂**K* is computed first, which is the rightmost change point. Then the time series data from 1 to *τ̂**K* will be divided into *K* − 1 parts, and we will estimate *τ**K* − 1 recursively. However, the computation complexity is *O*(*K**N*2), and taking Discrete Fourier transformation and spectrum smoothing into consideration, it is time-consuming.\nTo reduce computational complexity, Zou *et al*. (2014) proposed a screening algorithm. For *X**j*, …, *X**j* + *l*, where *l* is some constant integer, calculate the location where the function below reaches its maximum.\n \n$$\\begin{aligned}\nr\\_j&=&\\arg\\max\\limits\\_rr\\int\\_{-\\pi}^\\pi \\hat{f}\\_{j,j+r}(u)\\log\\frac{st(\\hat{f}\\_{j,j+r})}{st(f)}du\\\\\n& &+(l-r)\\int\\_{-\\pi}^\\pi \\hat{f}\\_{j+r+1,j+l}\\log\\frac{st(\\hat{f}\\_{j+r+1,j+l})}{st(f)}du.\\end{aligned}$$ \nHere *f̂**j*, *j* + *r* and *f̂**j* + *r* + 1, *j* + *l* denote the estimated spectral density functions based on observations from *j* to *j* + *r*, and *j* + *r* + 1 to *j* + *l*. Let *j* change from 1 to *n* − *l*, then we have a set *A**s**c* containing all *r**j*, then apply Dynamic Programming on *A**s**c*. The main idea is that if a change point is included in *X**j*, …, *X**j* + *l*, then the equation above should reach its maximum at this true change point. That is, *A**s**c* contains true change points. Here we should choose *l* < *N*min so that *X**j*, …, *X**j* + *l* contain only one change point.\nHere *f̂**j*, *j* + *r* and *f̂**j* + *r* + 1, *j* + *l* denote the estimated spectral density function of samples from *j* to *j* + *r*, and *j* + *r* + 1 to *j* + *l*. Let *j* change from 1 to *n* − *l*, then we have a set *A**s**c* containing all *r**j*, then apply dynamic programming on *A**s**c*. The main idea is that if a change point is included in *X**j*, …, *X**j* + *l*, then the equation above should reach its maximum at this true change point. That is, *A**s**c* contains true change points. Here we should choose *l* < *N*min so that *X**j*, …, *X**j* + *l* contain only one change point.\nWhen calculating the spectrums within each segment of time series, the most widely used method is Fast Fourier Transformation (FFT). However, in FFT, a problem is that spectral density function is estimated on Nyguist frequency. If so, time series with different length is estimated on different set of Fourier frequency. So when calculating the integral in *R*(*τ*1, …, *τ**K*), we cannot align the frequencies where spectral density functions *f**k* and *f* are estimated. Our method is that we first choose a set of frequencies, denoted by Λ, then apply Discrete Fourier Transformation on Λ for every subset of samples, which will solve the alignment problem naturally.\nFor BIC criterion, usually Dynamic Programming will be applied first, then the values of BIC criterion for all *L* ≤ *K*max will be calculated. Obviously, this will increase complexity. Killick *et al*. (2012) proposed a method called Pruned Exact Linear Time (PELT), which would significantly reduce computational complexity. The main idea is that when a new sample is included, check all the remaining locations before new sample. If the objective function decreases to the extent that it is larger than the penalty term *C**N*, those locations which do not satisfy the condition will be removed and the next sample will be added into our calculation until the end. The cardinality of the set of all remaining locations is the estimated number of change points and the elements within will be the estimated change points. Under some assumptions, the computational complexity is linear with respect to sample size.\nIn BIC criterion, another problem is how to choose *C**N*. Although we can set *C**N* to satisfy conditions in Theorem [BICconsist3], this choice may be too large which leads to underestimation of *K*. To overcome this difficulty, we first choose a length, which equals *m**l*, then compute the median, denoted by *m**e*BIC, for all values of the function below:\n$\\displaystyle \\int \\hat{f}\\_{j,j+ml}(u)\\log\\frac{st(\\hat{f}\\_{j,j+ml})}{st(f)}du$, ∀*j* = 1, …, *K*.\n*f̂**j*, *j* + *m**l* is the estimated spectral density function from *X**j*, …, *X**j* + *m**l*, *F̂**j*, *j* + *m**l* is its integral. Finally, *C**n* = *m**e*BIC × *n**c*. Our simulation shows that an appropriate choice for *c* is 0.73 regardless of *m*.\nThe selection of spectral windows is an important topic in spectral estimation. In Priestley (1981), bandwidth is selected as follows\n$B\\_W=2\\sqrt{6}\\left(\\frac{1}{m^r}k^{(r)}\\right)^{1/r}$,\nwhere $k^{(r)}=\\lim\\limits\\_{u\\rightarrow0}\\frac{1-w(u)}{|u|^r}$, where *w*(*u*) is the inverse Fourier transformation of *W*(*u*), and *r* is the largest integer so that the limit aforementioned exists and is non-zero. *m* is the scale parameter in spectral windows. By the proofs of Theorem [esticonsis3], we can see that estimators of change points reach consistency because *N* dominate convergence rate. So bandwidth selection does not matter too much, which is verified by our simulations.\nIn application, sometimes sample size is large. Although we can apply some methods, such as screening, PELT, to boost the calculation, the computational complexity is still intolerable. Here, we set a change point searching unit, denoted by *n*su (Hawkins 2001). That is, when searching for change points, we add a unit of observations into our calculation each time, not just one observation. This unit is different from *m**l*, which is used to calculate spectrums since estimating spectral density function usually needs sufficient amount of observations. By setting this unit, change point can only be estimated at *n*su, 2*n*su, 3*n*su, and so on. If we set this unit equal to 1, the algorithm degenerates to the general scenario when no searching unit is given. This will dramatically increase the speed of our algorithm. The computational complexity of Dynamic Programming is *O*(*n*2) (Zou *et al*. 2014), so by setting a searching unit *n*su, the complexity will drop to *O*(*n*2/*n*su2). Apparently the estimation accuracy will be sacrificed, since the true change points and the maximizers of objective function, may not lie on the grid. We suggest the choice of *n*su by choosing the desired estimation accuracy first. Intuitively speaking, if we want the estimates having the accuracy of 1% of the total sample size, then we can set *n*su = 0.01*n*.\nSimulation [ch3s4]\n==================\nIn this section we show the finite sample properties of our method and compare it with AutoPARM, Wild Binary Segmentation (WBS), Binary Segmentation (BS), MuBred, and NMCD under several cases.\nFollowing Zou *et al*. (2014), we calculate the distance between two sets *G* and *Ĝ*, which are the true change point set and the estimated set, respectively, by\n$\\varrho(\\hat{G}||G)=\\sup\\limits\\_{b\\in G}\\inf\\limits\\_{a\\in \\hat{G}}|a-b|$, and $\\varrho(G||\\hat{G})=\\sup\\limits\\_{a\\in \\hat{G}}\\inf\\limits\\_{b\\in G}|a-b|$.\nThe first measurement shows if there is an estimator close enough to a true change point, while the second measurement reveals the distance between estimates and true change points. When *K* is known, both measures should give good performances in the sense that $\\varrho(\\hat{G}||G)$ and $\\varrho(G||\\hat{G})$ are small. For all the tables, $\\varrho(\\hat{G}||G)$ and $\\varrho(G||\\hat{G})$ are shown outside the parentheses while $\\varrho(\\hat{G}||G)/N$ and $\\varrho(G||\\hat{G})/N$ are shown inside, which can measure the estimation accuracy of *κ̂*. In the following subsections, $\\xi\\_t\\overset{\\mathrm{i.i.d}}{\\sim} N(0,1)$, if it is not mentioned. To guarantee the estimation accuracy of spectral density function, we should set the minimal length of a segment, which is denoted by *m**l*. In this simulation, we set *m**l* = 350 if it is not mentioned. By setting *m**l*, we also assume that the distance of two adjacent change points will not be smaller than *m**l*, so we set *K*max = 6 for each case. If *K* should be estimated, we report the percentage when *K* is accurately detected. All simulations are obtained with 1000 replications.\nAutoRegresstive Process [ch3simu1]\n----------------------------------\nFollowing the examples in Davis *et al.* (2006), we generate the non-stationary time series from the following.\nAutoregressive Processes (Case 1):\n* *X**t* − 0.9*X**t* − 1 = *ξ**t*, 1 ≤ *t* ≤ 1024,\n* *X**t* − 1.69*X**t* − 1 + 0.81*X**t* − 2 = *ξ**t*, 1025 ≤ *t* ≤ 1536,\n* *X**t* − 1.32*X**t* − 1 + 0.81*X**t* − 2 = *ξ**t*, 1537 ≤ *t* ≤ 2048.\nand their normalized spectral density functions are shown in Figure [Davis].\n[Davis]![AR(1) is in black dotted line, the second AR is in blue dashed line, the last AR is in red solid line.](truescalspecDavis.png \"fig:\")\nIn Table [Davissimu1], we show the results of our method when *K* is known with the spectral density function of total samples and white noise as the baseline functions, and different choices of bandwidths, respectively. In Table [Davissimu2], the simulation is conducted when *K* is unknown. We adopt Bartlett window since it guarantees the non-negativity of estimated spectral density function. To reduce computational complexity, screening algorithm is applied.\n[H]\n| | | $\\varrho(\\hat{G}||G)$ | $\\varrho(G||\\hat{G})$ |\n| --- | --- | --- | --- |\n| | *f̂*0 | 22.99 (0.011) | 22.99 (0.011) |\n| | $\\frac{1}{2\\pi}$ | 23.55 (0.012) | 23.55 (0.012) |\n| | *f̂*0 | 22.61 (0.011) | 22.61 (0.011) |\n| | $\\frac{1}{2\\pi}$ | 26.47 (0.013) | 26.47 (0.013) |\n[H]\n| | *K̂* | $\\varrho(\\hat{G}||G)$ | $\\varrho(G||\\hat{G})$ |\n| --- | --- | --- | --- |\n| AutoPARM | 93.3% | 7.61 (0.004) | 8.05 (0.004) |\n| NSCD | 98.8% | 25.00 (0.012) | 37.33 (0.018) |\n| MuBred | 97.3 % | 12.219 (0.006) | 16.229 (0.008) |\n| WBS | 37.9% | 66.65 (0.033) | 183.184 (0.089) |\n| BS | 62.7% | 84.875 (0.041) | 122.058 (0.060) |\n| NMCD | 0% | 147.4365 (0.072) | 554.9088 (0.271) |\nFrom Table [Davissimu1], the choice of baseline function does not make much difference. Also, under different bandwidths, results are quite similar. We may suggest to choose larger bandwidth for large sample size since the bias of estimation is small. From Table [Davissimu2], it is not surprising that AutoPARM outperforms NSCD, because AutoPARM knows the exact structure of data. WBS and BS perform weaker than NSCD, and cannot estimate the number of change points accurately. The performance of MuBred is better than NSCD, although it does not need the assumption of Autoregressive process. NMCD fails to capture the true change points, since it is designed for independent random variables. In fact, it prefers to overestimate the number of change points, which is 3.86.\nARMA Process and Invertible Moving Average Process [ch3simu2]\n-------------------------------------------------------------\nNext, we simulate from ARMA and invertible MA processes. Table [ARMA1] and Table [ARMA2] contain the results.\nARMA (Case 2):\n* *X**t* − *X**t* − 1 + 0.25*X**t* − 2 = *ξ**t* + 0.8*ξ**t* − 1, 1 ≤ *t* ≤ 500,\n* *X**t* − 0.5*X**t* − 1 = *ξ**t*, 501 ≤ *t* ≤ 1100,\n* *X**t* − 1.7*X**t* − 1 + 0.9*X**t* − 2 − 0.168*X**t* − 3 = *ξ**t* − 1.6*ξ**t* − 1 + 0.79*ξ**t* − 2 − 0.12*ξ**t* − 3, 1101 ≤ *t* ≤ 1800.\nInvertible MA (Case 3):\n* *X**t* = (3 + *B*)(2 − *B*)*ξ**t*, 1 ≤ *t* ≤ 500,\n* *X**t* = (3 − *B*)(2 − *B*)*ξ**t*, 501 ≤ *t* ≤ 1100,\n* *X**t* = (3 + *B*)(2 − *B*)*ξ**t*, 1101 ≤ *t* ≤ 1800.\n[H]\n|c|c|c|c|c|c| & & & \n& & & \n & *f̂*0 & 35.29 (0.020) & 34.74 (0.020) & 13.09 (0.007) & 13.09 (0.007) \n & $\\frac{1}{2\\pi}$ & 32.74 (0.018) & 32.07 (0.018) & 13.11 (0.007) & 13.11 (0.007) \n & *f̂*0 & 31.12 (0.017) & 31.01 (0.017) & 12.84 (0.007) & 12.84 (0.007) \n & $\\frac{1}{2\\pi}$ & 26.39 (0.015) & 26.38 (0.015) & 13.71 (0.008) & 13.71 (0.008) \n[H]\n|c|c|c|c|c|c|c| & & \n& *K̂* & $\\varrho(\\hat{G}||G)$ & $\\varrho(G||\\hat{G})$ & *K̂* & $\\varrho(\\hat{G}||G)$ & $\\varrho(G||\\hat{G})$ \nAutoPARM & 95.3% & 48.37 (0.027) & 27.62 (0.015) & 99% & 11.42(0.006) & 11.42 (0.006) \nNSCD & 99% & 31.42 (0.017) & 33.26 (0.018) & 99.6% & 14.59 (0.008) & 15.69 (0.009) \nMuBred & 93.3% & 40.436 (0.022) & 44.467 (0.025) & 97.7% & 49.518 (0.028) & 41.444 (0.023) \nWBS & 74.9% & 184.393 (0.102) & 66.288 (0.037) & 91.8% & 33.586 (0.019) & 45.362 (0.025) \nBS & 77.7% & 188.955 (0.105) & 69.194 (0.038) & 96.3% & 29.673 (0.016) & 36.611 (0.020) \nNMCD & 0% & 188.64 (0.105) & 324.31 (0.180) & 0.1% & 160.4434 (0.089) & 326.4282 (0.181) \nWe expect that AutoPARM still works since it is well known that ARMA and invertible MA processes can be approximated by causal AR processes. We see that under ARMA setting, the performance of NSCD is comparable to AutoPARM, while AutoPARM performs better under MA settings. Both NSCD and AutoPARM perform better than WBS and BS. MuBred gives a slightly worse performance since periodogram is not consistent. NMCD fails in both cases as it does in the previous section. The number of change points is overestimated by NMCD again, which is 3.005 and 3.00, respectively.\nNon-invertible MA Process [ch3simu3]\n------------------------------------\nWe will show simulation results when samples are generated from non-invertible MA process. In theory causal AR cannot approximate non-invertible MA process. Results are given in Table [MA1] and [MA2].\nNon-invertible MA (Case 4):\n* *X**t* = (1 + 2*B* + *B*2 + 5*B*3)*ξ**t*, 1 ≤ *t* ≤ 500,\n* *X**t* = (1 − 2*B* + 2*B*2 − 5*B*3)*ξ**t*, 501 ≤ *t* ≤ 1100,\n* *X**t* = (1 + 2*B* − *B*2 + 5*B*3)*ξ**t*, 1101 ≤ *t* ≤ 1800.\n[H]\n|c|c|c|c| & & \n& & $\\varrho(\\hat{G}||G)$ & $\\varrho(G||\\hat{G})$ \n & *f̂*0 & 33.60 (0.019) & 33.60 (0.019) \n & $\\frac{1}{2\\pi}$ & 36.00 (0.020) & 36.00 (0.020) \n & *f̂*0 & 40.11 (0.022) & 40.11 (0.022) \n & $\\frac{1}{2\\pi}$ & 41.00 (0.023) & 41.00 (0.023) \n[H]\n|c|c|c|c|c|c|c| & \n& *K̂* & $\\varrho(\\hat{G}||G)$ & $\\varrho(G||\\hat{G})$ \nAutoPARM & 36.7% & 408.60 (0.227) & 27.24 (0.015) \nNSCD & 99.7% & 37.49 (0.021) & 38.02 (0.021) \nMuBred & 8.4% & 216.203 (0.120) & 26.724 (0.015) \nWBS & 18.2% & 488.674 (0.271) & 90.493 (0.050) \nBS & 8.6% & 576.462 (0.320) & 56.975 (0.032) \nNMCD & 0% & 165.54 (0.092) & 326.701 (0.182) \nFrom Table [MA2], we can see that NSCD works for Case 4, while AutoPARM and MuBred fail since the number of change points are all underestimated. The possible reason is that the marginal variance of Case 4 does not vary much.\nRandom Noise without the Existence of Higher Moments\n----------------------------------------------------\nNext, we investigate the results when *ξ* does not have higher-order expectations. Here $\\xi\\sim \\frac{1}{\\sqrt{2}}t(4)$ so that the variance of *ξ* is still 1. The results for four cases are shown in Table [highorder], when *K* is known.\n[H]\n| | $\\varrho(\\hat{G}||G)$ | $\\varrho(G||\\hat{G})$ |\n| --- | --- | --- |\n| Case 1 | 24.24 (0.012) | 24.37 (0.012) |\n| Case 2 | 29.67 (0.017) | 28.91 (0.016) |\n| Case 3 | 11.74 (0.007) | 11.74 (0.007) |\n| Case 4 | 36.36 (0.020) | 36.36 (0.020) |\nApparently, from Table [highorder], it is clear that Assumption 1 can be slightly violated in application, while still achieving good performance.\nInvestigation of Smaller Sample Size\n------------------------------------\nIn previous sections, we discuss the performances when sample size is about 2000, which is large. Here we shrink the sample size by half, and investigate the estimate accuracy as well as the choice of bandwidth. We set *m**l* = 200, and *K*max = 5. Baseline function is chosen to be *f̂*0. Results are shown in Table [smallsample].\n[H]\n| | Bandwidth | $\\varrho(\\hat{G}||G)$ | $\\varrho(G||\\hat{G})$ |\n| --- | --- | --- | --- |\n| | *m* = *N*1/3 | 23.195 (0.023) | 23.263 (0.023) |\n| | *m* = *N*1/4 | 27.716 (0.027) | 28.075 (0.027) |\n| | *m* = *N*1/3 | 31.72 (0.035) | 30.718 (0.035) |\n| | *m* = *N*1/4 | 28.104 (0.031) | 27.734 (0.030) |\n| | *m* = *N*1/3 | 14.664 (0.016) | 14.158 (0.016) |\n| | *m* = *N*1/4 | 12.652 (0.014) | 12.652 (0.014) |\n| | *m* = *N*1/3 | 37.28 (0.041) | 37.272 (0.041) |\n| | *m* = *N*1/4 | 44.3 (0.05) | 43.877 (0.049) |\n[H]\n| | bandwidth | *K̂* | $\\varrho(\\hat{G}||G)$ | $\\varrho(G||\\hat{G})$ |\n| --- | --- | --- | --- | --- |\n| | *m* = *N*1/3 | 96% | 24.972 (0.024) | 30.44 (0.030) |\n| | *m* = *N*1/4 | 90.4% | 35.268 (0.040) | 39.494 (0.044) |\n| | *m* = *N*1/3 | 96.2% | 38.326 (0.043) | 28.1 (0.031) |\n| | *m* = *N*1/4 | 98.7% | 27.494 (0.031) | 26.384 (0.029) |\n| | *m* = *N*1/3 | 96.5% | 31.458 (0.035) | 12.73 (0.014) |\n| | *m* = *N*1/4 | 96% | 32.484 (0.036) | 11.436 (0.013) |\n| | *m* = *N*1/3 | 95% | 41.228 (0.046) | 40.246 (0.045) |\n| | *m* = *N*1/4 | 92% | 47.67 (0.053) | 53.254 (0.059) |\nAs shown in the table, the estimate accuracy is not affected by the choice of bandwidth when sample size is relatively small. The estimation accuracy of *κ* is worse than the results in Section [ch3simu1] to [ch3simu3], since the sample size is decreased by half.\nFurther Investigation of BIC Criterion\n--------------------------------------\nIn this section, we investigate the performance with different choice of *N**c*. *c* ranges from 0.1 to 0.9 and *K̂* will be plotted in all the cases mentioned above with two choices of bandwidth. All the results are shown from Figures [choicec1] to [choicec8] with baseline function *f̂*0. From figures, we can see that *c* = 0.73 is a good choice for all the four cases.\nNext, we are going to simulate the performances of BIC when *m**l* changes. Since from the previous section,we can see that the choice of *C**N* depends on *m**l*. Figures [investbic300p3c1] to [investbic300p4c4] show the results when *m**l* = 300, while Figures [investbic250p3c1] to [investbic250p4c4] give the performances when *m**l* = 250.\nAs we can see from the figures, *c* = 0.73 is not a good choice in general, and it will overestimate the number of change points. Although overestimating is tolerable since we do not want to miss the true change points, we still suggest that a sufficiently large choice for *m**l* is necessary. Since the maximum possible number of change points is [*N*/*m**l*]. We suggest that one should choose a reasonable *K*max depending on some prior information, then choose a *m**l* satisfying *m**l* ≤ *N*/*K*max.\n \n \n \n \n \n \nInfluence of Change Point Searching Unit\n----------------------------------------\nIn this section, we investigate the influence of *n*su. For all the four cases, *n*su = 10, which means that for Case 2-4, *τ*10, …*τ**K*0 are all a multiple of *n*su, while for Case 1, the true change points are not divisible by *n*su. We set *m* = *N*1/4 and baseline function *f* = *f̂*0 for four cases. The results are shown in Table [su1] and [su2].\n[H]\n|c|c|c|c| & $\\varrho(\\hat{G}||G)$ & $\\varrho(G||\\hat{G})$ \nCase 1 & 22.9 (0.011) & 22.9 (0.011) \nCase 2 & 29.26 (0.016) & 29.26 (0.016) \nCase 3 & 10.14 (0.006) & 10.14 (0.006) \nCase 4 & 40.52 (0.023) & 40.52 (0.023) \n[H]\n| | *K̂* | $\\varrho(\\hat{G}||G)$ | $\\varrho(G||\\hat{G})$ |\n| --- | --- | --- | --- |\n| Case 1 | 96.2% | 22.696 (0.011) | 39.72 (0.019) |\n| Case 2 | 97.2% | 42.02 (0.0233) | 30.68 (0.017) |\n| Case 3 | 99.6% | 12.54 (0.007) | 10.14 (0.006) |\n| Case 4 | 98.6% | 40.56 (0.023) | 43.74 (0.024) |\nWe can see that setting an *n**s**u* will not affect our results much even if the true change points are not a multiple of *n**s**u* in Case 1, so it is safe to apply this in application, which could boost the computation as well as give accurate estimations.\nCase Study [ch3s5]\n==================\nSimulated Data\n--------------\nIn the simulation section, we check the performances of our method based on 4 cases, and compare NMSD with other methods. Here we will investigate the change point detection of these 4 cases more directly. Figure [ARdata]-[nonMAdata] are the realizations of Case 1 to Case 4. The long-dashed lines are the locations of estimated change points while the dotted lines represent true change points. It is easy to see from Figure [ARdata] that the existence of change points are obvious, since marginal variance of the second segment is bigger. In Case 2, the second change point is far less obvious than the first one which can be seen in Figure [ARMAdata]. In Figure [MAdata], since the observations in the first two segments concentrate more around their mean compared to the third segment, the second change point may be captured by eyes. In Figure [nonMAdata], two change points are not apparent any more. However, from the perspective of spectrum, those change points can be easily detected. Figure [ARspectrum]-[nonMAspectrum] give the estimated spectral density functions. In Case 4, we can see that the power of spectrum of the first segment concentrates more at higher and lower frequencies, while the spectrum of the second part has more power at low frequency. The spectrum of the third part is similar to the first one, but it has more power in the low frequency range.\n \n \nElectroencephalography Recordings for Seizure\n---------------------------------------------\nIn this section, we investigate the performance of NSCD when applied to EEG data for seizure (Goldberger et al. 2000). The data is retrieved from https://www.physionet.org/pn6/chbmit/. For each subject, the EEG signal was recorded into several data files, which was one-hour long. Here all subjects were monitored for several days to trace their states, so we only analyze the data file with seizures. The sampling rate is 256 Hz, so the number of observations in each recording is 9216000, which is huge. In the data file we choose (which is chb01\\_16), seizure happened only once with duration 51 seconds, so the number of change points is 2. The duration of seizure is very short compared with the total length of this recording, so we analyze a particular part of the data which begins at 10 seconds before the seizure and ends at the 10 seconds after the seizure, so the sample size is 18176, which is still large. So here we will set a change point searching unit equal to 64, which is 0.25 second, to ease the computation burden. What is more, the minimal length *n*min is 256 which is 1 second. There are totally 23 channels in EEG recording, we apply our method on channel “FP1-F3” and “FP1-F7”. The results are shown in Figure [EEGF3] and [EEGF7]. The vertical dotted lines represent the locations of the beginning and ending of seizure, while the vertical dashed lines are the estimated change points. To demonstrate the changes in spectrums, we plot the estimated spectrums in Figure F3s and F7s. As we can see, NSCD can successfully detect the true change points, while almost all the other estimates fall between the true change points. This is because during seizure, the EEG recordings change abruptly, which will bring more non-stationarity into the time series.\nConclusion\n==========\nIn this article, we propose a change point detection method based on spectral density functions for non-stationary time series. We assume that non-stationary time series can be segmented into several linear processes. Then Kullback-Leibler divergence is applied to measure the discrepancy between different spectral density functions. A BIC criterion is suggested to estimate the number of change points. Due to the separable structure of objective function, we use Dynamic Programming to find the estimators. We also show the consistency of our estimators in theory and the estimate accuracy by simulations.\nProofs\n======\nIn the appendix, *B*1 to *B*8 are appropriate constant and *B*7*ε*, *B*8*ε* are constant with respect to *ε*.\n[ch3l1] Suppose $X\\_t=\\sum\\limits\\_{j=-\\infty}^{+\\infty}a\\_j\\xi\\_{t-j}$, 1 ≤ *t* ≤ *N*, where $\\sum\\limits\\_j|a\\_j|<+\\infty$. $\\xi\\_j\\overset{\\mathrm{i.i.d}}{\\sim}(0,\\sigma^2)$. Denote *f*(*λ*) as the spectral density function of *X**t*. *f̂* is the estimated spectral density function. Then under Assumptions 1-6,\n$\\displaystyle \\left|\\frac{\\hat{f}(\\lambda)-f(\\lambda)}{f(\\lambda)}\\right|^{2q}=O\\_p(\\frac{m^{2q}}{N^{q}})$ \n$\\displaystyle \\max\\limits\\_{\\lambda\\_i\\in \\Lambda}\\left|\\frac{\\hat{f}(\\lambda\\_i)-f(\\lambda\\_i)}{f(\\lambda\\_i)}\\right|^{2q}=O\\_p(\\frac{m^{2q}}{N^{q-1}})$ \n**Proof**: Following the method of Woodroofe and Van Ness (1967), we separate our proofs into 3 parts.\nPart 1: we show that for given *λ* ∈ [ − *π*, *π*], $\\displaystyle E|g\\_N(\\lambda)-1|^{2q}=O\\_p\\left(\\frac{m^{2q}}{N^q}\\right)$. Here *g**N*(*λ*) is the smoothed spectral density estimation for {*ξ*1, ⋯, *ξ**N*}, *ξ*\n \n$$\\begin{aligned}\n\\left| g\\_N(\\lambda)-1\\right|^{2q}&=&\\left(m\\int W(m(u-\\lambda))\\frac{1}{N}\\left|\\sum\\limits\\_{t=1}^N\\xi\\_te^{-itu}\\right|^2du-\\sigma^2\\right)^{2q}\\\\\n&=&\\frac{1}{N^{2q}}\\left(\\int W(u)\\left(\\left(\\sum\\limits\\_{t=1}^N\\xi\\_t e^{-it(um^{-1}+\\lambda )}\\right)^2-N\\sigma^2\\right)du\\right)^{2q}\\\\\n&=&\\frac{1}{N^{2q}}\\left(2\\sum\\limits\\_{t=1}^N\\sum\\limits\\_{v=1}^{N-t}\\xi\\_{t}\\xi\\_{t+v}\\cos(v\\lambda)w(vm^{-1})+\\sum\\limits\\_{t=1}^N\\xi^2\\_t-N\\sigma^2\\right)^{2q}\\\\\n&=&\\frac{1}{N^{2q}}\\left(Z\\_N(\\lambda)+r\\_N(\\lambda)+r\\_N\\right)^{2q},\\end{aligned}$$ \nwhere $Z\\_N(\\lambda)=\\sum\\limits\\_{t=1}^NZ\\_{N,t}(\\lambda)$,\n$Z\\_{N,t}(\\lambda)=2\\sum\\limits\\_{v=1}^{m-1}\\xi\\_t\\xi\\_{t+s}w(vm^{-1})\\cos(v\\lambda)$, \n$r\\_N(\\lambda)=2\\sum\\limits\\_{t=N-m+2}^N\\sum\\limits\\_{v=N-t+1}^{m-1}\\xi\\_t\\xi\\_{t+v}w(vm^{-1})\\cos(v\\lambda)$, \n$r\\_N=\\sum\\limits\\_{t=1}^N(\\xi\\_t^2-\\sigma^2)$.\nSince for any real number *b*1, …, *b**p*, (*b*1 + … + *b**p*)2*q* ≤ 22*p**q*(*b*12*q* + … + *b**p*2*q*). So we only need to prove that (*Z**N*(*λ*))2*q*, (*r**N*(*λ*))2*q*, (*r**N*)2*q* are *O**p*(*N**q**m*2*q*). By Markov’s inequality, it suffices to show that *E*(*Z**N*(*λ*))2*q*, *E*(*r**N*(*λ*))2*q*, *E*(*r**N*)2*q* are *O*(*N**q**m*2*q*).\nAfter expanding (*r**N*(*λ*))2*q*, we have\n \n$$\\begin{aligned}\n\\left(r\\_N(\\lambda)\\right)^{2q}&=&\\sum\\limits\\_{p=1}^{2q}\\sum\\limits\\_{t\\_1+\\ldots+t\\_p=2q}\\sum\\limits\\_{j\\_1\\cdots j\\_p}\\sum\\limits\\_{v\\_1\\cdots v\\_p}\\left(\\xi\\_{j\\_1}\\xi\\_{j\\_1+v\\_1}\\right)^{t\\_1}\\cdots\\left(\\xi\\_{j\\_p}\\xi\\_{j\\_p+v\\_p}\\right)^{t\\_p}\\\\\n& &w^{t\\_1}(v\\_1m^{-1})\\cdots w^{t\\_p}(v\\_pm^{-1})\\cos^{t\\_1}(v\\_1\\lambda)\\cdots\\cos^{t\\_p}(v\\_p\\lambda),\\end{aligned}$$ \nwhere {*j*1, …, *j**p*} ⊂ {*N* − *m* + 2, …, *N*}. If *t**k* ≥ 2 for any *k*, then expectation of (*ξ**j*1*ξ**j*1 + *v*1)*t*1⋯(*ξ**j**p**ξ**j**p* + *v**p*)*t**p* is not zero. If one of *t**k* is 1, for example, *t**k*0 = 1, since *j**k* ≤ *N* < *j**k* + *v**k* for any *k*, it is impossible to find *ξ**j**k*0 in {*ξ**j*1 + *v*1, …, *ξ**j**p* + *v**p*}, so the expectation would be zero. To sum up, for all non-zero terms in *E*(*r**N*(*λ*))2*q*, *t**k* should be greater than 2. Since for any *s* ≤ 2*q*, $\\sum\\limits\\_{k=1}^s t\\_k=2q$, *w**j*(*u*)cos*j*(*u*), *E**ξ*1*t*1⋯*ξ**s**t**s* can be bounded by a real number, denoted by *B*1, so we only need to count the number of non-zero terms. When *t**k* = 2 for any *k*, then *p* = *q*, and the number of non-zero terms are no more than (*C*2*q*2*C*2*q* − 22⋯*C*22*C**m**q*)2 = *O*(*m*2*q*). When at least one of *t**k* > 2, say *t**k*0 = 3, the number of non-zero terms is no more than (*C*2*q*2*C*2*q* − 22⋯*C*43*C**m**q* − 1)2 = *O*(*m*2*q* − 2). So\n*E*(*r**N*(*λ*))2*q* ≤ *B*1*m*2*q*.\nFor\n$E(r\\_N)^{2q}=E(\\sum\\limits\\_{t=1}^N(\\xi\\_t^2-\\sigma^2))^{2q}=\\sum\\limits\\_{p=1}^{2q}\\sum\\limits\\_{t\\_1+\\ldots+t\\_p=2q}\\sum\\limits\\_{j\\_1\\cdots j\\_p}(\\xi\\_{j\\_1}^2-\\sigma^2)^{t\\_1}\\ldots(\\xi\\_{j\\_p}^2-\\sigma^2)^{t\\_p}\\leq B\\_1 m^{2q}$,\nwe can see that *t**k* ≥ 2 for any *k* in all non-zero terms, so following the discussions above, we have\n$E(\\sum\\limits\\_{t=1}^N(\\xi\\_t^2-\\sigma^2))^{2q}\\leq B\\_2 N^q$.\nBy far, we have proven that (*r**N*(*λ*))2*q* and *r**N*2*q* are *O**p*(*N**q**m*2*q*).\n \n$$\\begin{aligned}\n\\left(Z\\_N(\\lambda)\\right)^{2q}&=&\\left(\\sum\\limits\\_{t=1}^NZ\\_{N,t}(\\lambda)\\right)^{2q}\\\\\n&=&\\left(2\\sum\\limits\\_{t=1}^N\\sum\\limits\\_{v=1}^{m-1}\\xi\\_t\\xi\\_{t+v}w(vm^{-1})\\cos(v\\lambda)\\right)^{2q}\\\\\n&=&\\sum\\limits\\_{p=1}^{2q}\\sum\\limits\\_{t\\_1+\\ldots+t\\_p=2q}\\sum\\limits\\_{j\\_1\\cdots j\\_p}Z\\_{N,j\\_1}^{t\\_1}(\\lambda)\\cdots Z\\_{N,j\\_p}^{t\\_p}(\\lambda).\\end{aligned}$$ \nFirst, when *t**k* ≥ 2 for any *k*, *p* ≤ *q*, the number of terms after expanding *Z**N*, *j*1*t*1 is (*m* − 1)*t*1. So for fixed *t*1, …, *t**p*, *Z**N*, *j*1*t*1⋯*Z**N*, *j**p**t**p* contains no more than (*m* − 1)*t*1 + ⋯ + *t**p* = (*m* − 1)2*q* terms. What is more, for any fixed *p*, the number for all possible *j*1, …, *j**p*, of which the power satisfy *t*1 + ⋯ + *t**p* = 2*q*, is no more than *p*! *C**N**p*. So there are no more than *N**q**m*2*q* terms, which means that $E\\sum\\limits\\_{p=1}^{q}\\sum\\limits\\_{t\\_1+\\ldots+t\\_p=2q}\\sum\\limits\\_{j\\_1\\cdots j\\_p}Z\\_{N,j\\_1}^{t\\_1}(\\lambda)\\cdots Z\\_{N,j\\_p}^{t\\_p}(\\lambda)\\leq B\\_1 N^qm^{2q}$.\nWhen *p* ≤ *q* and some of *t**k* are equal to 1, we assume that *t**k**s*1 = ⋯ = *t**k**s**c* = 1. For *t**k**s*1, *j**k**s*1 should satisfy *j**k**s*1 − *j**k**s*1 − 1 < *m*, that is, *ξ**j**k**s*1 ∈ {*ξ**j**k**s*1 − 1 + 1, …, *ξ**j**k**s*1 − 1 + *m* − 1}. If not, since *ξ**j**k**s*1 is independent of (*Z**N*, *j*1)*t*1, …, (*Z**N*, *j**k**s*1 − 1)*t**j**k**s*1 − 1, (*ξ**j**k**s*1 + 1 + ⋯ + *ξ**j**k**s*1 + *m* − 1), \n(*Z**N*, *j**k**s*1 + 1)*t**j**k**s*1 + 1, …, (*Z**N*, *j**p*)*t**p*, we have\n*E*(*Z**N*, *j*1)*t*1⋯(*Z**N*, *j**p*)*t**p* = 0.\nSo the number of all non-zero terms is no more than *C**N**p* − *c**m**c*. For (*Z**N*, *j**k**s*1 − 1)*t**j**k**s*1 − 1, there are (*m* − 2)*t**j**k**s*1 − 1 terms which contain *ξ**j**k**s*1. So there are (*m* − 1)*t**j**k**s*1 − 1 − (*m* − 2)*t**j**k**s*1 − 1 = *O*((*m* − 2)*t**j**k**s*1 − 1) terms which do not contain *ξ**j**k**s*1. Then the number of non-zero terms is O(*N**p* − *c**m**c**m**t*1 + ⋯ + *t**p* − *c*) ≤ *O*(*N**q**m*2*q*). For *ξ**j**k**s*1, if *Z**N*, *j**k*1*t**k*1, …, *Z**N*, *j**k**d**t**k**d*, contain *ξ**j**k**s*1, then it is easy to see that there are totally no more than *m**t**k*1 + ⋯ + *t**k**d* − 1 terms which do not contain *ξ**j**k**s*1. However, now for all *j**k*1, …, *j**k**d*, ∣*j**k**v* − *j**k**s*1∣ ≤ *m* − 1. So the number of all possible non-zero terms is *O*(*N**p* − *c* − *d* + 1*m**c* + *d**m**t*1 + ⋯ + *t**p* − *c*) ≤ *O*(*N**q**m*2*q*). Hence *E*(*Z**N*, *j*1)*t*1⋯(*Z**N*, *j**p*)*t**p* ≤ *B*1*N**q**m*2*q*.\nWhen *p* > *q*, there are at least *p* − *q* of *t**j* = 1, so following the discussions above, *E*(*Z**N*, *j*1)*t*1⋯(*Z**N*, *j**p*)*t**p* ≤ *B*1*N**q**m*2*q*. Therefore summarizing discussions above, $E(Z\\_N(\\lambda))^{2q}=O\\_p(\\frac{m^{2q}}{N^q})$.\nPart 2, we prove that $\\displaystyle \\left|\\frac{f(\\lambda)-Ef\\_N(\\lambda)}{f(\\lambda)}\\right|^{2q}=O(\\frac{m^{2q}}{N^q})$.\n \n$$\\begin{aligned}\n\\left|\\frac{f(\\lambda)-Ef\\_N(\\lambda)}{f(\\lambda)}\\right|^{2q}&\\leq& B\\left|f(\\lambda)-m\\int W(m(\\lambda-u))f(u)du\\right|^{2q}\\\\\n&&+B\\left|m\\int W(m(\\lambda-u))(f(u)-EI\\_N(u))du\\right|^{2q}\\end{aligned}$$ \nAnd\n \n$$\\begin{aligned}\n\\left|f(\\lambda)-m\\int W(m(\\lambda-u))f(u)du\\right|^{2q}&=&\\left|\\int (f(\\lambda)-f(\\lambda-um^{-1}))W(u)du\\right|^{2q}\\\\\n&\\leq&\\left(\\int\\left|f(\\lambda)-f(\\lambda-um^{-1})\\right|W(u)du\\right)^{2q}\\\\\n&\\leq& B\\_f m^{-2q}\\left|\\int |u|W(u)du\\right|^{2q}\\end{aligned}$$ \nSince $\\alpha\\geq \\frac{1}{4}$, we have $\\displaystyle \\lim\\limits\\_{N\\rightarrow\\infty}\\frac{1}{m^{2q}}/\\frac{m^{2q}}{N^{q}}<+\\infty$.\nAnd\n \n$$\\begin{aligned}\n\\left|m\\int W(m(\\lambda-u))(f(u)-E I\\_N(u))du\\right|^{2q}&=&\\left|\\int W(u)(f(\\lambda-um^{-1})-E I\\_N(\\lambda-um^{-1}))du\\right|^{2q}\\\\\n&\\leq& \\left(\\int W(u)\\left|f(\\lambda-um^{-1})-E I\\_N(\\lambda-um^{-1})\\right|du\\right)^{2q}\\end{aligned}$$ \nSince by Woodroofe and Van Ness (1967),\nmax∣*λ*∣ ≤ *π*∣*f*(*λ*) − *E**I**N*(*λ*)∣ ≤ *B**p**e*log*N*/*N*,\nwhere *B**p**e* is a constant. So\n \n$$\\begin{aligned}\n\\left|m\\int W(m(\\lambda-u))(f(u)-E I\\_N(u))du\\right|^{2q}&\\leq& \\left|\\int W(u)\\frac{B\\_{pe}\\log N}{N}du\\right|^{2q}\\\\\n&\\leq& B\\_{pe}^{2q}\\frac{(\\log N)^{2q}}{N^{2q}}.\\end{aligned}$$ \nAnd $\\displaystyle \\lim\\limits\\_{N\\rightarrow\\infty}\\frac{(\\log N)^{2q}}{N^{2q}}/\\frac{m^{2q}}{N^{q}}\\rightarrow0$. So we complete Part 2.\nPart 3: we show that $E\\left|\\frac{f\\_N(\\lambda)-Ef\\_N(\\lambda)}{f(\\lambda)}-( g\\_N(\\lambda)-\\sigma^2)\\right|^{2q}=O\\_p(\\frac{m^{2q}}{N^q})$.\n \n$$\\begin{aligned}\n&&\\left|\\frac{f\\_N(\\lambda)-Ef\\_N(\\lambda)}{f(\\lambda)}-(g\\_N(\\lambda)-\\sigma^2)\\right|^{2q}\\\\\n&=&\\left|\\frac{f\\_N(\\lambda)-E f\\_N(\\lambda)-f(\\lambda)(g\\_N(\\lambda)-\\sigma^2)}{f(\\lambda)}\\right|^{2q}\\\\\n&\\leq& B\\_f\\left|f\\_N(\\lambda)-E f\\_N(\\lambda)-f(\\lambda)(g\\_N(\\lambda)-\\sigma^2)\\right|^{2q}\\\\\n&=&B\\_f\\left|m\\int W(m(\\lambda-u))\\left(I\\_N(u)-E I\\_N(u)-f(u)(J\\_N(u)-E J\\_N(u))\\right)du\\right.\\\\\n&&+\\left.m\\int W(m(\\lambda-u))\\left(f(\\lambda)-f(u)\\right)(J\\_N(u)-E J\\_N(u))du\\right|^{2q}\\\\\n&\\leq&B\\_3\\left|m\\int W(m(\\lambda-u))\\left(I\\_N(u)-E I\\_N(u)-f(u)(J\\_N(u)-E J\\_N(u))\\right)du\\right|^{2q}\\\\\n&&+B\\_3\\left|m\\int W(m(\\lambda-u))\\left(f(\\lambda)-f(u)\\right)(J\\_N(u)-E J\\_N(u))du\\right|^{2q}\\\\\n&=& R\\_1(\\lambda)+R\\_2(\\lambda),\\end{aligned}$$ \nwhere *J**N*(*u*) is the periodogram for *ξ*1, …, *ξ**N*, *B*3 is a constant. After some manipulations (see Woodroofe and Van Ness 1967, Grenander and Rosenblatt 1957).\n \n$$\\begin{aligned}\nE R\\_1(\\lambda)&=&E\\frac{1}{N^{2q}}\\left|\\sum\\limits\\_{r,s=-\\infty}^{+\\infty} a\\_ra\\_s d\\_{rs}(\\lambda)\\right|^{2q}\\\\\n&\\leq&\\frac{1}{N^{2q}} \\sum\\limits\\_{p=1}^{2q}\\sum\\limits\\_{t\\_1+\\cdots+t\\_p}\\sum\\limits\\_{\\substack{r\\_1,s\\_1\\\\ \\cdots \\\\ r\\_p,s\\_p}}E\\left(\\left|a\\_{r\\_1}a\\_{s\\_1}d\\_{r\\_1s\\_1}\\right|^{t\\_1}\\cdots\\left|a\\_{r\\_p}a\\_{s\\_p}d\\_{r\\_ps\\_p}\\right|^{t\\_p}\\right)\\\\\n&=&\\frac{1}{N^{2q}}\\sum\\limits\\_{p=1}^{2q}\\sum\\limits\\_{t\\_1+\\cdots+t\\_p}\\sum\\limits\\_{\\substack{r\\_1,s\\_1\\\\ \\cdots \\\\ r\\_p,s\\_p}}\\left|a\\_{r\\_1}a\\_{s\\_1}\\right|^{t\\_1}\\cdots\\left|a\\_{r\\_p}a\\_{s\\_p}\\right|^{t\\_p}E\\left(\\left|d\\_{r\\_1s\\_1}\\right|^{t\\_1}\\cdots\\left|d\\_{r\\_ps\\_p}\\right|^{t\\_p}\\right)\\\\\n&\\leq&\\frac{1}{N^{2q}}\\sum\\limits\\_{p=1}^{2q}\\sum\\limits\\_{t\\_1+\\cdots+t\\_p}\\sum\\limits\\_{\\substack{r\\_1,s\\_1\\\\ \\cdots \\\\ r\\_p,s\\_p}}\\left|a\\_{r\\_1}a\\_{s\\_1}\\right|^{t\\_1}\\cdots\\left|a\\_{r\\_p}a\\_{s\\_p}\\right|^{t\\_p}\\left(E\\left|d\\_{r\\_1s\\_1}\\right|^{2t\\_1}\\cdots\\left|d\\_{r\\_ps\\_p}\\right|^{2t\\_p}\\right)^{\\frac{1}{2}}\\end{aligned}$$ \nNow let us focus on ∣*d**r**s*∣4*q*, that is *r*1 = ⋯ = *r**p*, *s*1 = ⋯ = *s**p*, and denote *R**ξ*(*v*) as the autocovariance function of *ξ**j*.\n \n$$\\begin{aligned}\n\\left|d\\_{rs}\\right|^{4q}&=&\\left|\\sum\\limits\\_{v\\_1=1-r}^{N-r}\\sum\\limits\\_{v\\_2=1-s}^{N-s} w((v\\_1-v\\_2+r-s)m^{-1})e^{-i(v\\_1-v\\_2+r-s)\\lambda}(\\xi\\_{v\\_1}\\xi\\_{v\\_2}-R\\_\\xi(v\\_1-v\\_2))\\right.\\\\\n& &\\left.-\\sum\\limits\\_{v\\_1=1}^N\\sum\\limits\\_{v\\_2=1}^Nw((v\\_1-v\\_2+r-s)m^{-1})e^{-i(v\\_1-v\\_2+r-s)\\lambda}(\\xi\\_{v\\_1}\\xi\\_{v\\_2}-R\\_\\xi(v\\_1-v\\_2))\\right|^{4q}\\\\\n&=&\\left|\\sum\\limits\\_{v\\_1=1-r}^{N-r}\\sum\\limits\\_{v\\_2=1-s}^{N-s}w((v\\_1-v\\_2+r-s)m^{-1})\\cos((v\\_1-v\\_2+r-s)\\lambda)(\\xi\\_{v\\_1}\\xi\\_{v\\_2}-R\\_\\xi(v\\_1-v\\_2))\\right.\\\\\n&&+i\\sum\\limits\\_{v\\_1=1-r}^{N-r}\\sum\\limits\\_{v\\_2=1-s}^{N-s}w((v\\_1-v\\_2+r-s)m^{-1})\\sin((v\\_1-v\\_2+r-s)\\lambda)(\\xi\\_{v\\_1}\\xi\\_{v\\_2}-R\\_\\xi(v\\_1-v\\_2))\\\\\n& &-\\sum\\limits\\_{v\\_1=1}^N\\sum\\limits\\_{v\\_2=1}^Nw((v\\_1-v\\_2+r-s)m^{-1})\\cos((v\\_1-v\\_2+r-s)\\lambda)(\\xi\\_{v\\_1}\\xi\\_{v\\_2}-R\\_\\xi(v\\_1-v\\_2))\\\\\n& &\\left.+i\\sum\\limits\\_{v\\_1=1}^N\\sum\\limits\\_{v\\_2=1}^Nw((v\\_1-v\\_2+r-s)m^{-1})\\sin((v\\_1-v\\_2+r-s)\\lambda)(\\xi\\_{v\\_1}\\xi\\_{v\\_2}-R\\_\\xi(v\\_1-v\\_2))\\right|^{4q}\\\\\n&=&\\left|Q\\_{rs}+iT\\_{rs}\\right|^{4q}\\leq 2^{2q}\\left(Q\\_{rs}^{4q}+T\\_{rs}^{4q}\\right)\\end{aligned}$$ \nAgain, we only need to prove that the expectation of *Q**r**s*4*q* and *T**r**s*4*q* are *O*(*N*2*q**m*4*q*). And\n \n$$\\begin{aligned}\nE Q\\_{rs}^{4q}&=& E\\left(\\sum\\limits\\_{v\\_1=1-r}^{N-r}\\sum\\limits\\_{v\\_2=1-s,v\\_2\\neq v\\_1}^{N-s}w((v\\_1-v\\_2+r-s)m^{-1})\\cos((v\\_1-v\\_2+r-s)\\lambda)\\xi\\_{v\\_1}\\xi\\_{v\\_2}\\right.\\\\\n& &-\\sum\\limits\\_{v\\_1=1}^N\\sum\\limits\\_{v\\_2=1,v\\_2\\neq v\\_1}^Nw((v\\_1-v\\_2+r-s)m^{-1})\\cos((v\\_1-v\\_2+r-s)\\lambda)\\xi\\_{v\\_1}\\xi\\_{v\\_2}\\\\\n& &+\\sum\\limits\\_{v\\_1=\\max(1-r,1-s)}^{\\min(N-r,N-s)}\\left(\\xi\\_{v\\_1}^2-1\\right)\\left.-\\sum\\limits\\_{v\\_1=1}^{N}(\\xi\\_{v\\_1}^2-1)\\right)^{4q}\\\\\n&\\leq&2^{16q}\\left(E\\left(\\sum\\limits\\_{v\\_1=1-r}^{N-r}\\sum\\limits\\_{v\\_2=1-s,v\\_2\\neq v\\_1}^{N-s}w((v\\_1-v\\_2+r-s)m^{-1})\\cos((v\\_1-v\\_2+r-s)\\lambda)\\xi\\_{v\\_1}\\xi\\_{v\\_2}\\right)^{4q}\\right.\\\\\n&+&E\\left(\\sum\\limits\\_{v\\_1=\\max(1-r,1-s)}^{N-r,N-s}\\left(\\xi\\_{v\\_1}^2-\\sigma^2\\right)\\right)^{4q}\\\\\n&+&E\\left(\\sum\\limits\\_{v\\_1=1}^N\\sum\\limits\\_{v\\_2=1,v\\_2\\neq v\\_1}^Nw((v\\_1-v\\_2+r-s)m^{-1})\\cos((v\\_1-v\\_2+r-s)\\lambda)\\xi\\_{v\\_1}\\xi\\_{v\\_2}\\right)^{4q}\\\\\n&+&\\left.E\\left(\\sum\\limits\\_{v\\_1=1}^{N}(\\xi\\_{v\\_1}^2-\\sigma^2)\\right)^{4q}\\right)\\end{aligned}$$ \nFor those four terms in the last inequality above, following the proofs in Part 1, we can show that each of them is *O*(*N*2*q**m*4*q*). Similarly to the discussions above, *E*(*T**r**s*(*λ*))4*q* = *O*(*N*2*q**m*4*q*). So we have *E*∣*d**r**s*(*λ*)∣4*q* ≤ *O*(*N*2*q**m*4*q*). When some of (*r**k*, *s**k*) are different from each other, for example, (*r*1, *s*1) ≠ (*r**k*, *s**k*) for *k* ≥ 2, then in *E*∣*d**r*1*s*1(*λ*)∣2*t*1⋯∣*d**r**p**s**p*(*λ*)∣2*t**p*, the number of non-zero terms should be less than *E*∣*d**r**s*(*λ*)∣4*q* because some *ξ**j* in *d**r*1*s*1(*λ*) are not in *d**r**k**s**k*(*λ*). And now, when the power of *ξ**j* is 1 in the expansion of (*d**r*1*s*1(*λ*))2*t*1, we can not find any *ξ**j* in (*d**r**k**s**k*(*λ*))2*t**k*, so the expectations of these terms are 0. So,\n \n$$\\begin{aligned}\nE R\\_1(\\lambda)\\leq \\frac{B N^{q}m^{2q}}{N^{2q}}\\sum\\limits\\_{p=1}^{2q}\\sum\\limits\\_{t\\_1+\\cdots+t\\_p}\\sum\\limits\\_{\\substack{r\\_1,s\\_1\\\\ \\cdots \\\\ r\\_p,s\\_p}} \\left|a\\_{r\\_1}a\\_{s\\_1}\\right|^{t\\_1}\\cdots\\left|a\\_{r\\_p}a\\_{s\\_p}\\right|^{t\\_p}\\end{aligned}$$ \nSince ∑*j*∣*a**j*∣ < + ∞,\n$\\sum\\limits\\_{\\substack{r\\_1,s\\_1\\\\ \\cdots \\\\ r\\_p,s\\_p}}\\left|a\\_{r\\_1}a\\_{s\\_1}\\right|^{t\\_1}\\cdots\\left|a\\_{r\\_p}a\\_{s\\_p}\\right|^{t\\_p}<+\\infty$\nfor any given *t*1, …, *t**p*. Since *t*1 + ⋯ + *t**p* = 2*q*, $\\displaystyle E R\\_1(\\lambda)\\leq B\\_1\\frac{N^{q}m^{2q}}{N^{2q}}=B\\_1\\frac{m^{2q}}{N^{q}}$.\n \n$$\\begin{aligned}\nR\\_2(\\lambda)&=&B\\_2\\left(\\int W(u)\\left(f(\\lambda)-f(\\lambda-um^{-1})\\right)\\left(J\\_N(\\lambda-um^{-1})-E\\left(J\\_N(\\lambda-um^{-1})\\right)\\right)du\\right)^{2q}\\\\\n&=&B\\_3\\left(\\int W(u)^{1-\\frac{1}{2q}}W(u)^{\\frac{1}{2q}}\\left(f(\\lambda)-f(\\lambda-um^{-1})\\right)\\times\\right.\\\\\n& &\\left.\\left(J\\_N(\\lambda-um^{-1})-E\\left(J\\_N(\\lambda-um^{-1})\\right)\\right)du\\right)^{2q}\\\\\n&\\leq&B\\_4\\left(\\int W(u)^{1-\\frac{1}{2q}}W(u)^{\\frac{1}{2q}}um^{-1}\\left(J\\_N(\\lambda-um^{-1})-E\\left(J\\_N(\\lambda-um^{-1})\\right)\\right)du\\right)^{2q}\\\\\n&\\leq& B\\_4\\left(\\int \\left(\\left(W(u)\\right)^{1-\\frac{1}{2q}}\\right)^{\\frac{2q}{2q-1}}du\\right)^{2q \\frac{2q-1}{2q}}\\times\\\\\n& &\\left(\\int u^{2q}m^{-2q}W(u)\\left(J\\_N(\\lambda-um^{-1})-E J\\_N(\\lambda-um^{-1})\\right)^{2q}du\\right)^{2q\\frac{1}{2q}}\\\\\n&=&B\\_4m^{-2q} \\left|\\int u^{2q}W(u)\\left(J\\_N(\\lambda-um^{-1})-E J\\_N(\\lambda-um^{-1})\\right)^{2q}du\\right|\\\\\n&=&B\\_4m^{-2q}\\left|i^{-2q}\\int (iu)^{2q} W(u)\\left(J\\_N(\\lambda-um^{-1})-E J\\_N(\\lambda-um^{-1})\\right)^{2q}du\\right|\\\\\n&=&B\\_4m^{-2q}\\left|\\int \\mathcal{F}(w^{(2q)})(u)\\left(J\\_N(\\lambda-um^{-1})-E J\\_N(\\lambda-um^{-1})\\right)^{2q}du\\right|\\\\\n&=&B\\_4m^{-2q}\\int \\mathcal{F}(w^{(2q)})(u)\\left(\\sum\\limits\\_{t=1}^N\\sum\\limits\\_{v=1}^{N-t}\\xi\\_t\\xi\\_{t+v}e^{-i(\\lambda-um^{-1})v}+\\sum\\limits\\_{t=1}^N(\\xi\\_{t}^2-1)\\right)^{2q}du\\end{aligned}$$ \nwhere the second inequality holds because of Hölder inequality, and F(*w*(2*q*))(*u*) is the Fourier transformation of the 2*q*th derivative of *w*(*v*). Since *w*(*v*) = 0 for *v* ≥ 1, we have *w*(2*q*) = 0 for *v* ≥ 1, and *w*(2*q*)(*v*) is bounded. So similar to Part 1, we have $\\displaystyle R\\_2(\\lambda)\\leq B\\_5m^{-2q}\\frac{m^{2q}}{N^q}=\\frac{B\\_5}{N^q}$. Here we complete Part 3.\nSince\n \n$$\\begin{aligned}\n\\left|\\frac{f\\_N(v)-f(v)}{f(v)}\\right|^{2q}&\\leq&\\left|\\frac{f\\_N(v)-E f\\_N(\\lambda)+E f\\_N(\\lambda)-f(\\lambda)}{f(\\lambda)}-(g\\_N(\\lambda)-1)+(g\\_N(\\lambda)-1)\\right|^{2q}\\\\\n&\\leq&2^{6q}\\left(\\left|\\frac{f\\_N(\\lambda)-Ef\\_N(\\lambda)}{f(\\lambda)}-(g \\_N(\\lambda)-\\sigma^2)\\right|^{2q}+\\left|\\frac{f(\\lambda)-Ef\\_N(\\lambda)}{f(\\lambda)}\\right|^{2q}\\right.\\\\\n& &\\left.+\\left|g\\_N(\\lambda)-\\sigma^2\\right|^{2q}\\vphantom{\\left|\\frac{f\\_N(\\lambda)-Ef\\_N(\\lambda)}{f(\\lambda)}\\right|}\\right),\\end{aligned}$$ \nwe have\n$\\displaystyle E\\left|\\frac{f\\_N(v)-f(v)}{f(v)}\\right|^{2q}\\leq B\\frac{m^{2q}}{N^q}$.\nThen, by Markov inequality,\n \n$$\\begin{aligned}\nP\\left(\\left|\\frac{\\hat{f}(\\lambda)-f(\\lambda)}{f(\\lambda)}\\right|\\geq \\epsilon \\frac{m}{N^{1/2}}\\right)&\\leq&\\frac{N^q}{m^{2q}\\epsilon^{2q}}E\\left|\\frac{f\\_N(v)-f(v)}{f(v)}\\right|^{2q}\\\\\n&\\leq&\\frac{N^q}{m^{2q}\\epsilon^{2q}}B\\frac{m^{2q}}{N^q}=\\frac{B}{\\epsilon^{2q}}\\end{aligned}$$ \n \n$$\\begin{aligned}\nP\\left(\\max\\_{\\lambda\\_j}\\left|\\frac{f\\_N(\\lambda\\_j)-f(\\lambda\\_j)}{f(\\lambda\\_j)}\\right|\\geq\\epsilon\\frac{m}{n^{(q-1)/(2q)}}\\right)&\\leq&\\sum\\limits\\_{j=1}^NP\\left(\\left|\\frac{f\\_N(\\lambda\\_j)-f(\\lambda\\_j)}{f(\\lambda\\_j)}\\right|\\geq\\epsilon\\frac{m}{N^{(q-1)/(2q)}}\\right)\\\\\n&\\leq&\\sum\\limits\\_{j=1}^N \\frac{N^{q-1}}{\\epsilon^{2q}m^{2q}}B\\_5\\frac{m^{2q}}{N^q}=\\frac{B\\_5}{\\epsilon^{2q}}.\\end{aligned}$$ \nSo\n \n$$\\begin{aligned}\nP(\\max\\limits\\_{1\\leq k< l\\leq N}\\max\\_{\\lambda\\_i}\\left|\\frac{f\\_N(\\lambda\\_i)-f(\\lambda\\_i)}{f(\\lambda\\_i)}\\right|\\geq\\epsilon\\frac{m}{N^{\\frac{q-3}{2q}}})\\leq\\sum\\limits\\_{k=1}^N\\sum\\limits\\_{l=1}^NB\\_5\\frac{m^{2q}N^{q-3}}{N^{q-1}\\epsilon^{2q}}=\\frac{B\\_5m^{2q}}{\\epsilon^{2q}}.\\end{aligned}$$ \nHere we complete the proof of Lemma [ch3l1].\n[ch3l2] Suppose ∀*k*, $X\\_t=\\sum\\limits\\_{j=-\\infty}^{+\\infty} a\\_j(k)\\epsilon\\_{t-j}$ satisfy Assumptions 1-8, then\n \n$$f\\_0(\\lambda)=\\sum\\limits\\_{k=1}^{K+1}\\frac{N\\_k}{N} f\\_k(\\lambda)\\\\$$ \n \n$$\\max\\limits\\_{\\lambda\\_i\\in\\Lambda}\\left|\\frac{\\hat{f}\\_0(\\lambda\\_i)-f\\_0(\\lambda\\_i)}{f\\_0(\\lambda\\_i)}\\right|=O\\_p\\left(\\frac{m}{N^{(q-1)/2q}}\\right)$$ \n**Proof**:\n \n$$\\begin{aligned}\n\\hat{f}\\_0(\\lambda)&=&m\\int\\_{-\\infty}^{+\\infty} W(m(u-\\lambda))\\hat{I}\\_0(u)du\\\\\n&=&m\\int\\_{-\\infty}^{+\\infty} W(m(u-\\lambda))\\frac{1}{N}\\left|\\sum\\limits\\_{j=1}^N e^{-iju}X\\_j\\right|^2du\\\\\n&=&m\\int\\_{-\\infty}^{+\\infty} W(m(u-\\lambda))\\frac{1}{N}\\left|\\sum\\limits\\_{k=1}^{K+1}\\sum\\limits\\_{j=\\tau\\_{k-1}^0+1}^{\\tau\\_k^0}e^{-iju}X\\_j\\right|^2du\\\\\n&=&m\\int\\_{-\\infty}^{+\\infty} W(m(u-\\lambda))\\frac{1}{N}\\left(\\sum\\limits\\_{k=1}^{K+1}\\left|\\sum\\limits\\_{j=\\tau\\_{k-1}^0+1}^{\\tau\\_k^0}e^{-iju}X\\_j\\right|^2\\right.\\\\\n& &\\left.+\\sum\\limits\\_{k\\_1\\neq k\\_2}\\sum\\limits\\_{j=\\tau\\_{k\\_1-1}^0+1}^{\\tau\\_{k\\_1}^0}e^{-iju}X\\_j\\sum\\limits\\_{j=\\tau\\_{k\\_2-1}^0+1}^{\\tau\\_{k\\_2}^0}e^{iju}X\\_j\\right)du\\\\\n&=&m\\int\\_{-\\infty}^{+\\infty} W(m(u-\\lambda))\\frac{1}{N}\\left(\\sum\\limits\\_{k=1}^{K+1}\\left|\\sum\\limits\\_{j=1}^{N\\_k}e^{-iju}X\\_j\\right|^2\\left|e^{i\\tau\\_{k-1}^0u}\\right|^2+\\sum\\limits\\_{k\\_1\\neq k\\_2}\\zeta\\_{k\\_1}(u)\\bar{\\zeta}\\_{k\\_2}(u)\\right)du\\\\\n&=&\\sum\\limits\\_{k=1}^{K+1}\\hat{f}\\_{k}(\\lambda)+\\sum\\limits\\_{k\\_1\\neq k\\_2}\\frac{m}{N}\\int\\_{-\\infty}^\\infty W(m(u-\\lambda))\\zeta\\_{k\\_1}(u)\\bar{\\zeta}\\_{k\\_2}(u)du,\\end{aligned}$$ \nwhere $\\zeta\\_{k\\_1}(u)=\\sum\\limits\\_{j=\\tau\\_{k\\_1-1}^0+1}^{\\tau\\_{k\\_1}^0}e^{-iju}X\\_j$. So\n \n$$\\begin{aligned}\n& &P\\left(\\left|\\frac{\\hat{f}\\_0(\\lambda)-f\\_0(\\lambda)}{f\\_0(\\lambda)}\\right|\\geq\\epsilon\\frac{m}{N^{1/2}}\\right)\\\\\n&\\leq& P\\left(B\\_f\\left|\\frac{\\hat{f}\\_0(\\lambda)-f\\_0(\\lambda)}{f\\_0(\\lambda)}\\right|\\geq\\epsilon\\frac{m}{N^{1/2}}\\right)\\\\\n&\\leq& \\sum\\limits\\_{k=1}^{K+1}P\\left(\\frac{N\\_k}{N}\\left|\\hat{f}\\_k(\\lambda)-f\\_k(\\lambda)\\right|\\geq\\epsilon\\frac{m}{N^{1/2}(K+1)^2}\\right)\\\\\n& &+\\sum\\limits\\_{k\\_1\\neq k\\_2}P\\left(\\left|\\frac{m}{N}\\int\\_{-\\infty}^{+\\infty} W(m(u-\\lambda))\\zeta\\_{k\\_1}(u)\\bar{\\zeta}\\_{k\\_2}(u)du\\right|\\geq \\epsilon\\frac{m}{N^{1/2}(K+1)^2}\\right).\\end{aligned}$$ \nBy Lemma [ch3l1], we only need to show that\n$\\displaystyle \\left|\\frac{m}{N}\\int\\_{-\\infty}^{+\\infty} W(m(u-\\lambda))\\zeta\\_{k\\_1}(u)\\bar{\\zeta}\\_{k\\_2}(u)du\\right|=O\\_p(\\frac{m}{N^{1/2}})$.\nWhat is more, assume *k*1 < *k*2 without loss of generality, then we have\n \n$$\\begin{aligned}\n& &\\left|\\frac{m}{N}\\int\\_{-\\infty}^{+\\infty} W(m(u-\\lambda))\\zeta\\_{k\\_1}(u)\\bar{\\zeta}\\_{k\\_2}(u)du\\right|\\\\\n&=&\\frac{1}{N}\\left|\\int W(u)\\left(\\zeta\\_{k\\_1}(um^{-1}+\\lambda)\\bar{\\zeta}\\_{k\\_2}(um^{-1}+\\lambda)\\right)du\\right|\\\\\n&=&\\frac{1}{N}\\left|\\int W(u)\\left(\\sum\\limits\\_{j=\\tau\\_{k\\_1-1}^0}^{\\tau\\_{k\\_1}^0}e^{-ij(um^{-1}+\\lambda)}X\\_j\\sum\\limits\\_{j=\\tau\\_{k\\_2-1}^0}^{\\tau\\_{k\\_2}^0}e^{ij(um^{-1}+\\lambda)}X\\_j\\right)du\\right|\\\\\n&=&\\frac{1}{N}\\left|\\sum\\limits\\_{l=\\tau\\_{k\\_2-1}^0+1-\\tau\\_{k\\_1}^0}^{\\tau\\_{k\\_2-1}+1-\\tau\\_{k\\_1}^0+N\\_{k\\_1}}w(um^{-1})e^{il\\lambda}\\sum\\limits\\_{j=\\tau\\_{k\\_1}^0-(l-\\tau\\_{k\\_2-1}^0+1-\\tau\\_{k\\_1}^0)}^{\\tau\\_{k\\_1}^0}X\\_jX\\_{j+l}\\right.\\\\\n& &+\\sum\\limits\\_{l=\\tau\\_{k\\_2-1}+1-\\tau\\_{k\\_1}+N\\_{k\\_1}+1}^{\\tau\\_{k\\_2}-\\tau\\_{k\\_1}}w(um^{-1})e^{il\\lambda}\\sum\\limits\\_{j=\\tau\\_{k\\_1-1}}^{\\tau\\_{k\\_1}}X\\_jX\\_{j+l}\\\\\n& &\\left.+\\sum\\limits\\_{l=\\tau\\_{k\\_2}-\\tau\\_{k\\_1}+1}^{\\tau\\_{k\\_2}-\\tau\\_{k\\_1}-1}w(um^{-1})e^{il\\lambda}\\sum\\limits\\_{j=\\tau\\_{k\\_1-1}+1}^{\\tau\\_{k\\_1}+1+N\\_{k\\_2}-N\\_{k\\_1}-(l-\\tau\\_{k\\_2}-\\tau\\_{k\\_1}+1)}X\\_jX\\_{j+l}\\right|.\\end{aligned}$$ \nSince *N**k* > *m*, ∀*k*, *w*(*u*) = 0 for ∣*u*∣ ≥ 1, so if *k*2 − *k*1 > 1, *ζ**k*1(*u*)*ζ̄**k*2(*u*) = 0. So without loss of generality, we set *k*1 = 1, *k*2 = 1. Next, we separate the proofs into 2 parts, as in Lemma [ch3l1].\nPart 1:\n \n$$\\begin{aligned}\n& &\\frac{1}{N}\\left|m\\int W(m(u-\\lambda))\\zeta\\_1(u)\\bar{\\zeta}\\_2(u)\\right|\\\\\n&=&\\frac{1}{N}\\left|\\int W(u)\\sum\\limits\\_{j=1}^{N\\_1}e^{-ij(um^{-1}+\\lambda)}\\xi\\_j\\sum\\limits\\_{j=N\\_1+1}^{N\\_2}e^{ij(um^{-1}+\\lambda)}\\xi\\_j\\right|\\\\\n&=&\\frac{1}{N}\\left|\\sum\\limits\\_{l=1}^{N\\_1}w(lm^{-1})e^{il\\lambda}\\sum\\limits\\_{j=N\\_1-l+1}^{N\\_1-m}\\xi\\_j\\xi\\_{j+l}\\right|.\\end{aligned}$$ \nSo\n \n$$\\begin{aligned}\n& &E\\frac{1}{N^{2q}}\\left|\\sum\\limits\\_{l=1}^{N\\_1}w(lm^{-1})e^{il\\lambda}\\sum\\limits\\_{j=N\\_1-l+1}^{N\\_1-m}\\xi\\_j\\xi\\_{j+l}\\right|^{2q}\\\\\n&\\leq&E\\left(\\frac{1}{N^{2q}}2^{2q}\\left(\\sum\\limits\\_{l=1}^mw(lm^{-1})\\cos(l\\lambda)\\sum\\limits\\_{j=N\\_1-l+1}^{N\\_1-m}\\xi\\_j\\xi\\_{j+l}\\right)^{2q}\\right.\\\\\n& &\\left.+\\frac{1}{N^{2q}}2^{2q}\\left(\\sum\\limits\\_{l=1}^mw(lm^{-1})\\sin(l\\lambda)\\sum\\limits\\_{j=N\\_1-l+1}^{N\\_1-m}\\xi\\_j\\xi\\_{j+l}\\right)^{2q}\\right).\\\\\\end{aligned}$$ \nSo for two terms in the inequality above, following the proofs in Part 1 of Lemma [ch3l1], we have\n$\\displaystyle E\\left(\\frac{1}{N^{2q}}2^{2q}\\left(\\sum\\limits\\_{l=1}^mw(lm^{-1})\\cos(l\\lambda)\\sum\\limits\\_{j=n\\_1-l+1}^{n\\_1-m}\\xi\\_j\\xi\\_{j+l}\\right)^{2q}\\right)=O\\left(\\frac{m^{3q}}{N^{2q}}\\right)$, \n$\\displaystyle E\\left(\\frac{1}{N^{2q}}2^{2q}\\left(\\sum\\limits\\_{l=1}^mw(lm^{-1})\\sin(l\\lambda)\\sum\\limits\\_{j=n\\_1-l+1}^{n\\_1-m}\\xi\\_j\\xi\\_{j+l}\\right)^{2q}\\right)=O\\left(\\frac{m^{3q}}{N^{2q}}\\right)$.\nPart 2: Set *f*11(*u*), *f*22(*u*) satisfying *f*1(*u*) = *f*11(*u*)*f̄*11(*u*), *f*2(*u*) = *f*22(*u*)*f̄*22(*u*), where *f̄*22 denotes the conjugate of *f*22. We show that $\\displaystyle \\left|\\frac{m\\int W(m(u-\\lambda))\\zeta\\_1(u)\\bar{\\zeta}\\_2(u)du}{f\\_{11}\\bar{f}\\_{22}}- g\\_n(\\lambda)\\right|^{2q}=O\\_p(\\frac{m^{2q}}{N^{q}})$.\n \n$$\\begin{aligned}\n& &\\left|\\frac{m\\int W(m(u-\\lambda))\\zeta\\_1(u)\\bar{\\zeta}\\_2(u)du}{f\\_{11}\\bar{f}\\_{22}}- g\\_n(\\lambda)\\right|^{2q}\\\\\n&\\leq&B\\_f\\left|m\\int W(m(u-\\lambda))\\zeta\\_1(u)\\bar{\\zeta}\\_2(u)du-f\\_{11}(\\lambda)\\bar{f}\\_{22}(\\lambda)g\\_n(\\lambda)\\right|^{2q}\\\\\n&\\leq&\\left|m\\int W(m(u-\\lambda))(\\zeta\\_1(u)\\bar{\\zeta}\\_2(u)-f\\_{11}(u)\\bar{f}\\_{22}(u))\\right|^{2q}\\\\\n& &+\\left|m\\int W(m(u-\\lambda))(f\\_{11}(\\lambda)\\bar{f}\\_{22}(\\lambda)-f\\_{11}(u)\\bar{f}\\_{22}(u))J\\_1(u)J\\_2(u)du\\right|^{2q}\\\\\n&=&R\\_1(\\lambda)+R\\_2(\\lambda).\\end{aligned}$$ \nSince ∀*k*, *f**k*(*λ*) all satisfy uniform Lipschitz condition, then it is easy to see that *f*11(*u*)*f̄*22(*u*) also satisfies uniform Lipschitz condition. Following the proofs in Part 3 of Lemma [ch3l1], we have\n$\\displaystyle R\\_2(\\lambda)=O\\_p(\\frac{1}{N^{q}})$.\nFollowing the proofs in Part 3 of Lemma [ch3l1] again, we have $\\displaystyle R\\_1(\\lambda)=|\\sum\\limits\\_{r=-\\infty,s=-\\infty}^{+\\infty} a\\_{r}(1)a\\_s(2)d\\_{rs}|^{2q}$, where\n \n$$\\begin{aligned}\nd\\_{rs}&=&\\sum\\limits\\_{v\\_1=1-r}^{\\tau\\_1-r}\\sum\\limits\\_{v\\_2=\\tau\\_1+1-s}^{\\tau\\_2-s}\\xi\\_{v\\_1}\\xi\\_{v\\_2}e^{i(v\\_1-v\\_2)(um^{-1}+\\lambda)}w((v\\_2-v\\_1)m^{-1})\\\\\n& &-\\sum\\limits\\_{v\\_1=1}^{\\tau\\_1}\\sum\\limits\\_{v\\_2=1}^{\\tau\\_2}\\xi\\_{v\\_1-r}\\xi\\_{v\\_2-s}e^{i(v\\_2-v\\_1)(um^{-1}+\\lambda)}w((v\\_2-v\\_1)m^{-1}).\\end{aligned}$$ \nSo following the proofs in Part 3 of Lemma [ch3l1] again. we have\n$\\displaystyle R\\_1(\\lambda)=O\\_p(\\frac{m^{2q}}{N^q})$.\nHere we complete the proof of Lemma [ch3l2].\n[ch3l3] Assume Assumption 1-8, ∀*s* = 1, ⋯, *K* + 1, $\\max\\limits\\_{\\tau\\_{s-1}^0\\leq k< l\\leq \\tau\\_{s}^0}\\vartheta\\_{kl}\\sim O\\_p(N^{-\\frac{q-3-2q\\alpha}{2q}})$, when *l* − *k* = *N**k**l* ≥ *m**l*. Here\n$\\displaystyle \\vartheta\\_{kl}=\\frac{N\\_{kl}}{N}\\int\\_{-\\pi}^\\pi \\hat{f}\\_k^l(v)\\log\\left(\\frac{st(\\hat{f}\\_k)}{st(f\\_s)}\\right)dv$\n**Proof**: Set *m**k**l* = *N**k**l**α*, and *B*1 to *B*6 are all constant. By Lemma [ch3l1], we have\n$\\displaystyle P\\left(\\max\\limits\\_{\\lambda\\_i}\\left|\\frac{\\hat{f}\\_{k}^l(\\lambda\\_i)-f(\\lambda\\_i)}{f(\\lambda\\_i)}\\right|^{2q}>\\epsilon\\frac{m^{2q}}{N^{(q-3)}}\\right)\\leq \\frac{B\\_5}{N^{2}\\epsilon^{2q}}$,\nSo denote $\\displaystyle A\\_{kl}=\\{\\max\\limits\\_{\\lambda\\_i}\\left|\\frac{\\hat{f}\\_{k}^l(\\lambda\\_i)-f(\\lambda\\_i)}{f(\\lambda\\_i)}\\right|\\leq\\epsilon\\frac{m}{N^{(q-3)/(2q)}}\\}$, then on set *A**k**l*,\n$\\displaystyle \\left|\\frac{\\hat{f}\\_{k}^l(\\lambda\\_i)-f(\\lambda\\_i)}{f(\\lambda\\_i)}\\right|\\leq\\epsilon\\frac{m}{N^{(q-3)/(2q)}}\n\\Rightarrow \\max(0,1-\\epsilon\\frac{B\\_5m}{ N^{(q-3)/(2q)}})\\leq \\frac{\\hat{f}\\_{kl}(\\lambda\\_i)}{f\\_s(\\lambda\\_i)}\\leq 1+\\epsilon\\frac{B\\_5m}{ N^{(q-3)/(2q)}}$, ∀*k*, *l*, *λ**i*.\nSince $\\frac{\\hat{f}\\_{kl}(\\lambda\\_i)}{f\\_s(\\lambda\\_i)}\\geq0$,\n$\\displaystyle \\max(0,1-\\epsilon\\frac{B\\_5m}{ N^{(q-3)/(2q)}})\\leq \\frac{\\hat{f}\\_{kl}(\\lambda\\_i)}{f\\_s(\\lambda\\_i)}\\leq 1+\\epsilon\\frac{B\\_5m}{ N^{(q-3)/(2q)}}$.\nSo on set *A**k**l*, we have\n \n$$\\begin{aligned}\n& &0\\leq \\frac{\\hat{f}\\_{kl}(\\lambda\\_i)}{f\\_s(\\lambda\\_i)}\\leq 1+\\epsilon\\frac{B\\_5m}{ N^{(q-3)/(2q)}}\\\\\n&\\Rightarrow& 0\\leq \\hat{f}\\_{kl}(\\lambda\\_i)\\leq f\\_s(\\lambda\\_i)\\left(1+\\epsilon\\frac{B\\_5m}{ N^{(q-3)/(2q)}}\\right)\\\\\n&\\Rightarrow& 0\\leq \\hat{F}\\_{kl}(\\pi)\\leq F\\_s(\\pi)\\left(1+\\epsilon\\frac{B\\_6m}{ N^{(q-3)/(2q)}}\\right).\\end{aligned}$$ \nThen on $\\bigcap\\limits\\_{kl}A\\_{kl}$,\n \n$$\\begin{aligned}\n\\max\\limits\\_{kl}\\vartheta\\_{kl}&=&\\max\\limits\\_{kl}\\frac{N\\_{kl}}{N}\\int\\_{-\\pi}^\\pi \\hat{f}\\_k^l(u)\\log\\left(\\frac{st(\\hat{f}\\_k)}{st(f\\_s)}\\right)du\\\\\n&=&\\max\\limits\\_{kl}\\frac{N\\_{kl}}{N}\\int\\_{-\\pi}^\\pi \\hat{f}\\_k^l(u)\\log\\left(\\frac{F\\_s(\\pi)}{\\hat{F}\\_k(\\pi)}\\times\\frac{\\hat{f}\\_k(u)}{f\\_s(u)}\\right)du\\\\\n&\\leq&\\max\\limits\\_{kl}\\frac{N\\_{kl}}{N}\\int\\_{-\\pi}^\\pi \\max\\limits\\_{u}\\hat{f}\\_k^l(u)\\log\\left(\\frac{F\\_s(\\pi)}{\\hat{F}\\_k(\\pi)}\\times\\max\\limits\\_{u}\\frac{\\hat{f}\\_k(u)}{f\\_s(u)}\\right)du\\\\\n&\\leq&\\max\\limits\\_{kl}\\frac{N\\_{kl}}{N}\\int\\_{-\\pi}^\\pi \\max\\limits\\_{u}f\\_s(u)\\left(1+\\frac{B\\_5m\\epsilon}{ N^{(q-3)/(2q)}}\\right)\\\\\n& &\\log\\left(\\left(1+\\frac{B\\_6m\\epsilon}{ N^{(q-3)/(2q)}}\\right)\\left(1+\\frac{B\\_5m\\epsilon}{ N^{(q-3)/(2q)}}\\right)\\right)du\\\\\n&\\overset{\\log(1+x)\\leq x}{\\leq}&\\max\\limits\\_{kl}\\frac{N\\_{kl}}{N}\\int\\_{-\\pi}^\\pi \\max\\limits\\_{u}f\\_s(u)\\left(1+\\frac{B\\_5m\\epsilon}{ N^{(q-3)/(2q)}}\\right)\\left(\\frac{B\\_6m\\epsilon}{ N^{(q-3)/(2q)}}+\\frac{B\\_5m\\epsilon}{ N^{(q-3)/(2q)}}\\right)du\\\\\n&\\leq& B\\_{7\\epsilon}\\frac{m}{N^{(q-3)/(2q)}}=B\\_{7\\epsilon}N^{-\\frac{q-3-2q\\alpha}{2q}}.\\end{aligned}$$ \nHere *B*7*ε* is some constant containing *ε*. So set *B*8*ε* > *B*7*ε*,\n \n$$\\begin{aligned}\nP\\left(\\max\\limits\\_{kl}\\vartheta\\_{kl}\\geq B\\_{8\\epsilon}N^{-\\frac{q-3-2q\\alpha}{2q}}\\right)&=&P\\left(\\max\\limits\\_{kl}\\vartheta\\_{kl}\\geq B\\_{8\\epsilon} N^{-\\frac{q-3-2q\\alpha}{2q}}|\\bigcap\\limits\\_{kl}A\\_{kl}\\right)P\\left(\\bigcap\\limits\\_{kl}A\\_{kl}\\right)\\\\\n& &+P\\left(\\max\\limits\\_{kl}\\vartheta\\_{kl}\\geq B\\_{8\\epsilon}N^{\\frac{q-3-2q\\alpha}{2q}}|\\bigcup\\limits\\_{kl}A^c\\_{kl}\\right)P\\left(\\bigcup\\limits\\_{kl}A\\_{kl}^c\\right)\\\\\n&\\leq&P\\left(\\max\\limits\\_{kl}\\vartheta\\_{kl}\\geq B\\_{8\\epsilon}N^{-\\frac{q-3-2q\\alpha}{2q}}|\\bigcap\\limits\\_{kl}A\\_{kl}\\right)+P\\left(\\bigcup\\limits\\_{kl}A\\_{kl}^c\\right)\\\\\n&\\leq&0+\\frac{B\\_5}{\\epsilon^{2q}},\\end{aligned}$$ \nand the last inequality can be arbitrarily small by choosing sufficiently large *ε*. Here we complete proofs of Lemma [ch3l3].\n[ch3l4] Assume Assumption 1-8, ∀*s* = 1, ⋯, *K* + 1, $\\displaystyle \\max\\limits\\_{1\\leq k< l\\leq N}\\vartheta\\_{kl}\\sim O\\_p(N^{-\\frac{q-3-2q\\alpha}{2q}})$, when *l* − *k* = *N**k**l* ≥ *m**l*. Here\n$\\displaystyle \\vartheta\\_{kl}=\\frac{N\\_{kl}}{N}\\int\\_{-\\pi}^\\pi \\hat{f}\\_k^l(v)\\log\\left(\\frac{st(\\hat{f}\\_k)}{st(f\\_s)}\\right)dv$\n**Proof**: Following the proofs of Lemma [ch3l3], this lemma can be easily obtained.\n**Proofs of Theorem [esticonsis3]:** \nDenote $\\displaystyle A\\_{kl}=\\left\\{\\omega: \\max\\limits\\_{\\lambda\\_i}\\left|\\frac{\\hat{f}\\_{k}^l(\\lambda\\_i)-f(\\lambda\\_i)}{f(\\lambda\\_i)}\\right|\\leq\\epsilon\\frac{m}{N^{(q-3)/(2q)}},\\forall j=1,\\ldots,K\\right\\}$, where *ω* is the event in probability space (Ω, F, *P*). Then $\\forall \\omega\\in\\bigcap\\limits\\_{kl}A\\_{kl}$, we prove that *κ̂**k* → *κ**k*0, ∀*j*.\nFor *κ̂**k*, since *κ̂**k*(*ω*) is bounded, ∀*k*, then there exists {*n**s*} such that *κ̂**n**s*(*ω*) → *κ**k*\\* on the subsequence. It follows from Lemma [ch3l2] and [ch3l3], that\n$\\displaystyle \\frac{1}{N} R(\\kappa\\_1^\\*,\\ldots,\\kappa\\_K^\\*)\\leq\\sum\\limits\\_{i=1}^{K+1}(\\kappa\\_i^\\*-\\kappa\\_{i-1}^\\*)\\int f\\_{\\kappa\\_{i-1}^\\*\\kappa\\_i^\\*}(v)\\log \\frac{st(f\\_{\\kappa\\_{i-1}^\\*\\kappa\\_i^\\*})}{st(f)}dv+O(N^{-\\frac{q-3-2q\\alpha}{2q}})$,\nwhere *λ*0\\* = 0, *λ**K* + 1\\* = 1. If *κ**i* − 1 ≤ *κ**j* − 1\\* < *κ**k* < ⋯ < *κ**i* + *k* < *κ**j*\\*, then by Lemma [ch3l3], we have\n$\\displaystyle f\\_{\\kappa\\_{j-1}^\\*\\kappa\\_j^\\*}=\\frac{\\kappa\\_i-\\kappa\\_{j-1}^\\*}{\\kappa\\_j^\\*-\\kappa\\_{j-1}^\\*}f\\_i+\\frac{\\kappa\\_{i+1}-\\kappa\\_i}{\\kappa\\_j^\\*-\\kappa\\_{j-1}^\\*}f\\_{i+1}+\\ldots+\\frac{\\kappa\\_j^\\*-\\kappa\\_{j+k}}{\\kappa\\_j^\\*-\\kappa\\_{j-1}^\\*}f\\_{j+k+1}$ \n$\\displaystyle F\\_{\\kappa\\_{j-1}^\\*\\kappa\\_j^\\*}=\\frac{\\kappa\\_i-\\kappa\\_{j-1}^\\*}{\\kappa\\_j^\\*-\\kappa\\_{j-1}^\\*}F\\_i+\\frac{\\kappa\\_{i+1}-\\kappa\\_i}{\\kappa\\_j^\\*-\\kappa\\_{j-1}^\\*}F\\_{i+1}+\\ldots+\\frac{\\kappa\\_j^\\*-\\kappa\\_{j+k}}{\\kappa\\_j^\\*-\\kappa\\_{j-1}^\\*}F\\_{j+k+1}$.\nSince $\\frac{1}{N}R(\\kappa\\_1,\\ldots,\\kappa\\_K)=\\sum\\limits\\_{k=1}^{K+1}\\int f\\_k(u)\\log\\frac{st(f\\_k)}{st(f)} du$.\nSo\n \n$$\\begin{aligned}\n& &\\frac{1}{N}R(\\lambda\\_{j-1}^\\*,\\lambda\\_{j}^\\*)-\\frac{1}{N}R(\\lambda\\_{j-1}^\\*,\\lambda\\_i)-\\ldots-\\frac{1}{N}R(\\lambda\\_{i+k-1}^\\*,\\lambda\\_{i+k})\\\\\n&=&(\\lambda\\_j^\\*-\\lambda\\_i^0)\\int f\\_i(u)\\log\\frac{st(f\\_{\\lambda\\_{j-1}^\\*\\lambda\\_j^\\*})}{st(f\\_i)}du+(\\lambda\\_{i+1}^0-\\lambda\\_i^0)\\int f\\_{i+1}(u)\\log\\frac{st(f\\_{\\lambda\\_{j-1}^\\*\\lambda\\_j^\\*})}{st(f\\_{i+1})}du\\\\\n& &+\\cdots+(\\lambda\\_j^\\*-\\lambda\\_{i+k}^0)\\int f\\_{i+k}(u)\\log\\frac{st(f\\_{\\lambda\\_{j-1}^\\*\\lambda\\_j^\\*})}{st(f\\_{i+k})}du+O(N^{-\\frac{q-3-2q\\alpha}{2q}})<0\\end{aligned}$$ \nas *N* → ∞, since every term above is always negative. If *κ**j* − 1 < *κ**k*\\* < *κ**k* + 1\\* < *κ**j*, then\n \n$$\\begin{aligned}\n\\frac{1}{N}R(\\hat{\\lambda}\\_{k-1},\\hat{\\lambda}\\_k)=(\\lambda\\_j^\\*-\\lambda\\_{j-1}^\\*)\\int f\\_j(v)\\log \\frac{st(f\\_j)}{st(f)}dv+O(N^{-\\frac{q-3-2q\\alpha}{2q}}).\\end{aligned}$$ \nSo we have\n \n$$\\begin{aligned}\n\\frac{1}{N}R(\\hat{\\kappa}\\_1,\\ldots,\\hat{\\kappa}\\_K)=\\frac{1}{N}R(\\kappa^\\*\\_1,\\ldots,\\kappa\\_K^\\*)+O(N^{-\\frac{q-3-2q\\alpha}{2q}})-\\frac{1}{N}R(\\kappa\\_1^0,\\ldots,\\kappa\\_K^0)<0\\end{aligned}$$ \nas *N* → ∞. This is a contradiction because *κ̂**k* are the maximizers of *R*(*κ*1, …, *κ**K*). Since\n$\\displaystyle P\\left(\\bigcap\\limits\\_{k,l}A\\_{kl}\\right)=1-P\\left(\\bigcup\\limits\\_{k,l}A^c\\_{kl}\\right)\\geq 1-\\sum\\limits\\_{l-k\\geq ml}^NP\\left(A\\_{kl}^c\\right)=1-\\frac{B\\_1}{\\epsilon^{2q}}$.\nso $\\hat{\\kappa}\\_j\\overset{p}{\\rightarrow} \\kappa\\_j^0$.\n**Proofs of Theorem [BICconsist3]:**\nIf *L* < *K*, then there should be a change point *λ**j*0 that can not estimated consistently. Then following the proof in Theorem [esticonsis3],\n \n$$\\begin{aligned}\n&&-\\frac{1}{N}R(\\hat{\\kappa}\\_1,\\ldots,\\hat{\\kappa}\\_L)+LC\\_N/N+\\frac{1}{N}R(\\hat{\\kappa}\\_1,\\ldots,\\hat{\\kappa}\\_K)-KC\\_N/N\\\\\n&<&-c+O\\_p(N^{-\\frac{q-3-2q\\alpha}{2q}})-(K-L)C\\_N/N<0\\end{aligned}$$ \nas *N* → ∞.\nIf *K̂* > *K*, then still every change point *λ**i* should be estimated consistently. So, there should be a change point *κ**j* − 1 < *κ**k*\\* < *κ**j*. Then by Lemma [ch3l2] and [ch3l3], $\\displaystyle R(\\kappa\\_{j-1},\\kappa\\_k^\\*,\\kappa\\_j)-R(\\kappa\\_{j-1},\\kappa\\_{j})=O\\_p(N^{-\\frac{q-3-2q\\alpha}{2q}})$. So $BIC\\_L-BIC\\_K=O\\_p(N^{-\\frac{q-3-2q\\alpha}{2q}})+(K-L)C\\_N/N<0$, as *N* → ∞. Here we complete the proofs of Theorem [BICconsist3].\n99 Brockwell, P. J., and Davis, R. A. (1991), *Time Series: Theory and Methods (2nd Edition)*, Springer-Verlag. Davis, R. A., Lee, T. C. M., and Rodriguez-Yam, G. A. (2006), “Structural Break Estimation for Nonstationary Time Series Models,” *Journal of the American Statistical Association*, 101(473), 223-239. Fan, J. and Yao, Q., (2003), *Nonlinear Time Series*, Springer. Goldberger, A. L., Amaral, L. A. N., Glass, L., Hausdorff, J. M., Ivanov, P. Ch., Mark, R. G., Mietus, J. E., Moody, G. B., Peng, C-K., Stanley, H. E. (2000), “PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals”, *Circulation* 101(23), e215-e220. 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(2001), “Automatic Statistical Analysis of Bivariate Nonstationary Time Series,” *Journal of the American Statistical Association*, 96(454), 543-560. Parzen, E. (1982), “Maximum Entropy Interpretation of Autoregressive Spectral Densities,” *Statistics and Probability*, 1, 2-6. Parzen, E. (1983), “Time Series ARMA Model Identification by Estimating Information,” Technical Report, Texas A&M Research Foundation Project No. 4226T, N-37. Preuss, P., Puchstein, R., and Dette, H. (2015), “Detection of Multiple Structural Breaks in Multivariate Time Series,” *Journal of the American Statistical Association*, 110(510), 654-668. Priestley, M. B. (1981), *Spectral Analysis and Time Series*, Academic Press. Rao, T. S. (1993), *Developments in Time Series Analysis*, Chapman and Hall/CRC. Shore, J. E. (1981), “Minimum Cross-Entropy Spectral Analysis,” *IEEE Transsactions on Acoustics, Speech, and Signal Processing*, ASSP-29(2), 230-237. Woodroofe, M. B., and Van Ness, J. W. (1967), “The Maximum Deviation of Sample Spectral Densities,” *The Annals of Mathematical Statistics*, 38(5), 1558-1569. Yao, Y.-C. (1988), “Estimating the Number of Change-Points Via Schwarz’ Criterion,” *Statistics & Probability Letters*, 6(3), 181-189. Yao, Y.-C., and Au, S. T. (1989), “Least-Squares Estimation of a Step Function,” *Sankhyā*, 51(3), 370-381. Zou, C., Yin, G., Feng, L., and Wang, Z. (2014), Nonparametric Maximum Likelihood Approach to Multiple Change-Point Problems. *The Annals of Statistics*, 42(3), 970-1002.\n"},"data_id":{"kind":"string","value":"arxiv_0183860"}}},{"rowIdx":45994,"cells":{"content":{"kind":"string","value":"Tri-hypercharge: a separate gauged weak hypercharge for each fermion family as the origin of flavour\n====================================================================================================\nfpheader\nIntroduction\n============\nThe existence of the three families of quarks and leptons is one of the fundamental mysteries of the Standard Model (SM). Indeed many of the parameters of the SM are associated with the resulting quark and lepton mass patterns and mixings. The low energy masses of quarks and charged leptons may be expressed approximately as \n \n$$\\begin{aligned}\n{2}\n & m\\_{t}\\sim\\frac{v\\_{\\mathrm{SM}}}{\\sqrt{2}}\\,,\\qquad & m\\_{c}\\sim\\lambda^{3.3}\\frac{v\\_{\\mathrm{SM}}}{\\sqrt{2}}\\,,\\qquad & m\\_{u}\\sim\\lambda^{7.5}\\frac{v\\_{\\mathrm{SM}}}{\\sqrt{2}}\\,,\\\\\n & m\\_{b}\\sim\\lambda^{2.5}\\frac{v\\_{\\mathrm{SM}}}{\\sqrt{2}}\\,,\\qquad & m\\_{s}\\sim\\lambda^{5.0}\\frac{v\\_{\\mathrm{SM}}}{\\sqrt{2}}\\,,\\qquad & m\\_{d}\\sim\\lambda^{7.0}\\frac{v\\_{\\mathrm{SM}}}{\\sqrt{2}}\\,,\\\\\n & m\\_{\\tau}\\sim\\lambda^{3.0}\\frac{v\\_{\\mathrm{SM}}}{\\sqrt{2}}\\,,\\qquad & m\\_{\\mu}\\sim\\lambda^{4.9}\\frac{v\\_{\\mathrm{SM}}}{\\sqrt{2}}\\,,\\qquad & m\\_{e}\\sim\\lambda^{8.4}\\frac{v\\_{\\mathrm{SM}}}{\\sqrt{2}}\\,,\\end{aligned}$$ \nwhere *v*SM ≃ 246 GeV is the SM vacuum expectation value (VEV) and *λ* ≃ 0.224 is the Wolfenstein parameter which parameterises the CKM matrix as\n \n$$\\begin{aligned}\n{2}\n & V\\_{us}\\sim\\lambda\\,,\\qquad & V\\_{cb}\\sim\\lambda^{2}\\,,\\qquad & V\\_{ub}\\sim\\lambda^{3}\\,.\\end{aligned}$$ \nThe hierarchical patterns of masses and CKM mixing, and the possibility that they might be understood in the form of a more fundamental *theory of flavour* beyond the SM, has classically been denoted as the *flavour puzzle*. The discovery of neutrino oscillations, which proves that at least two neutrinos have non-zero mass, has made the flavour puzzle difficult to ignore, enlarging the flavour sector with extra neutrino mixing angles \n \n$$\\begin{aligned}\n{2}\n & \\tan\\theta\\_{23}\\sim1\\,,\\qquad & \\tan\\theta\\_{12}\\sim\\frac{1}{\\sqrt{2}}\\,,\\qquad & \\theta\\_{13}\\sim\\frac{\\lambda}{\\sqrt{2}}\\,,\\end{aligned}$$ \nplus very tiny neutrino masses which follow either a normal or inverted ordering. The need for a theory of flavour to understand all fermion masses and mixings is now more urgent than ever.\nTheories of flavour may involve new symmeties structures (global, local, continuous, discrete, abelian, non-abelian...) beyond the SM group, possibly broken at some high scale down to the SM. Traditionally, new gauge structures beyond the SM have been considered to be flavour universal, as grand unified theories (GUTs) which usually embed all three families in an identical way, or even extended GUTs which embed all three families in a single representation (usually along with extra exotic fermions). Alternatively, there exists the well-motivated case of family symmetries which commute with the SM gauge group, and are then spontaneously broken, leading to family structure. However, there are other less explored ways in which the SM gauge group could be embedded into a larger gauge structure in a flavour non-universal way. In particular, the *family decomposition* of the SM gauge group (including a hierarchical symmetry breaking pattern down to the SM) was first proposed during the 80s and 90s, with the purpose of motivating lepton non-universality or assisting technicolor model building. However, the natural origin of flavour hierarchies in such a framework was not explored until more recently in. Here it was proposed that the flavour non-universality of Yukawa couplings in the SM might well find its origin in a flavour non-universal gauge sector, broken in a hierarchical way down to the SM. Interestingly, model building in this direction has received particular attention in recent years. With the exception of Ref. , the remaining recent attempts have been motivated by the need to obtain a TeV scale vector leptoquark from Pati-Salam unification in order to address the so-called *B*-anomalies[1](#fn1). Therefore, all these setups share a similar feature: a low scale *S**U*(4) gauge group under which only the third family of SM fermions transforms in a non-trivial way. In contrast, in this work we want to explore the capabilities of flavour non-universality to address the flavour puzzle in a more minimal, simple and bottom-up approach.\nWe propose that the SM symmetry originates from a larger gauge group in the UV that contains three separate weak hypercharge gauge factors,\n \n*S**U*(3)*c* × *S**U*(2)*L* × *U*(1)*Y*1 × *U*(1)*Y*2 × *U*(1)*Y*3 , \nwhich we will denote as the *tri-hypercharge* (TH) *U*(1)*Y*3 gauge group. We will associate each of the three hypercharge gauge groups with a separate SM family, such that each fermion family *i* only carries hypercharge under the corresponding *U*(1)*Y**i* factor. This ensures that each family transforms differently under the gauge group *U*(1)*Y*3, which avoids the family repetition of the SM, and provides the starting point for a theory of flavour. For example, assuming that a single Higgs doublet only carries third family hypercharge, then only the third family Yukawa couplings are allowed at renormalisable level. With two Higgs doublets carrying third family hypercharge, we show that the naturalness of the scheme increases. This simple and economical framework naturally explains the heaviness of the third family, the smallness of *V**c**b* and *V**u**b*, and delivers Yukawa couplings that preserve an accidental and global *U*(2)5 flavour symmetry acting on the light families, which is known to provide a good first order description of the SM spectrum plus an efficient suppression of flavour-violating effects for new physics. Remarkably, this appears to be the simplest way to provide the *U*(2)5 flavour symmetry[2](#fn2). The masses of first and second family fermions, along with the CKM mixing, then appear as small breaking sources of *U*(2)5 that arise after the cascade spontaneous symmetry breaking of *U*(1)*Y*3 down to SM hypercharge, which can be parameterised in a model-independent way in terms of spurions. In a realistic model, the spurions will be realised by a choice of “hyperon” scalars which transform under the different family hypercharge groups, breaking the tri-hypercharge symmetry. We will motivate a specific symmetry breaking chain where dynamics at a low scale, which could be as low as the TeV, explain the flavour hierarchies *m*2/*m*3, while dynamics at a heavier scale explain *m*1/*m*2. This symmetry breaking pattern will sequentially recover the approximate flavour symmetry of the SM, and provide a natural suppression of FCNCs for TeV new physics, while the rest of flavour-violating effects are suppressed by a naturally heavier scale.\nThe paper is organised as follows. In Section [sec:The-Tri-Hypercharge-model] we introduce the TH gauge group, along with the fermion and Higgs doublet content of the model. We discuss the implications for third family fermion masses along with the mass hierarchy between the top and bottom/tau fermions. In Section [sec:Charged-fermion-masses-mixing] we study the origin of charged fermion masses and mixing in the TH model, firstly via a spurion formalism which reveals model-independent considerations, and secondly by introducing example models with hyperons. In Section [sec:Neutrino-masses-and-Mixing] we study the origin of neutrino masses and mixing in the TH model. In particular, we discuss the impact of the *U*(2)5 flavour symmetry over the dimension-5 Weinberg operator, and afterwards we provide an example type I seesaw model where neutrino masses and mixing can be accommodated. In Section [sec:Phenomenology] we perform a preliminary exploration of the phenomenological implications and discovery prospects of the *U*(1)*Y*3 theory of flavour. Finally, Section [sec:Conclusions] outlines our main conclusions.\nTri-hypercharge gauge theory[sec:The-Tri-Hypercharge-model]\n===========================================================\nThe tri-hypercharge gauge group is based on assigning a separate gauged weak hypercharge to each fermion family,\n[t]\nlccccc Field & *S**U*(3)*c* & *S**U*(2)*L* & *U*(1)*Y*1 & *U*(1)*Y*2 & *U*(1)*Y*3 \n*Q*1 & $\\mathrm{\\mathbf{3}}$ & **2** & 1/6 & 0 & 0 \n*u*1*c* & $\\bar{\\mathbf{3}}$ & **1** & − 2/3 & 0 & 0 \n*d*1*c* & $\\bar{\\mathbf{3}}$ & **1** & 1/3 & 0 & 0 \n*L*1 & **1** & **2** & − 1/2 & 0 & 0 \n*e*1*c* & **1** & **1** & 1 & 0 & 0 \n*Q*2 & $\\mathrm{\\mathbf{3}}$ & **2** & 0 & 1/6 & 0 \n*u*2*c* & $\\bar{\\mathbf{3}}$ & **1** & 0 & − 2/3 & 0 \n*d*2*c* & $\\bar{\\mathbf{3}}$ & **1** & 0 & 1/3 & 0 \n*L*2 & **1** & **2** & 0 & − 1/2 & 0 \n*e*2*c* & **1** & **1** & 0 & 1 & 0 \n*Q*3 & $\\mathrm{\\mathbf{3}}$ & **2** & 0 & 0 & 1/6 \n*u*3*c* & $\\bar{\\mathbf{3}}$ & **1** & 0 & 0 & − 2/3 \n*d*3*c* & $\\bar{\\mathbf{3}}$ & **1** & 0 & 0 & 1/3 \n*L*3 & **1** & **2** & 0 & 0 & − 1/2 \n*e*3*c* & **1** & **1** & 0 & 0 & 1 \n \n*S**U*(3)*c* × *S**U*(2)*L* × *U*(1)*Y*1 × *U*(1)*Y*2 × *U*(1)*Y*3 , \nin such a way that the *i*th fermion family only carries *Y**i* hypercharge, with the other hypercharges set equal to zero (see Table [tab:Fieldcontent]), where *Y* = *Y*1 + *Y*2 + *Y*3 is equal to SM weak hypercharge. Anomalies cancel separately for each family, as in the SM, but without family replication. The TH gauge group is broken down to the SM via appropriate SM singlet scalars, which however carry family hypercharges. We denote these fields linking the family hypercharges as *hyperons*. The TH group could be broken down to the SM in different ways, however we motivate the following symmetry breaking pattern,\n \n$$\\begin{aligned}\n{1}\n & SU(3)\\_{c}\\times SU(2)\\_{L}\\times{\\displaystyle U(1)\\_{Y\\_{1}}\\times U(1)\\_{Y\\_{2}}\\times U(1)\\_{Y\\_{3}}}\\nonumber \\\\\n & {\\displaystyle \\overset{v\\_{12}}{\\rightarrow}SU(3)\\_{c}\\times SU(2)\\_{L}\\times U(1)\\_{Y\\_{1}+Y\\_{2}}\\times U(1)\\_{Y\\_{3}}}\\label{eq:Symmetry\\_Breaking}\\\\\n & {\\displaystyle \\overset{v\\_{23}}{\\rightarrow}SU(3)\\_{c}\\times SU(2)\\_{L}\\times U(1)\\_{Y\\_{1}+Y\\_{2}+Y\\_{3}}\\,,}\\nonumber \\end{aligned}$$ \nThis choice is well supported by symmetry arguments that will have phenomenological consequences: At high energies, the TH group discriminates between the three SM fermion families, explicitly breaking the approximate *U*(3)5 flavour symmetry of the SM. At a heavy scale *v*12, the first and second hypercharges are broken down to their diagonal subgroup, and the associated *Z*ʹ boson potentially mediates dangerous 1-2 flavour-changing neutral currents (FCNCs). Nevertheless, the gauge group below the scale *v*12 preserves an accidental *U*(2)5 flavour symmetry. The groups *U*(1)*Y*1 + *Y*2 × *U*(1)*Y*3 are broken down to their diagonal subgroup (SM hypercharge) at a scale *v*23, and the associated *Z*ʹ boson is protected from mediating the most dangerous FCNCs thanks to the *U*(2)5 symmetry. In this manner, the most dangerous FCNCs are suppressed by the heavier scale *v*12, while the scale *v*23 can be very low with interesting phenomenological implications. We will see that dynamics connected to the scale *v*12 will play a role in the origin of the family hierarchy *m*1/*m*2, while dynamics connected to the scale *v*23 will play a role in the origin of *m*2/*m*3. The distribution of the various scales and the approximate flavour symmetries that apply at each scale are summarised in the diagram of Fig. [fig:-Multiscale-picture-flavour].\n\nProvided that the SM Higgs only carries third family hypercharge, $H(\\mathbf{1},\\mathbf{2})\\_{(0,0,-\\frac{1}{2})}$, then the third family Yukawa couplings are allowed at renormalisable level and an accidental *U*(2)5 flavour symmetry acting on the light families emerges in the Yukawa sector,\n \nL = *y**t**Q*3*H̃**u*3*c* + *y**b**Q*3*H**d*3*c* + *y**τ**L*3*H**e*3*c* + h.c. \nwhere *H̃* is the *C**P*-conjugate of *H*. This setup already provides an explanation for the smallness of light fermion masses with respect to the third family, along with the smallness of quark mixing, as they all must arise from non-renormalisable operators which minimally break the *U*(2)5 symmetry. Although this is a good first order description of the SM spectrum, the question of why the bottom and tau fermions are much lighter than the top remains unanswered, and assuming only a single Higgs doublet, a tuning of order 2% for the bottom coupling and of 1% for the tau coupling would be required. Given that *m**s*, *μ* ∝ *λ*5*m**t* while *m**c* ∝ *λ*3*m**t*, this setup also requires to generate a stronger fermion hierarchy in the down and charged lepton sectors with respect to the up sector, unless the tuning in the bottom and tau couplings is extended to the second family. As we shall see shortly, the *U*(1)*Y*3 model (and very likely a more general set of theories of flavour based on the family decomposition of the SM group) predicts a similar mass hierarchy for all charged sectors.\nDue to the above considerations, it seems natural to consider a type II two Higgs doublet model (2HDM), where both Higgs doublets only carry third family hypercharge,\n \n$$H\\_{u}(\\mathbf{1},\\mathbf{2})\\_{(0,0,\\frac{1}{2})},\\ \\ \\ \\ H\\_{d}(\\mathbf{1},\\mathbf{2})\\_{(0,0,-\\frac{1}{2})}\\,,$$ \nwhere as usual for a type II 2HDM, FCNCs can be forbidden by e.g. a *Z*2 discrete symmetry or by supersymmetry (not necessarily low scale), which we however do not specify in order to preserve the bottom-up spirit of this work. In any case, tan*β* = *v**u*/*v**d* ∼ *λ*− 2 ≈ 20, which is compatible with current data (see e.g. ), will provide the hierarchy between the top and bottom/tau masses with all dimensionless couplings being O(1). Such an overall hierarchy between the down and charged leptons sector with respect to the up sector is extended to all families, providing a better description of second family charged fermion masses as we shall see.\nCharged fermion masses and mixing[sec:Charged-fermion-masses-mixing]\n====================================================================\nLessons from the spurion formalism[subsec:Lessons-from-the]\n-----------------------------------------------------------\nIn all generality, we introduce *U*(2)5-breaking *spurions* Φ in the Yukawa matrices of charged fermions\nL & =\nQ1 & Q2 & Q3\n(,0,-) & (-,,-) & (-,0,) \n(,-,-) & (0,,-) & (0,-,) \n(,0,-) & (0,,-) & 1\nu1c \nu2c \nu3c\nHu \n& +\nQ1 & Q2 & Q3\n(-,0,) & (-,-,) & (-,0,) \n(-,-,) & (0,-,) & (0,-,) \n(-,0,) & (0,-,) & 1\nd1c \nd2c \nd3c\nHd[eq:Spurions] \n& +\nL1 & L2 & L3\n(-,0,) & (,-1,) & (,0,-) \n(-1,,) & (0,-,) & (0,,-) \n(-1,0,1) & (0,-1,1) & 1\ne1c \ne2c \ne3c\nHd+,\nwhere each spurion carries non-trivial charge assignments under *U*(1)*Y*3. In an effective field theory (EFT) approach, each spurion above can be matched to specific ratios of hyperons *ϕ**i* over EFT cut-off scales Λ*i*, i.e.\n \n$$\\Phi=\\frac{\\phi\\_{1}...\\phi\\_{n}}{\\Lambda\\_{1}...\\Lambda\\_{n}}\\,,$$ \nwhere we have suppressed dimensionless couplings. The choice of hyperons and Λ*i* above carries all the model dependence.\nAssuming that the cut-off scales Λ*i* of the EFT are universal, i.e. that all Λ*i* are common to all charged sectors, then the spurion formalism reveals some general considerations about the origin of charged fermion masses and mixing:\n* The same spurions (up to conjugation) appear in the diagonal entries of all matrices. Therefore, unless texture zeros are introduced in specific models, this means that the masses of second family fermions are likely to be degenerate up to dimensionless couplings, and the same discussion applies to first family fermions. This motivates again the addition of the second Higgs doublet or an alternative mechanism in order to generate the hierarchy between the charm mass and the lighter strange and muon masses.\n* The same spurions appear in the (2,3) entries of the up and down Yukawa matrices. Therefore, the 2-3 mixing in both the up and down sectors is expected to be of a similar size, giving no predictions about the alignment of the CKM element *V**c**b*. The similar argument applies to 1-3 mixing and *V**u**b*.\n* The spurions in the (1,2) entry of the up and down matrices are different. Therefore, specific models have the potential to give predictions about the alignment of the CKM element *V**u**s*.\n* The same spurion (up to conjugation) that enters in all (2,2) entries also populates the (2,3) entry of the charged lepton Yukawa matrix. Similarly, the same spurion (up to conjugation) that enters in all (1,1) entries also populates the (1,3) entry of the charged lepton Yukawa matrix. In general, this predicts left-handed *μ* − *τ* (*e* − *τ*) mixing of O(*m*2/*m*3) (O(*m*1/*m*3)), unless texture zeros are introduced in specific models (see Section [subsec:From-Spurions]). This leads to a sizable enhancement of LFV *τ* → *μ* and *τ* → *e* transitions above the SM predictions, mediated by heavy *Z*ʹ bosons in the model (see Section [sec:Phenomenology]).\n* The spurions in the lower off-diagonal entries of the Yukawa matrices all carry independent charge assignments, so right-handed fermion mixing is model-dependent and can be different in all charged sectors.\nIn the following, we go beyond the spurion formalism and introduce different sets of hyperons. As we shall wee, the hyperons will provide small *U*(2)5-breaking effects via non-renormalisable operators, leading to the masses of first and second family charged fermions, along with CKM mixing. In the next few subsections, we will describe example scenarios which provide a good description of charged fermion masses and mixing.\nFrom spurions to hyperons [subsec:From-Spurions]\n------------------------------------------------\nThe physical origin of the spurions of the previous subsection will correspond to new Higgs scalar fields that break the *U*(1)*Y*3 symmetry, which we call *hyperons*. The hyperons induce small *U*(2)5-breaking effects at the non-renormalisable level that will lead to the masses and mixings of charged fermions. As the most straightforward scenario, we could promote the spurions in the diagonal entries of the matrices in Eq. to hyperons, along with the off-diagonal spurions in the upper half of the down matrix[3](#fn3). In an EFT approach, the set of hyperons that we have assumed generates the following Yukawa matrices,\nLd5 & =\nQ1 & Q2 & Q3\n13(,0,-)/& 0 & q13(-,0,)/ \n0 & 23(0,,-)/& q23(0,-,)/ \n0 & 0 & 1\nu1c \nu2c \nu3c\nHu \n& +\nQ1 & Q2 & Q3\n13(-,0,)/& d12(-,-,)/& q13(-,0,)/ \n0 & 23(0,-,)/& q23(0,-,)/ \n0 & 0 & 1\nd1c \nd2c \nd3c\nHd \n& +\nL1 & L2 & L3\n13(-,0,)/& 0 & 13(,0,-)/ \n0 & 23(0,-,)/& 23(0,,-)/ \n0 & 0 & 1\ne1c \ne2c \ne3c\nHd+,\nwhere the universal scale Λ is the high cut-off scale of the EFT, and we ignore the O(1) dimensionless couplings of each entry. Although we have chosen only the specific set of hyperons shown, leaving some zeros in the matrices, these zeros may be filled in by higher order operators with dimension larger than 5, which so far we are ignoring.\nWhen the hyperons develop VEVs, assumed to be smaller than the cut-off scale Λ, then each entry of the matrix will receive a suppressed numerical effective coupling given by ratios of the form ⟨*ϕ*⟩/Λ, whose values can be assumed arbitrarily. Having the freedom to choose arbitrary VEVs for each hyperon, the Yukawa matrices above could provide a good first order description of charged fermion masses and CKM mixing. We choose to fix the ⟨*ϕ*⟩/Λ ratios in terms of powers of the Wolfenstein parameter *λ* ≃ 0.224, obtaining\nL & =\nu1 & u2 & u3\n6 & 0 & 3 \n0 & 3 & 2 \n0 & 0 & 1\nu1c \nu2c \nu3c\n[eq:Texturea] \n& +\nd1 & d2 & d3\n6 & 4 & 3 \n0 & 3 & 2 \n0 & 0 & 1\nd1c \nd2c \nd3c\n2[eq:Textureb] \n& +\ne1 & e2 & e3\n6 & 0 & 6 \n0 & 3 & 3 \n0 & 0 & 1\ne1c \ne2c \ne3c\n2+[eq:Texturec]\nAs anticipated from the spurion formalism, the alignment of *V**c**b* and *V**u**b* is not predicted by the model. In contrast, the model predicts a relevant left-handed *μ* − *τ* (*e* − *τ*) mixing connected to the same hyperon that provides the second family (first family) effective Yukawa couplings. Thanks to the addition of the second Higgs doublet, the model successfully explains third and second family fermion masses with O(1) dimensionless couplings. The down-quark and electron masses are also reasonably explained, although the up-quark mass is naively a factor O(*λ*1.5) larger than current data. Notice that so far we are only assuming one universal cut-off scale Λ, while in realistic models several cut-off scales Λ may be associated to different messengers in the UV theory, which could provide a larger suppression for the up-quark effective coupling. Therefore, within the limitations of our EFT approach, the description of charged fermion masses given by the set of Eqs. ([eq:Texturea]-[eq:Texturec]) is very successful.\nAs an alternative example, one could also consider a model where the (1,1) spurion in Eq. is not promoted to hyperon, but instead all the spurions in the (1,2) and (2,1) entries are promoted, so that the Yukawa matrices show an exact texture zero in the (1,1) entry,\nLd5 & =\nQ1 & Q2 & Q3\n0 & u12(-,,-) & q13(-,0,) \nu21(,-,-) & 23(0,,-) & q23(0,-,) \n0 & 0 & 1\nu1c \nu2c \nu3c\nHu \n& +\nQ1 & Q2 & Q3\n0 & d12(-,-,) & q13(-,0,) \nd21(-,-,) & 23(0,-,) & q23(0,-,) \n0 & 0 & 1\nd1c \nd2c \nd3c\nHd \n& +\nL1 & L2 & L3\n0 & e12(,-1,) & 0 \ne21(-1,,) & 23(0,-,) & 23(0,,-) \n0 & 0 & 1\ne1c \ne2c \ne3c\nHd+,\nwhere we have omitted the high cut-off Λ suppressing each dimension-5 operator above. The VEV over Λ ratios of the new hyperons can be fixed by the requirement of addressing first family fermion masses, obtaining Yukawa matrices with texture zeros given by\nL & =\nu1 & u2 & u3\n0 & 5 & 3 \n5.5 & 3 & 2 \n0 & 0 & 1\nu1c \nu2c \nu3c\n \n& +\nd1 & d2 & d3\n0 & 4 & 3 \n4 & 3 & 2 \n0 & 0 & 1\nd1c \nd2c \nd3c\n2 \n& +\ne1 & e2 & e3\n0 & 5 & 0 \n4.4 & 3 & 3 \n0 & 0 & 1\ne1c \ne2c \ne3c\n2+,\nwhich provide an even better description of first family fermion masses than the original simplified model. Notice that in this scenario, a sizable left-handed *e* − *τ* mixing is no longer predicted.\nWe conclude that the most straightforward choices of hyperons, motivated by the spurion formalism, already provide a good description of charged fermion masses and mixings. However, these simplified models leave some questions unanswered. Given that we are assuming the symmetry breaking highlighted in Eq. and Fig. [fig:-Multiscale-picture-flavour], we notice that there are no hyperons breaking the first and second hypercharges down to their diagonal subgroup, and we would expect those to play a role in the origin of fermion hierarchies and mixing. Moreover, in the simplified models introduced so far, several hyperons display unexplained large hierarchies of VEVs whose values are assumed *a posteriori* to fit the fermion masses. Given that all these hyperons participate in the 23-breaking step of Eq. , we would expect all of them to develop VEVs at a similar scale, rather than the hierarchical scales assumed. This motivates further model building. In the following subsections we discuss a couple of example models which address these issues.\nModel 1: Minimal case with three hyperons[subsec:Model-1:-Minimal]\n------------------------------------------------------------------\nWe introduce here the following set of three hyperons,\n \n$${\\phi}\\_{\\ell23}^{(0,\\frac{1}{2},-\\frac{1}{2})}\\,,\\qquad{\\phi}\\_{q23}^{(0,-\\frac{1}{6},\\frac{1}{6})}\\,,\\qquad{\\phi}\\_{q12}^{(-\\frac{1}{6},\\frac{1}{6},0)}\\,.$$ \nFollowing the EFT approach of the previous subsection, we now analyse the effective Yukawa matrices obtained by combining the SM charged fermions, the Higgs doublets and the hyperons, in a tower of non-renormalisable operators preserving the *U*(1)*Y*3 gauge symmetry,\nL & =\nQ1 & Q2 & Q3\nq12323 & q1223 & q12q23 \nq12423 & 23 & q23 \nq12423q23 & 23q23 & 1\nu1c \nu2c \nu3c\nHu \n& +\nQ1 & Q2 & Q3\nq12323 & q1223 & q12q23 \nq12223 & 23 & q23 \nq122q232 & q232 & 1\nd1c \nd2c \nd3c\nHd \n& +\nL1 & L2 & L3\nq12323 & q12323 & q12323 \nq12623 & 23 & 23 \nq126232 & 232 & 1\ne1c \ne2c \ne3c\nHd+,\nwhere the powers of Λ in the denominator and the dimensionless couplings of each entry are not shown. Once the hyperons above develop VEVs, we obtain very economical and efficient Yukawa textures for modeling the observed pattern of SM Yukawa couplings. In particular, the masses of second family fermions arise at dimension 5 in the EFT, while first family masses have an extra suppression as they arise from dimension-8 operators. Regarding CKM mixing, 2-3 quark mixing leading to *V**c**b* arises from dimension-5 operators, while *V**u**b* has an extra mild suppression as it arises from dimension-6 operators. In all cases, right-handed fermion mixing is suppressed with respect to left-handed mixing. This is a highly desirable feature, given the strong phenomenological constraints on right-handed flavour-changing currents, which may be mediated by heavy *Z*ʹ bosons arising from the symmetry breaking of *U*(1)*Y*3 (see Section [sec:Phenomenology]).\nIn good approximation, quark mixing leading to *V**u**s* arises as the ratio of the (1,2) and (2,2) entries of the quark matrices above, therefore we expect\n \n$$\\frac{\\left\\langle {\\phi}\\_{q12}\\right\\rangle }{\\Lambda}\\sim V\\_{us}\\simeq\\lambda\\,,$$ \nwhere *λ* = sin*θ**C* ≃ 0.224. In a similar manner, we can fix the ratio ⟨*ϕ**q*23⟩/Λ by reproducing the observed *V**c**b*\n \n$$\\frac{\\left\\langle {\\phi}\\_{q23}\\right\\rangle }{\\Lambda}\\sim V\\_{cb}\\simeq\\lambda^{2}\\,.$$ \nGiven that both ⟨*ϕ**q*23⟩ and ⟨*ϕ*ℓ23⟩ play a role in the last step of the symmetry breaking cascade (see Fig. [fig:-Multiscale-picture-flavour]), it is expected that both VEVs live at a similar scale, although they are not expected to be degenerate but differ by an O(1) factor. This way, we are free to choose\n \n$$\\frac{\\left\\langle {\\phi}\\_{\\ell23}\\right\\rangle }{\\Lambda}\\sim\\frac{m\\_{c}}{m\\_{t}}\\sim\\lambda^{3}\\,,$$ \nwhich, given that ⟨*H**d*⟩ provides an extra suppression of O(*λ*2) for down-quarks and charged lepton Yukawas, allows to predict all second family masses with O(1) dimensionless couplings. In contrast with the simplified models of Section [subsec:From-Spurions], this model provides all the 23-breaking VEVs at the same scale, plus a larger 12-breaking VEV, following a mild hierarchy given by *v*23/*v*12 ∼ *λ*. This way, the symmetry breaking of the *U*(1)*Y*3 gauge group proceeds just like in Eq. and Fig. [fig:-Multiscale-picture-flavour], as desired.\nHaving fixed all the hyperon VEVs with respect to Λ, now we are able to write the full mass matrices for each sector in terms of the Wolfenstein parameter *λ*,\nL & =\nu1 & u2 & u3\n6 & 4 & 3 \n7 & 3 & 2 \n9 & 5 & 1\nu1c \nu2c \nu3c\n \n& +\nd1 & d2 & d3\n6 & 4 & 3 \n5 & 3 & 2 \n6 & 4 & 1\nd1c \nd2c \nd3c\n2 \n& +\ne1 & e2 & e3\n6 & 6 & 6 \n9 & 3 & 3 \n12 & 6 & 1\ne1c \ne2c \ne3c\n2+.\nWe can see that this setup provides a reasonable description of charged fermion masses and mixing. Although the up and down-quark masses are slightly off by O(*λ*) factors, we remember that we are only assuming one universal cut-off scale Λ, while in realistic models several scales Λ may be associated to different messengers in the UV theory, further improving the fit of first family quark masses, as discussed in Section [subsec:From-Spurions]. All things considered, the description of fermion masses seems very efficient, considering the limitations of our EFT framework.\nHowever, the model does not predict the alignment of *V**u**s*. Moreover, we also notice that right-handed *s* − *d* mixing is just mildly suppressed as *s*12*d**R* ≃ O(*λ*2) in this model. Given the stringent bounds over left-right scalar operators contributing to *K*0 − *K̄*0 meson mixing (which might arise in this kind of models as we shall see in Section [sec:Phenomenology]), the scale *v*12 can be pushed far above the TeV if *V**u**s* originates from the down sector. From a phenomenological point of view, it would be interesting to find models which give clear predictions about the alignment of *V**u**s*, and ideally provide a more efficient suppression of right-handed quark mixing. We shall see in the next subsection that this can be achieved by minimally extending the set of hyperons of this model.\nModel 2: Five hyperons for a more predictive setup[subsec:Model-2:-Five]\n------------------------------------------------------------------------\nModel 1 proposed in the previous subsection, despite its simplicity and minimality, does not give clear predictions about the alignment of the CKM matrix. Heavy *Z*ʹ bosons arising from the symmetry breaking of *U*(1)*Y*3 have the potential to mediate contributions to *K*0 − *K̄*0 meson mixing, which could set a lower bound over the scale of *U*(1)*Y*3-breaking, but such contributions depend on the alignment of *V**u**s*. Moreover, the largest contributions to *K*0 − *K̄*0 mixing depend both on the alignment of *V**u**s* and on right-handed *s* − *d* mixing, which is just mildly suppressed in Model 1. Therefore, we propose here a similar model with slightly extended hyperon content that can account for a clear prediction about the alignment of *V**u**s*, plus a more efficient suppression of right-handed fermion mixing. We consider here the hyperons\n \n$${\\phi}\\_{\\ell23}^{(0,\\frac{1}{2},-\\frac{1}{2})},\\qquad{\\phi}\\_{q23}^{(0,-\\frac{1}{6},\\frac{1}{6})},\\qquad{\\phi}\\_{q13}^{(-\\frac{1}{6},0,\\frac{1}{6})},\\qquad{\\phi}\\_{d12}^{(-\\frac{1}{6},-\\frac{1}{3},\\frac{1}{2})},\\qquad{\\phi}\\_{e12}^{(\\frac{1}{4},-\\frac{1}{4},0)}.$$ \nWith this set of hyperons, the effective Yukawa couplings in the EFT are (suppressing as usual powers of Λ and dimensionless couplings)\nL & =\nQ1 & Q2 & Q3\ne12223 & q13q2323 & q13 \ne122q13q2323 & 23 & q23 \ne122q1323 & 23q23 & 1\nu1c \nu2c \nu3c\nHu \n& +\nQ1 & Q2 & Q3\ne12223 & d12 & q13 \nq132q23 & 23 & q23 \nq132 & q232 & 1\nd1c \nd2c \nd3c\nHd \n& +\nL1 & L2 & L3\ne12223 & e12223 & e12223 \ne12423 & 23 & 23 \ne124232 & 232 & 1\ne1c \ne2c \ne3c\nHd+.\nFollowing the same approach as with Model 1, we assign the following powers of *λ* to the VEV over Λ ratios in order to reproduce fermion masses and CKM mixing,\n \n$$\\frac{\\left\\langle {\\phi}\\_{\\ell23}\\right\\rangle }{\\Lambda}=\\frac{\\left\\langle {\\phi}\\_{q13}\\right\\rangle }{\\Lambda}\\simeq\\lambda^{3}\\,,\\quad\\frac{\\left\\langle {\\phi}\\_{q23}\\right\\rangle }{\\Lambda}\\simeq\\lambda^{2}\\,,\\quad\\frac{\\left\\langle {\\phi}\\_{d12}\\right\\rangle }{\\Lambda}\\simeq\\lambda^{4}\\,,\\quad\\frac{\\left\\langle {\\phi}\\_{e12}\\right\\rangle }{\\Lambda}\\simeq\\lambda\\,.$$ \nAlthough it would seem that in this scenario there exists a mild hierarchy between 23-breaking VEVs of O(*λ*2), since *ϕ**d*12 only appears in the 12 entry of the down matrix, it would be very reasonable that the dimensionless coupling in that entry provides a factor *λ* suppression, such that all 23-breaking VEVs live at the same scale. The largest VEV is still the 12-breaking one, which is now associated to the hyperon *ϕ**e*12, and the mild hierarchy between scales remains as *v*12/*v*23 ≃ *λ*. With these assignments of VEVs over Λ ratios, the Yukawa textures are given by\nL & =\nu1 & u2 & u3\n5 & 8 & 3 \n10 & 3 & 2 \n7 & 5 & 1\nu1c \nu2c \nu3c\n \n& +\nd1 & d2 & d3\n5 & 4 & 3 \n8 & 3 & 2 \n6 & 4 & 1\nd1c \nd2c \nd3c\n2 \n& +\ne1 & e2 & e3\n5 & 5 & 5 \n7 & 3 & 3 \n10 & 6 & 1\ne1c \ne2c \ne3c\n2+.\nJust like in Model 1, this model provides a compelling description of all charged fermion masses and mixing. Notice that this scenario provides a very efficient suppression of right-handed fermion mixing. Moreover, it is clear that here *V**u**s* mixing originates from the down sector, providing a more predictive setup than Model 1, which will be useful for phenomenological purposes as discussed in Section [sec:Phenomenology].\nFinally we comment that the Higgs doublets and hyperons can mediate FCNCs such as *K*0 − *K̄*0 mixing. In the present model, this may arise from tree-level exchange of *ϕ**d*12 hyperons which can mediate down-strange transitions. Such a coupling originates from *Q*1*ϕ**d*12*d*2*c**H**d*/Λ which will lead to a suppressed down-strange coupling of order *λ*2*v*SM/Λ ≈ 10− 5 (taking Λ ≈ 100 TeV as expected if *v*23 ≈ O(TeV)). Assuming the mass of the hyperons to be at the scale *v*23, then we expect these FCNCs to be under control. Since we assume a type II 2HDM, tree-level Higgs doublets exchange contributions are forbidden, and FCNCs mediated by the Higgs doublets can only proceed via their mixing with hyperons, carrying therefore an extra suppression via the mixing angle, along with the suppression of hyperon couplings already discussed. In more general models of this kind, such contributions to *K*0 − *K̄*0 mixing could be even further suppressed depending on the order of the operator and the alignment of *V**u**s*, and in all generality we expect all hyperon couplings to be suppressed by at least a factor *v*SM/Λ ≈ 10− 3. A more detailed study of the phenomenology of the scalar sector in this general class of models is beyond the scope of this work.\nNeutrino masses and mixing[sec:Neutrino-masses-and-Mixing]\n==========================================================\nGeneral considerations and spurion formalism\n--------------------------------------------\nThe origin of neutrino masses and mixing requires a dedicated analysis due to their particular properties. We start by introducing *U*(2)5-breaking spurions (carrying inverse of mass dimension) for the Weinberg operator\n \n$$\\mathcal{L}\\_{\\mathrm{Weinberg}}=\\begin{pmatrix}L\\_{1} & L\\_{2} & L\\_{3}\\end{pmatrix}\\begin{pmatrix}{\\Phi}(1,0,-1) & {\\Phi}(\\frac{1}{2},\\frac{1}{2},-1) & {\\Phi}(\\frac{1}{2},0,-\\frac{1}{2})\\\\\n{\\Phi}(\\frac{1}{2},\\frac{1}{2},-1) & {\\Phi}(0,1,-1) & {\\Phi}(0,\\frac{1}{2},-\\frac{1}{2})\\\\\n{\\Phi}(\\frac{1}{2},0,-\\frac{1}{2}) & {\\Phi}(0,\\frac{1}{2},-\\frac{1}{2}) & 1\n\\end{pmatrix}\\begin{pmatrix}L\\_{1}\\\\\nL\\_{2}\\\\\nL\\_{3}\n\\end{pmatrix}H\\_{u}H\\_{u}\\,,$$ \nwhich reveals that, as expected, the *U*(2)5 approximate symmetry is naively present in the neutrino sector as well. As a consequence, one generally expects one neutrino to be much heavier than the others, displaying tiny mixing with the other neutrino flavours. In the spirit of the type I seesaw mechanism, one could think of adding a *U*(1)*Y*3 singlet neutrino as *N*(0, 0, 0). Such a singlet neutrino can only couple to the third family active neutrino at renormalisable level, i.e. L*N* ⊃ *L*3*H**u**N* + *m**N**N**N*, where all fermion fields are written in a left-handed 2-component convention. This way, the coupling *L*2*H**u**N*, which is required for large atmospheric neutrino mixing, can only arise at the non-renormalisable level. Therefore, it is expected to be suppressed with respect to *L*3*H**u**N*. This seems to be inconsistent with large atmospheric neutrino mixing, at least within the validity of our EFT framework. As anticipated before, this is a consequence of the accidental *U*(2)5 flavour symmetry delivered by the TH model.\nGiven such general considerations, we conclude that in order to obtain neutrino masses and mixing from the type I seesaw mechanism, it is required to add SM singlet neutrinos that carry tri-hypercharges (but whose hypercharges add up to zero). These neutrinos will allow to introduce *U*(2)5-breaking operators similar for all neutrino flavours, providing a mechanism to obtain the adequate neutrino mixing in a natural way. In order to cancel gauge anomalies, the most simple option is that these singlet neutrinos are vector-like. As a consequence, they can obtain their mass from the unspecified vector-like mass terms and from the VEVs of hyperons in the specific model.\nRemarkably, if a given SM singlet neutrino *N*atm carrying non-trivial hypercharges provides the couplings L*N*atm ⊃ *L*3*H**u**N*atm + *L*2*H**u**N*atm, as required to explain atmospheric mixing, then the conjugate neutrino will always couple to *L*3 as $L\\_{3}H\\_{u}\\overline{N}\\_{\\mathrm{atm}}$, but not necessarily to *L*2. As shown in Appendix [sec:General-formalism-for-seesaw], when the terms $L\\_{3}H\\_{u}\\overline{N}\\_{\\mathrm{atm}}$ enter the seesaw mechanism, they lead to a hierarchical effective neutrino mass matrix proportional to the vector-like mass terms, while the terms involving SM singlet neutrinos like *N*atm lead to a neutrino mass matrix where all entries are O(1) and proportional to the VEVs of the hyperons. Therefore, the simplest way to explain the observed pattern of large neutrino mixing requires that the vector-like masses are of similar or smaller order than the VEVs of the hyperons, leading to a *low scale* seesaw mechanism if the VEVs of the hyperons are not very large (which is consistent with current data as discussed in Section [sec:Phenomenology]).\nExample of successful neutrino mixing from the seesaw mechanism[subsec:Example-of-seesaw]\n-----------------------------------------------------------------------------------------\nIn the following, we provide an example scenario which reproduces the observed pattern of neutrino mixing, as a proof of principle. According to the discussion in the previous subsection, in order to implement a type I seesaw mechanism that delivers large neutrino mixing, we need to add vector-like neutrinos that carry tri-hypercharges (but whose hypercharges add up to zero). We also need to introduce hyperons that will provide small Dirac mass terms for the active neutrinos in the form of non-renormalisable operators. Under these considerations, we start by adding the following vector-like neutrino[4](#fn4) and hyperon\n \n$$N\\_{\\mathrm{atm}}^{(0,\\frac{1}{4},-\\frac{1}{4})}\\,,\\qquad\\overline{N}\\_{\\mathrm{atm}}^{(0,-\\frac{1}{4},\\frac{1}{4})}\\,,\\qquad\\phi\\_{\\mathrm{atm}}^{(0,\\frac{1}{4},-\\frac{1}{4})}\\,,$$ \nwhere the charge assignments are chosen to provide large *atmospheric neutrino mixing*. This way, we can write the following non-renormalisable operators along with the Majorana and vector-like masses of *N*atm,\n \n$$\\begin{aligned}\n\\mathcal{L}\\_{N\\_{\\mathrm{atm}}}\\supset & \\frac{1}{\\Lambda\\_{\\mathrm{atm}}}(\\phi\\_{\\mathrm{atm}}L\\_{2}+\\tilde{\\phi}\\_{\\mathrm{atm}}L\\_{3})H\\_{u}N\\_{\\mathrm{atm}}+\\frac{\\phi\\_{\\mathrm{atm}}}{\\Lambda\\_{\\mathrm{atm}}}L\\_{3}H\\_{u}\\overline{N}\\_{\\mathrm{atm}}\\label{eq:Latm}\\\\\n & +\\phi\\_{\\ell23}N\\_{\\mathrm{atm}}N\\_{\\mathrm{atm}}+\\tilde{\\phi}\\_{\\ell23}\\overline{N}\\_{\\mathrm{atm}}\\overline{N}\\_{\\mathrm{atm}}+M\\_{N\\_{\\mathrm{atm}}}\\overline{N}\\_{\\mathrm{atm}}N\\_{\\mathrm{atm}}\\,,\\nonumber \\end{aligned}$$ \nwhere we have ignored the O(1) dimensionless couplings, and the hyperon $\\phi\\_{\\ell23}^{(0,\\frac{1}{2},-\\frac{1}{2})}$ is already present in both Model 1 and Model 2 for the charged fermion sector. In a similar spirit, we introduce another vector-like neutrino and other hyperons in order to obtain large *solar neutrino mixing*\n \n$$N\\_{\\mathrm{sol}}^{(\\frac{1}{4},\\frac{1}{4},-\\frac{1}{2})}\\,,\\qquad\\overline{N}\\_{\\mathrm{sol}}^{(-\\frac{1}{4},-\\frac{1}{4},\\frac{1}{2})}\\,,\\qquad\\phi\\_{\\mathrm{sol}}^{(-\\frac{1}{2},-\\frac{1}{2},1)},\\qquad\\phi\\_{\\nu13}^{(-\\frac{1}{4},-\\frac{1}{4},\\frac{1}{2})}\\,,$$ \nwhich provide the following non-renormalisable operators and mass terms,\n \n$$\\begin{aligned}\n\\mathcal{L}\\_{N\\_{\\mathrm{sol}}}\\supset & \\frac{1}{\\Lambda\\_{\\mathrm{sol}}}(\\phi\\_{e12}L\\_{1}+\\tilde{\\phi}\\_{e12}L\\_{2}+\\phi\\_{\\nu13}L\\_{3})H\\_{u}N\\_{\\mathrm{sol}}+\\frac{\\phi\\_{\\nu13}}{\\Lambda\\_{\\mathrm{sol}}}L\\_{3}H\\_{u}\\overline{N}\\_{\\mathrm{sol}}\\label{eq:Lsol}\\\\\n & +\\phi\\_{\\mathrm{sol}}N\\_{\\mathrm{sol}}N\\_{\\mathrm{sol}}+\\tilde{\\phi}\\_{\\mathrm{sol}}\\overline{N}\\_{\\mathrm{sol}}\\overline{N}\\_{\\mathrm{sol}}+M\\_{N\\_{\\mathrm{sol}}}\\overline{N}\\_{\\mathrm{sol}}N\\_{\\mathrm{sol}}\\,,\\nonumber \\end{aligned}$$ \nwhere we have ignored again the O(1) dimensionless couplings, and the hyperon $\\phi\\_{e12}^{(\\frac{1}{2},-\\frac{1}{2},0)}$ is already present in Model 2 for the charged fermion sector. The hyperon *ϕ**ν*13 (which will eventually populate the (1,3) entry of the effective neutrino mass matrix) is not required to obtain non-zero reactor mixing, which would already arise from the other operators, but it is required in order to have enough free parameters to fit all observed neutrino mixing angles and mass splittings.\nNotice that the vector-like neutrinos *N*atm and *N*sol get contributions to their masses from the VEVs of the hyperons *ϕ*ℓ23 and *ϕ*sol, respectively, which we denote generically as *v*23 since they both take part in the 23-breaking step of Eq. . In addition, *N*atm and *N*sol get contributions to their masses from the unspecified vector-like mass terms, that we generically denote as *M*VL. As shown in Appendix [sec:General-formalism-for-seesaw] (see Eq. ), given that the conjugate neutrinos $\\overline{N}\\_{\\mathrm{atm}}$ and $\\overline{N}\\_{\\mathrm{sol}}$ only couple to the third family, the seesaw formula reveals that the effective neutrino mass matrix *m**ν* receives two main contributions:\n* A contribution proportional to *v*23 which populates all entries of *m**ν* with O(1) terms.\n* A contribution proportional to the vector-like masses *M*VL, which populates only the third row and column entries of *m**ν* with O(1) terms and the others are zero.\nTherefore, if *M*VL ≫ *v*23, then after the seesaw mechanism the light effective Majorana neutrino mass matrix *m**ν* will have only the third row and column being non-zero (to good approximation), which is inconsistent with the observed pattern of neutrino mixing and mass splittings. Instead, if $M\\_{\\mathrm{VL}}\\apprle v\\_{23}$, then the contribution proportional *v*23 dominates the seesaw mechanism. In this case, the conjugate neutrinos $\\overline{N}\\_{\\mathrm{atm}}$ and $\\overline{N}\\_{\\mathrm{sol}}$ become irrelevant for the seesaw mechanism, and *m**ν* can be obtained by considering only the presence of the SM singlet neutrinos *N*atm and *N*sol and applying the seesaw formula. We construct the Dirac and Majorana matrices (ignoring O(1) dimensionless couplings) as\n \n$$m\\_{D}=\\left(\n\\global\\long\\def\\arraystretch{0.7}\\begin{array}{@{}llc@{}}\n & \\multicolumn{1}{c@{}}{\\phantom{\\!\\,}N\\_{\\mathrm{sol}}} & \\phantom{\\!\\,}N\\_{\\mathrm{atm}}\\\\\n\\cmidrule(l){2-3}\\left.L\\_{1}\\right| & \\frac{\\phi\\_{e12}}{\\Lambda\\_{\\mathrm{sol}}} & 0\\\\\n\\left.L\\_{2}\\right| & \\frac{\\tilde{\\phi}\\_{e12}}{\\Lambda\\_{\\mathrm{sol}}} & \\frac{\\phi\\_{\\mathrm{atm}}}{\\Lambda\\_{\\mathrm{atm}}}\\\\\n\\left.L\\_{3}\\right| & \\frac{\\phi\\_{\\nu13}}{\\Lambda\\_{\\mathrm{sol}}} & \\frac{\\tilde{\\phi}\\_{\\mathrm{atm}}}{\\Lambda\\_{\\mathrm{atm}}}\n\\end{array}\\right)H\\_{u}\\,,\\qquad M\\_{N}=\\left(\n\\global\\long\\def\\arraystretch{0.7}\\begin{array}{@{}llc@{}}\n & \\multicolumn{1}{c@{}}{\\phantom{\\!\\,}N\\_{\\mathrm{sol}}} & \\phantom{\\!\\,}N\\_{\\mathrm{atm}}\\\\\n\\cmidrule(l){2-3}\\left.\\;\\,N\\_{\\mathrm{sol}}\\right| & \\phi\\_{\\mathrm{sol}} & 0\\\\\n\\left.N\\_{\\mathrm{atm}}\\right| & 0 & \\phi\\_{\\ell23}\n\\end{array}\\right)\\,.\\label{seesaw}$$ \nWe could have included the *U*(1)*Y*3 singlet neutrino *N*(0, 0, 0), but as discussed in the previous subsection, its contributions to the Weinberg operator may be suppressed by its large Majorana mass *m**N*, resulting in possibly negligible contributions to the seesaw mechanism. We are therefore free to assume that such a neutrino *N*(0, 0, 0), if exists, is in any case decoupled from the seesaw, while the atmospheric and solar SM singlet neutrinos *N*atm and *N*sol could yield dominant and subdominant contributions, resulting in a natural normal neutrino mass hierarchy as in sequential dominance.\nMore generally, after applying the seesaw formula using Eq. , we obtain the light effective Majorana neutrino mass matrix (ignoring again O(1) dimensionless couplings) as\nm & mDMN-1mD \n& =(\nccc & & \n & + & + \n & + & +\n)HuHu.\nGiven the symmetry breaking pattern of the model shown in Eq. , we take ⟨*ϕ**e*12⟩ ≃ O(*v*12) and ⟨*ϕ**ν*13⟩ ≈ ⟨*ϕ*ℓ23⟩ ≈ ⟨*ϕ*atm⟩ ≈ ⟨*ϕ*sol⟩ ≈ O(*v*23)[5](#fn5). Motivated by our discussion of the charged fermion sector (see Section [sec:Charged-fermion-masses-mixing]), we consider the relation *v*23/*v*12 ≃ *λ*. By inserting such VEVs, we obtain\n \n$$m\\_{\\nu}\\simeq\\left(\\begin{array}{ccc}\n0 & 0 & 0\\\\\n0 & 1 & 1\\\\\n0 & 1 & 1\n\\end{array}\\right)v\\_{23}\\frac{H\\_{u}H\\_{u}}{\\Lambda\\_{\\mathrm{atm}}^{2}}\\,+\\left(\\begin{array}{ccc}\n1 & 1 & \\lambda\\\\\n1 & 1 & \\lambda\\\\\n\\lambda & \\lambda & \\lambda^{2}\n\\end{array}\\right)v\\_{23}\\frac{H\\_{u}H\\_{u}}{\\lambda^{2}\\Lambda\\_{\\mathrm{sol}}^{2}}\\,.$$ \nIf Λsol = Λatm, we observe that there exists a mild hierarchy of order *λ*2 between the 12 and 23 sectors in the matrix above. Considering the dimensionless coefficients that we have ignored so far, the numerical diagonalisation of *m**ν* would require some parameters of O(0.01) in order to explain the observed neutrino mixing angles and mass splittings. The situation can be improved if we assume a mild hierarchy between cut-off scales Λatm/Λsol ≃ *λ*, obtaining to leading order for each entry (ignoring dimensionless coefficients),\n \n$$m\\_{\\nu}\\simeq\\left(\\begin{array}{ccc}\n1 & 1 & \\lambda\\\\\n1 & 1 & 1\\\\\n\\lambda & 1 & 1\n\\end{array}\\right)v\\_{23}\\frac{v\\_{\\mathrm{SM}}^{2}}{\\Lambda\\_{\\mathrm{atm}}^{2}}\\,,$$ \nwhere we have introduced the SM VEV as ⟨*H**u*⟩ = *v*SM (ignoring the factor $1/\\sqrt{2}$). Considering now the dimensionless coefficients in the matrix above, we find that numerical diagonalisation can accommodate all the observed neutrino mixing angles and mass splittings with O(1) parameters, and we are able to reproduce both normal and inverted ordered scenarios.\nNotice that we have been driven to a scenario where the vector-like neutrinos get Majorana masses from the VEVs of hyperons in the model. Furthermore, the vector-like masses necessarily have to be of the same or smaller order than the VEVs of the hyperons in order to explain the observed pattern of neutrino mixing. Therefore, in the particular example included in this section, the vector-like neutrinos get a mass at the scale *v*23 of the 23-breaking step in Fig. , which could happen at a relatively low scale as we shall see in Section [sec:Phenomenology]. As a consequence, the vector-like neutrinos involved in the seesaw mechanism are expected to be relatively light, and the high energy cut-offs of the EFT Λatm and Λsol are expected to provide most the suppression for tiny neutrino masses. As anticipated before, due to the *U*(2)5 flavour symmetry provided by the TH model, we have been driven to a low scale seesaw in order to predict the observed pattern of neutrino mixing.\nPhenomenology[sec:Phenomenology]\n================================\nCouplings of the heavy *Z*ʹ bosons to fermions [subsec:CouplingsZprime]\n-----------------------------------------------------------------------\nIn Sections [sec:Charged-fermion-masses-mixing] and [sec:Neutrino-masses-and-Mixing] we have discussed examples of *U*(1)*Y*3 models which provide a compelling description of all fermion masses and mixings, and we have highlighted model-independent features which are intrinsic to the *U*(1)*Y*3 framework. Under well-motivated arguments, we have assumed that the symmetry breaking pattern of the *U*(1)*Y*3 group down to the SM is described by Eq. and Fig. [fig:-Multiscale-picture-flavour], in such a way that at a high scale *v*12, the group *U*(1)*Y*1 × *U*(1)*Y*2 is broken down to its diagonal subgroup. The remaining group *U*(1)*Y*1 + *Y*2 × *U*(1)*Y*3 is broken down to SM hypercharge at a lower scale *v*23. The hierarchy between the scales *v*12 and *v*23 generally plays a role on the origin of flavour hierarchies in the SM, although in specific models we have found that a mild hierarchy *v*23/*v*12 ≃ *λ* is enough.\nA massive gauge boson *Z*ʹ12 is predicted to live at the higher scale *v*12, displaying *intrinsically* flavour non-universal couplings to the first two families of SM fermions. Similarly, another massive boson *Z*ʹ23 lives at the lower scale *v*23. The pattern of symmetry breaking is such that *Z*ʹ23 has flavour universal couplings to first and second family fermions, while the couplings to the third family are intrinsically different. In the following, we include the coupling matrices in family space (from the covariant derivatives in Appendices [sec:High-scale-symmetry] and [sec:Low-scale-symmetry]), ignoring fermion mass mixing,\n& Z’12YL,RL,R(\nccc -g112 & 0 & 0 \n0 & g212 & 0 \n0 & 0 & 0\n)L,RZ’12, &\n12=,[eq:Z12couplings]\n& Z’23YL,RL,R(\nccc -g1223 & 0 & 0 \n0 & -g1223 & 0 \n0 & 0 & g323\n)L,RZ’23, &\n23=,[eq:Z23couplings]\nwhere *Y**ψ**L*, *R* is the SM hypercharge of *ψ**L*, *R*[6](#fn6), where *ψ* is a 3-component column vector containing the three families *ψ* = *u**i*, *d**i*, *e**i*, *ν**i*. Explicitly, *ψ**L* = *u**L**i*, *d**L**i*, *e**L**i*, *ν**L**i* with *Y**ψ**L* = 1/6, 1/6, − 1/2, − 1/2, and *ψ**R* = *u**R**i*, *d**R**i*, *e**R**i* with *Y**ψ**R* = 2/3, − 1/3, − 1, respectively, ignoring couplings to the SM singlet neutrinos discussed in the previous section[7](#fn7).\nIncluding fermion mass mixing, we would have *ψ**L*, *R* = *V**ψ**L*, *R**ψ̂**L*, *R* with *ψ̂**L*, *R* containing the mass eigenstates and *V**ψ**L*, *R* being the *non-generic* mixing matrices obtained after diagonalising the Yukawa matrices for a given model. Notice that the couplings to *right-handed* fermions are larger since their hypercharges are generally larger in magnitude than those of left-handed fermions. The SM hypercharge gauge coupling *g**Y*(*M**Z*) ≃ 0.36 is entangled to the *g**i* couplings via the relations\n \n$$g\\_{Y}=\\frac{g\\_{12}g\\_{3}}{\\sqrt{g\\_{12}^{2}+g\\_{3}^{2}}}\\,,\\qquad\\qquad\\qquad\\qquad g\\_{12}=\\frac{g\\_{1}g\\_{2}}{\\sqrt{g\\_{1}^{2}+g\\_{2}^{2}}}\\,.\\label{eq:gauge\\_couplings}$$ \nThe expressions above reveal a lower bound on the gauge couplings $g\\_{i}\\apprge g\\_{Y}$. Throughout this work we have considered a bottom-up approach where the *U*(1)*Y*3 model is just the next step in our understanding of Nature, which reveals information about the origin of flavour, but nevertheless is an EFT remnant of a more fundamental UV-complete theory. In this spirit, we have studied the RGE evolution of the gauge couplings *g**i*, obtaining that for *g**i*(TeV) ≃ 1 the model can be extrapolated to the Planck scale (and beyond). Instead, for *g**i*(TeV) ≃ 2, a Landau pole is found at a scale O(104 TeV), which anyway seems like a reasonable scale for an UV embedding, given that we expect the cut-off scale of the effective Yukawa operators (see Section [sec:Charged-fermion-masses-mixing]) to be around O(102 TeV) in order to provide the required suppression for charged fermion masses. Therefore, in order to protect the perturbativity of the model, we avoid considering *g**i* > 2 in the phenomenological analysis. Nevertheless, we highlight a natural scenario where the three gauge couplings have a similar size $g\\_{1}\\simeq g\\_{2}\\simeq g\\_{3}\\simeq\\sqrt{3}g\\_{Y}$, which could be connected to a possible *gauge unification*. This benchmark is depicted as a dashed horizontal line in Figs. [fig:Zp12plot] and [fig:Zp23plot].\nThe high scale boson *Z*ʹ12\n---------------------------\nIn any implementation of the *U*(1)*Y*3 model, *Z*ʹ12 is expected to mediate sizable tree-level transitions between first and second generation left-handed quarks, either in the up or down sector depending on the alignment of the CKM matrix predicted by the specific model. Furthermore, our analysis in Section [sec:Charged-fermion-masses-mixing] reveals that *U*(1)*Y*3 models generally predict non-vanishing charged lepton mixing and mixing among right-handed quarks. This way, contributions to *K*0 − *K̄*0 and *D*0 − *D̄*0 meson mixing, along with CLFV processes such as *μ* → *e**γ*, have the potential to push the scale *v*12 far above the TeV.\n\nBeing more specific, for Model 1 described in Section [subsec:Model-1:-Minimal] we find the stringent bounds over *v*12 to come from the scalar and coloured operator (*s̄**L**α**d**R**β*)(*s̄**R**β**d**L**α*) obtained after integrating out *Z*ʹ12 at tree-level (and applying a Fierz rearrangement), which contributes to *K*0 − *K̄*0 mixing. Model 1 predicts the up and down left-handed mixings to be similar up to dimensionless couplings, which must therefore play some role in the alignment of the CKM matrix. In either case, mildly suppressed right-handed *s* − *d* mixing *s*12*d**R* ≃ O(*λ*2) is predicted. If *V**u**s* originates mostly from the down sector, then *K*0 − *K̄*0 mixing imposes the stringent bound *M**Z*ʹ12 > 170 TeV for gauge couplings of O(0.5). Instead, if the dimensionless coupling provides a mild suppression of O(0.1) in left-handed *s* − *d* mixing, such that *V**u**s* originates mostly from the up sector, then the bound is relaxed to *M**Z*ʹ12 > 55 TeV. We find bounds from *D*0 − *D̄*0 mixing to be always weaker, even if *V**u**s* originates from the up sector, since right-handed up mixing is strongly suppressed in Model 1.\nIn contrast with Model 1, Model 2 described in Section [subsec:Model-2:-Five] provides a more predictive scenario where *V**u**s* originates unambiguously from the down sector. Here right-handed quark mixing is more suppressed, obtaining *s*12*d**R* ≃ O(*λ*5). Nevertheless, *K*0 − *K̄*0 mixing still imposes the strongest bounds over the parameter case, as can be seen in Fig. [fig:Zp12plot]. In this case, the lower bound over the mass of *Z*ʹ12 can be as low as 10-50 TeV, depending on the values of the gauge couplings. We find the CLFV process *μ* → *e**γ* to provide a slightly weaker bound over the parameter space, because charged lepton mixing is generally suppressed with respect to quark mixing in Model 2. We find the bound from *μ* → 3*e* to be very similar to the bound from *μ* → *e**γ*.\nThe low scale boson *Z*ʹ23\n--------------------------\nGiven that the high scale symmetry breaking can be as low as 10-20 TeV for specific models, and considering the hierarchy of scales *v*23/*v*12 ≃ *λ* suggested by these specific models in Section [sec:Charged-fermion-masses-mixing], it is possible to find the low scale breaking *v*23 near the TeV. Since *Z*ʹ23 features flavour universal couplings to the first and second families, the stringent bounds from *K*0 − *K̄*0 mixing and *μ* → *e**γ* are avoided, in the spirit of the *GIM mechanism*. This way, *Z*ʹ23 can live at the TeV scale, within the reach of the LHC and future colliders.\nAny implementation of the *U*(1)*Y*3 model predicts small *mixing* between *Z*ʹ23 and the SM *Z* boson given by the mixing angle (see Appendix [sec:Low-scale-symmetry])[8](#fn8)\n \n$$\\mathrm{sin}\\theta\\_{Z-Z'\\_{23}}=\\frac{g\\_{3}\\mathrm{cos}\\theta\\_{23}}{\\sqrt{g\\_{Y}^{2}+g\\_{L}^{2}}}\\left(\\frac{M\\_{Z}^{0}}{M\\_{Z'\\_{23}}^{0}}\\right)^{2}\\,,$$ \nwhere *M**Z*0 and *M**Z*ʹ230 are the masses of the *Z* and *Z*ʹ23 bosons in the absence of mixing, respectively, and *g**L* is the gauge coupling of *S**U*(2)*L*. This mixing leads to a *small shift* on the mass of the *Z* boson, which has an impact on the so-called *ρ* parameter\n \n$$\\rho=\\frac{M\\_{W}^{2}}{M\\_{Z}^{2}\\cos^{2}\\theta\\_{W}}=\\frac{1}{1-g\\_{3}^{2}\\cos^{2}\\theta\\_{23}\\left(\\frac{v\\_{\\mathrm{SM}}}{2M\\_{Z'\\_{23}}^{0}}\\right)^{2}}\\,,$$ \nwhich is predicted as *ρ* = 1 at tree-level in the SM. This is a consequence of custodial symmetry in the Higgs potential, which is explicitly broken in our model via *Z* − *Z*ʹ23 mixing leading to deviations in *ρ*. The fact that in our model *M**Z* is always shifted to smaller values leads to *ρ* > 1 at tree-level. Given that *M**Z* is commonly an input experimental parameter of the SM used in the determination of *g**Y* and *g**L*, the downward shift of *M**Z* with respect to the SM prediction would be seen from the experimental point of view as an upward shift of *M**W* with respect to the SM prediction. Nevertheless, the experimental picture of *M**W* is puzzling after the recent measurement by CDF. This measurement points towards *M**W* being larger than the SM prediction with high significance, but it is in tension with the combination of measurements by LHC, LEP and Tevatron D0. Neglecting the recent CDF measurement for the moment, current data[9](#fn9) provides *ρ* = 1.0003 ± 0.0005 (assuming that both the oblique parameters *T* and *S* are non-zero, as we expect in our model). We obtain the approximate bound *g*3/*M**Z*ʹ < 3.1 TeV at 95% CL, which translates to an approximate bound over the mixing angle of sin*θ**Z* − *Z*ʹ23 < 0.001.\n\n*Z* − *Z*ʹ23 mixing also shifts the couplings of the *Z* boson to fermions, leading to an important impact over *Z*-pole electroweak precision observables (EWPOs) if *Z*ʹ23 lives at the TeV scale. We find bounds coming from tests of *Z* boson lepton universality and flavour-violating *Z* decays not to be competitive with the bound from *ρ*. The electron asymmetry parameter *A**e*, which already deviates from the SM by almost 2*σ*, is expected to deviate further in our model. Nevertheless, we expect our model to improve the fit of *A**b*FB, which is in tension with the SM prediction by more than 2*σ*. In conclusion, the global effect of our model over EWPOs can only by captured by performing a global fit, which we leave for future work. In this direction, a dedicated phenomenological analysis along the lines of would be much needed. Global fits of EWPOs in the context of other *Z*ʹ models have been performed in the literature, see e.g. , which obtain 95% CL maximum values of sin*θ**Z* − *Z*ʹ ranging from 0.002 to 0.0006 depending on the model. We expect our model to lie on the more restrictive side of that range. In principle, our model could explain the anomalous CDF *M**W* measurement, however in this case we expect the contributions to other EWPOs to be intolerably large, worsening the global fit.\nThe massive *Z*ʹ23 boson has sizable couplings to light quarks and light charged leptons unless *g*3 is very large, which we do not expect based on naturalness arguments and also to protect the extrapolation of the model in the UV, as discussed in Section [subsec:CouplingsZprime]. Consequently, on general grounds we expect a significant production of a TeV-scale *Z*ʹ23 at the LHC, plus a sizable branching fraction to electrons and muons. We have prepared the UFO model of *Z*ʹ23 using `FeynRules`, and then we have computed the *Z*ʹ23 production cross section for 13 TeV *p**p* collisions using `Madgraph5` with the default PDF `NNPDF23LO`. We estimated analytically the branching fraction to electrons and muons, and we computed the total decay width via the narrow width approximation. We confront our results with the limits from the most recent dilepton resonance searches by ATLAS and CMS in order to obtain 95% CL exclusion bounds. The bounds from ditau and ditop searches turn out not to be competitive even for the region of large *g*3, where the bound from the *ρ* electroweak parameter is stronger. Our results are depicted as the black-shaded region in Fig. [fig:Zp23plot]. As expected, the bounds become weaker in the region *g*3 > 1 where the couplings to light fermions become mildly suppressed. In the region of small *g*3 we find the opposite behavior, such that LHC limits can exclude *Z*ʹ23 as heavy as 6-7 TeV. After combining the LHC exclusion with the bounds coming from the *ρ* electroweak parameter, we conclude that we can find *Z*ʹ23 as light as 3.5 TeV for *g*3 = 1, while for the benchmark $g\\_{i}\\simeq\\sqrt{3}g\\_{Y}$ we obtain $M\\_{Z'\\_{23}}\\apprge5\\;\\mathrm{TeV}$.\ncc *Z*ʹ23 decay mode & B \n*t̄**t* & ∼ 0.28 \n*ū**u* + *c̄**c* & ∼ 0.14 \n*t̄**c* + *c̄**t* & ∼ 10− 4 \n*b̄**b* & ∼ 0.08 \n*d̄**d* + *s̄**s* & ∼ 0.04 \n*b̄**s* + *s̄**b* & ∼ 10− 4 \n*τ*+*τ*− & ∼ 0.25 \n*e*+*e*− + *μ*+*μ*− & ∼ 0.12 \n*τ*+*μ*− + *τ*−*μ*+ & ∼ 10− 5 \n*ν̄**ν* & ∼ 0.08 \nGiven that *Z*ʹ23 has sizable couplings to electrons, we have studied the bounds over contact interactions obtained at LEP. For our model, the most competitive bounds arise from contact interactions involving only electrons. Assuming vector-like interactions, the bounds by LEP are only sensitive to regions with very small *g*3, but can exclude *Z*ʹ23 masses beyond 10 TeV. However, we expect this bound to be slightly overestimated for our model, since the interactions of *Z*ʹ23 are not exactly vector-like due to the different hypercharge of *e**L* and *e**R*, as depicted in Eq. . Nevertheless, the bounds over chiral operators are much weaker than the bound over the vector-like operator, and a dedicated reanalysis of the data would be required in order to obtain the proper bound for our model, which is beyond the scope of this work. Therefore, we prefer to be conservative and depict the largest bound of the vector-like operator as the purple region in Fig. [fig:Zp23plot].\nWe have also considered implications for *B*-physics. The heavy boson *Z*ʹ23 has a sizable left-handed *b**L**s**L* coupling and an approximately vector-like and universal coupling to electron and muon pairs. Given these features, a *Z*ʹ23 with a mass of 2 TeV mediates a meaningful contribution to the effective operator $\\mathcal{O}\\_{9}^{\\ell\\ell}=(\\bar{s}\\_{L}\\gamma\\_{\\mu}b\\_{L})(\\bar{\\ell}\\gamma^{\\mu}\\ell)$ (with ℓ = *e*, *μ*), where sizable NP contributions are preferred according to the most recent global fits, without contributing to the SM-like *R**K*( \\* ) ratios. However, as depicted in Fig. [fig:Zp23plot], the region where the model could address the anomalies in *b* → *s**μ**μ* data are in tension with the bounds obtained by dilepton searches, as expected for a *Z*ʹ which has sizable couplings to light quarks. Nevertheless, we can see that a relevant *C*9ℓℓ ∼ 0.1 can be obtained for a heavier *Z*ʹ23 in the region where *g*3 < 0.5, as in this region couplings to light fermions are enhanced.\nB(*B**s* → *μ**μ*) is also enhanced above the SM prediction due to both *Z*ʹ23 and *Z* exchange diagrams. In the region of small *g*3 the *Z*ʹ23 exchange dominates, while for large *g*3 the *Z* exchange dominates. As anticipated before, the couplings of the *Z*ʹ23 boson to muons are approximately, but not completely, vector-like. Therefore, a small contribution to the operator O10*μ**μ* = (*s̄**L**γ**μ**b**L*)(*μ̄**γ**μ**γ*5*μ*) is generated in the region of small *g*3, where *Z*ʹ23 couplings to muons are larger, leading to *Z*ʹ23 exchange being dominant. However, in the region of large *g*3, the flavour-violating coupling *s̄**L**b**L**Z*ʹ23 mixes into *s̄**L**b**L**Z* via *Z* − *Z*ʹ23 mixing. This provides an effective contribution to O10*μ**μ* mediated by the *Z* boson that enhances B(*B**s* → *μ**μ*) above the SM prediction. Overall, the region excluded at 95% CL by the current HFLAV average (including the latest measurement by CMS ) is depicted as blue-shaded in Fig. [fig:Zp23plot]. It is clear that the resulting bound is not currently competitive with that from the *ρ* electroweak parameter which constrains the size of *Z* − *Z*ʹ23 mixing.\nThe flavour-violating structure of *Z*ʹ23 fermion couplings leads to sizable contributions to *B**s* − *B̄**s* meson mixing and CLFV processes involving *τ* → *μ* transitions and *τ* → *e* transitions, although well below existing experimental limits. *τ* → *μ*(*e*) transitions arise from mixing angles connected to the flavour hierarchy O(*m*2/*m*3) (O(*m*1/*m*3)), see Section [subsec:Lessons-from-the]. As an example, in Fig. [fig:Zp23plot] we depict the contour for B(*τ* → 3*μ*) = 10− 12. We find *τ* → *μ**γ* to be more competitive than *τ* → 3*μ* only in the region *g*3 > 1.\nBeyond indirect detection, in the near future *Z*ʹ23 could be directly produced at the LHC, HL-LHC and future colliders such as FCC or a high energy muon collider. The particular pattern of *Z*ʹ23 fermion couplings will allow to disentangle our model from all other proposals. For the natural benchmark $g\\_{1}\\simeq g\\_{2}\\simeq g\\_{3}\\simeq\\sqrt{3}g\\_{Y}$, *Z*ʹ23 preferentially decays to top pairs and ditaus, as can be seen in Table [tab:Main-decay-modes]. Furthermore, *Z*ʹ23 preferentially couples and decays to *right-handed* charged fermions, given their larger hypercharge with respect to left-handed charged fermions. Similarly, decays to down-type quarks are generally suppressed with respect to (right-handed) up-type quarks and charged leptons, given the smaller hypercharge of the former. Due to the modification of EWPOs, our model can also be tested in an electroweak precision machine such as FCC-*e**e*. An alternative way of discovery would be the detection of the hyperon scalars breaking the *U*(1)*Y*3 group down to SM hypercharge, however we leave a study about the related phenomenology for future work. In the same spirit, the model naturally predicts SM singlet neutrinos which could be as light as the TeV scale, with phenomenological implications yet to be explored in a future work.\nConclusions[sec:Conclusions]\n============================\nWe have proposed a tri-hypercharge (TH) embedding of the Standard Model, based on assigning a separate gauged weak hypercharge to each family. The idea is that each fermion family *i* only carries hypercharge under a corresponding *U*(1)*Y**i* factor. This ensures that each family transforms differently under the TH gauge group *U*(1)*Y*3, which avoids the family repetition of the SM and provides the starting point for a theory of flavour.\nThe three family specific hypercharge groups are spontaneously broken in a cascade symmetry breaking down to SM hypercharge. We have motivated a particular symmetry breaking pattern, where in a first step *U*(1)*Y*1 × *U*(1)*Y*2 is broken down to its diagonal subgroup at a high scale *v*12. The remaining group *U*(1)*Y*1 + *Y*2 × *U*(1)*Y*3 is broken down to SM hypercharge at a scale *v*23. This symmetry breaking pattern sequentially recovers the accidental flavour symmetry of the SM, providing protection versus FCNCs that allows the NP scales to be relatively low. Dynamics connected to the scale *v*12 play a role in the origin of the family hierarchy *m*1/*m*2, while dynamics connected to the scale *v*23 play a role in the origin of *m*2/*m*3. The hierarchy of scales *v*23/*v*12 generally plays a role on the origin of flavour hierarchies as well, although we have found that a mild hierarchy *v*23/*v*12 ≃ *λ* is enough for specific implementations of the model, where *λ* ≃ 0.224 is the Wolfenstein parameter.\nAssuming that the SM Higgs only carries third family hypercharge, then only the third family Yukawa couplings are allowed at renormalisable level. This explains the heaviness of the third family, the smallness of *V**c**b* and *V**u**b* quark mixing, and delivers an accidental *U*(2)5 flavour symmetry in the Yukawa sector acting on the light families, which provides a reasonable first order description of the SM spectrum. However, *U*(2)5 does not explain the hierarchical heaviness of the top quark with respect to the bottom and tau fermions. Furthermore, we have proven that the model generates a similar mass hierarchy for all charged sectors, being unable to explain the heaviness of the charm quark with respect to the strange and muon without small couplings. We have motivated the addition of a second Higgs doublet as a natural and elegant solution, which allows a more natural description of the hierarchies between the different charged fermion sectors.\nWe have explored the capabilities of the *U*(1)*Y*3 model to explain the observed hierarchies in the charged fermion sector, via the addition of non-renormalisable operators containing *U*(1)*Y*3-breaking scalars which act as small breaking effects of *U*(2)5. After extracting model-independent considerations from the spurion formalism, we have presented example models where all charged fermion masses and mixings are addressed. Following a similar methodology, we have studied the origin of neutrino masses and mixing in the TH model. We have shown that due to the *U*(1)*Y*3 gauge symmetry, the implementation of a type I seesaw mechanism naturally leads to a low scale seesaw, where the SM singlet neutrinos in the model may be as light as the TeV scale. We have provided an example model compatible with the observed pattern of neutrino mixing.\nFinally, we have performed a preliminary exploration of the phenomenological implications and discovery prospects of the *U*(1)*Y*3 theory of flavour. The heavy gauge boson *Z*ʹ12 arising from the 12-breaking displays completely flavour non-universal couplings to fermions, and generally contributes to Δ*F* = 2 and CLFV processes. The size of the most dangerous contributions are however model-dependent. In selected specific models provided in this manuscript, we have found that the most dangerous contributions to *K*0 − *K̄*0 mixing and *μ* → *e**γ* are strongly suppressed, allowing for *Z*ʹ12 to be as light as 10-50 TeV. Therefore, the lightest gauge boson *Z*ʹ23 arising from the 23-breaking can live at the TeV scale, within the reach of LHC and future colliders, since *Z*ʹ23 avoids bounds from *K*0 − *K̄*0 mixing and *μ* → *e**γ* thanks to an accidental GIM mechanism for light fermions.\nWe find the gauge boson *Z*ʹ23 to have a rich low energy phenomenology: mixing with the SM *Z* boson leads to implications for the *W* boson mass and EWPOs, plus we expect contributions to flavour-violating processes involving the third family, such as *τ* → 3*μ*(*e*) and *B**s* − *B̄**s* meson mixing. After our preliminary analysis, we find that current data allows *Z*ʹ23 to be as light as 3-4 TeV in some regions of the parameter space, the strongest bounds coming from dilepton searches at LHC along with the contribution to the *ρ* electroweak parameter. In the case of discovery, the particular pattern of *Z*ʹ23 couplings and decays to fermions will allow to disentangle our model from all other proposals. However, most of the phenomenological consequences are yet to be explored in detail: a global fit to EWPOs and flavour observables will allow to properly confront our model versus current data. An alternative way of discovery would be the detection of the Higgs scalars (hyperons) breaking the *U*(1)*Y*3 down to SM hypercharge, however we leave a discussion about the related phenomenology for future work. In the same spirit, the model naturally predicts SM singlet neutrinos which could be as light as the TeV, with phenomenological implications yet to be explored. The tri-hypercharge gauge group may be the first step towards understanding the origin of three fermion masses in Nature, the hierarchical charged fermion masses and CKM mixing, revealing the existence of a flavour non-universal gauge structure encoded in Nature at energies above the electroweak scale.\nAcknowledgements\n================\nWe would like to thank the CERN Theory group for hospitality and financial support during an intermediate stage of this work. This work has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under Marie Skłodowska-Curie grant agreement HIDDeN European ITN project (H2020-MSCA-ITN-2019//860881-HIDDeN). SFK acknowledges the STFC Consolidated Grant ST/L000296/1.\nGeneral formalism for the seesaw mechanism[sec:General-formalism-for-seesaw]\n============================================================================\nIn Section [sec:Neutrino-masses-and-Mixing] we have motivated that in order to explain the observed pattern of neutrino mixing via a type I seesaw mechanism, it is required to add SM singlet neutrinos which carry family hypercharges (but are SM singlets). Therefore, the latter have to be vector-like in order to cancel gauge anomalies. In all generality, such vector-like neutrinos can get contributions to their mass from the unspecified vector-like mass terms and from Majorana masses given by VEVs of hyperons breaking the *U*(1)*Y*3 group. Both the SM singlet neutrinos *N* and the conjugate partners $\\overline{N}$ (both LH in our convention) can couple to the active neutrinos *ν* through non-renormalisable operators involving the hyperons. This way, one can write the general neutrino matrix,\n \n$$M\\_{\\nu}=\\left(\n\\global\\long\\def\\arraystretch{0.7}\\begin{array}{@{}llcc@{}}\n & \\multicolumn{1}{c@{}}{\\phantom{\\!\\,}\\nu} & \\phantom{\\!\\,}\\overline{N} & \\phantom{\\!\\,}N\\\\\n\\cmidrule(l){2-4}\\left.\\:\\,\\nu\\right| & 0 & m\\_{D\\_{L}} & m\\_{D\\_{R}}\\\\\n\\left.\\overline{N}\\right| & m\\_{D\\_{L}}^{\\mathrm{T}} & M\\_{L} & M\\_{LR}\\\\\n\\left.N\\right| & m\\_{D\\_{R}}^{\\mathrm{T}} & M\\_{LR}^{\\mathrm{T}} & M\\_{R}\n\\end{array}\\right)\\equiv\\left(\\begin{array}{cc}\n0 & m\\_{D}\\\\\nm\\_{D}^{\\mathrm{T}} & M\\_{N}\n\\end{array}\\right)\\,.\\label{eq:Full\\_Mnu}$$ \nIn order to match Eq. to our example seesaw model of Section [subsec:Example-of-seesaw], we define *ν* as a 3-component vector containing the weak eigenstates of active neutrinos, while *N* and $\\overline{N}$ are 2-component vectors containing the SM singlet *N* and conjugate neutrinos $\\overline{N}$, respectively. Similarly, in the following we define each of the sub-matrices of Eq. in terms of the matter content of Section [subsec:Example-of-seesaw], ignoring the O(1) dimensionless couplings:\n& & & mDL=(\n@llc@ & & \n(l)2-3.L1| & 0 & 0 \n.L2| & 0 & 0 \n.L3| & &\n)Hu, & & mDR=(\n@llc@ & & N \n(l)2-3.L1| & & 0 \n.L2| & & \n.L3| & &\n)Hu,[eq:mDLmDR] \n \n& & & ML=(\n@llc@ & & \n(l)2-3.| & & 0 \n.| & 0 & 23\n)v2322, & & MR(\n@llc@ & & N \n(l)2-3.N| & & 0 \n.N| & 0 & 23\n)v2322,[eq:MLMR]\n \n$$M\\_{LR}=\\left(\n\\global\\long\\def\\arraystretch{0.7}\\begin{array}{@{}llc@{}}\n & \\multicolumn{1}{c@{}}{\\phantom{\\!\\,}N\\_{\\mathrm{sol}}} & \\phantom{\\!\\,}N\\_{\\mathrm{atm}}\\\\\n\\cmidrule(l){2-3}\\left.\\:\\:\\overline{N}\\_{\\mathrm{sol}}\\right| & M\\_{N\\_{\\mathrm{sol}}} & 0\\\\\n\\left.\\overline{N}\\_{\\mathrm{atm}}\\right| & 0 & M\\_{N\\_{\\mathrm{atm}}}\n\\end{array}\\right)\\approx M\\_{\\mathrm{VL}}\\mathbb{I}\\_{2\\times2}\\,,\\label{eq:MLR}$$ \nwhere I2 × 2 is the 2 × 2 identity matrix. In Eqs. and above, we have considered the following approximations and assumptions:\n* We have neglected O(1) dimensionless couplings generally present for each non-zero entry of Eq. . With this consideration, we find *M**L* = *M**R* after the hyperons develop their VEVs. Furthermore, since the two hyperons appearing in *M**L* and *M**R* participate in the 23-breaking step of Eq. , we have assumed that they both develop a similar VEV ⟨*ϕ*ℓ23⟩ ≈ ⟨*ϕ*sol⟩ ≈ O(*v*23). For simplicity we take them to be equal, although the same conclusions hold as long as they just differ by O(1) factors, as naturally expected.\n* For simplicity, we have assumed a similar vector-like mass for both neutrinos in Eq. , i.e. *M**N*sol ≈ *M**N*atm ≡ *M*VL.\nDirac-type masses in *m**D**L*, *R* may be orders of magnitude smaller than the electroweak scale, because they arise from non-renormalisable operators proportional to the SM VEV. In contrast, the eigenvalues of *M**N* are not smaller than O(*v*23), which is at least TeV. Therefore, the condition *m**D* ≪ *M**N* is fulfilled in Eq. and we can safely apply the seesaw formula as\n \n$$\\begin{aligned}\nm\\_{\\nu} & \\simeq m\\_{D}M\\_{N}^{-1}m\\_{D}^{\\mathrm{T}}\\label{eq:Seesaw\\_General}\\\\\n & =\\left(\\begin{array}{cc}\nm\\_{D\\_{L}} & m\\_{D\\_{R}}\\end{array}\\right)\\left(\\begin{array}{cc}\nv\\_{23} & -M\\_{\\mathrm{VL}}\\\\\n-M\\_{\\mathrm{VL}} & v\\_{23}\n\\end{array}\\right)\\left(\\begin{array}{c}\nm\\_{D\\_{L}}^{\\mathrm{T}}\\\\\nm\\_{D\\_{R}}^{\\mathrm{T}}\n\\end{array}\\right)\\frac{1}{v\\_{23}^{2}-M\\_{\\mathrm{VL}}^{2}}\\nonumber \\\\\n & =\\left[m\\_{D\\_{L}}m\\_{D\\_{L}}^{\\mathrm{T}}v\\_{23}-m\\_{D\\_{L}}m\\_{D\\_{R}}^{\\mathrm{T}}M\\_{\\mathrm{VL}}-m\\_{D\\_{R}}m\\_{D\\_{L}}^{\\mathrm{T}}M\\_{\\mathrm{VL}}+m\\_{D\\_{R}}m\\_{D\\_{R}}^{\\mathrm{T}}v\\_{23}\\right]\\frac{1}{v\\_{23}^{2}-M\\_{\\mathrm{VL}}^{2}}\\,.\\nonumber \\end{aligned}$$ \nGiven the structure of *m**D**L* and *m**D**R* in Eq. ([eq:mDLmDR]), the products above involving *m**D**L* lead to a hierarchical *m**ν* matrix where only the entries in the third row and column are populated and the others are zero. In contrast, the product *m**D**R**m**D**R*T provides a matrix *m**ν* where all entries are populated. Therefore, if *M*VL ≫ *v*23, then the effective neutrino matrix becomes hierarchical, rendering impossible to explain the observed pattern of neutrino mixing and mass splittings with O(1) parameters. Instead, if *M*VL is of the same order or smaller than *v*23, i.e. $M\\_{\\mathrm{VL}}\\apprle v\\_{23}$, then the resulting matrix is in any case a matrix where all entries are populated, which has the potential to explain the observed patterns of neutrino mixing. As shown in the main text, Section [subsec:Example-of-seesaw], by further assuming the mild hierarchies *v*23/*v*12 ≃ *λ* and Λsol/Λatm ≃ *λ* the product *m**D**R**m**D**R*T above provides a texture for *m**ν* that can reproduce the observed neutrino oscillation data with O(1) parameters.\nThis argument holds as long as *m**D**L* is populated by zeros in at least some of the entries involving *L*1 and *L*2, like in our example model of Section [subsec:Example-of-seesaw]. Instead, in the very particular case where both *m**D**L* and *m**D**R* are similarly populated matrices and of the same order, then the terms proportional to *M*VL in Eq. can provide an effective neutrino mass matrix with O(1) coefficients in each entry. In this case, *M*VL > *v*23 is possible. Nevertheless, even in this scenario we expect *M*VL not to be very large, since the smallness of *m**D**L* and *m**D**R* (that arise from non-renormalisable operators) may potentially provide most of the suppression for the small neutrino masses. Furthermore, this scenario involves the addition of several extra hyperons with very particular charges, making the model more complicated, so we do not consider it.\nHigh scale symmetry breaking[sec:High-scale-symmetry]\n=====================================================\nAssuming that the 12-breaking scale is far above the electroweak scale, at very high energies we consider only the factors *U*(1)*Y*1 × *U*(1)*Y*2 with renormalisable Lagrangian (neglecting fermion content and any kinetic mixing[10](#fn10) for simplicity),\n \n$$\\begin{aligned}\n\\mathcal{L} & =-\\frac{1}{4}F\\_{\\mu\\nu}^{(1)}F^{\\mu\\nu(1)}-\\frac{1}{4}F\\_{\\mu\\nu}^{(2)}F^{\\mu\\nu(2)}+(D\\_{\\mu}\\phi\\_{12})^{\\*}D^{\\mu}\\phi\\_{12}-V(\\phi\\_{12})\\,,\\label{eq:Lagrangian\\_12}\\end{aligned}$$ \nwhere for simplicity we assume only one hyperon *ϕ*12(*q*, − *q*), which develops a VEV $\\left\\langle \\phi\\_{12}\\right\\rangle =v\\_{12}/\\sqrt{2}$ spontaneously breaking *U*(1)*Y*1 × *U*(1)*Y*2 down to its diagonal subgroup. The covariant derivative reads\n \n*D**μ* = ∂*μ* − *i**g*1*Y*1*B*1*μ* − *i**g*2*Y*2*B*2*μ* . \nExpanding the kinetic term of *ϕ*12, we obtain mass terms for the gauge bosons as\n \n$${\\displaystyle \\mathcal{M}^{2}=\\frac{q^{2}v\\_{12}^{2}}{2}\\left(\n\\global\\long\\def\\arraystretch{0.7}\\begin{array}{@{}lcc@{}}\n & \\phantom{\\!\\,}B{}\\_{1}^{\\mu} & \\phantom{\\!\\,}B{}\\_{2}^{\\mu}\\\\\n\\cmidrule(l){2-3}\\left.B\\_{1\\mu}\\right| & g\\_{1}^{2} & -g\\_{1}g\\_{2}\\\\\n\\left.B\\_{2\\mu}\\right| & -g\\_{1}g\\_{2} & g\\_{2}^{2}\n\\end{array}\\right)\\,.}$$ \nThe diagonalisation of the matrix above reveals\n \n$$\\hat{\\mathcal{M}}^{2}=\\frac{q^{2}v\\_{12}^{2}}{2}\\left(\n\\global\\long\\def\\arraystretch{0.7}\\begin{array}{@{}lcc@{}}\n & \\phantom{\\!\\,}Y{}\\_{12}^{\\mu} & \\phantom{\\!\\,}Z'{}\\_{12}^{\\mu}\\\\\n\\cmidrule(l){2-3}\\left.\\;\\,Y\\_{12\\mu}\\right| & 0 & 0\\\\\n\\left.\\,\\,Z'\\_{12\\mu}\\right| & 0 & g\\_{1}^{2}+g\\_{2}^{2}\n\\end{array}\\right)\\,,\\label{eq:MassMatrixGauge\\_Simple}$$ \nin the basis of mass eigenstates given by\n \n$$\\left(\\begin{array}{c}\nY\\_{12\\mu}\\\\\nZ'\\_{12\\mu}\n\\end{array}\\right)=\\left(\\begin{array}{cc}\n\\cos\\theta\\_{12} & \\sin\\theta\\_{12}\\\\\n-\\sin\\theta\\_{12} & \\cos\\theta\\_{12}\n\\end{array}\\right)\\left(\\begin{array}{c}\nB\\_{1\\mu}\\\\\nB\\_{2\\mu}\n\\end{array}\\right)\\,,\\qquad\\sin\\theta\\_{12}=\\frac{g\\_{1}}{\\sqrt{g\\_{1}^{2}+g\\_{2}^{2}}}\\,.$$ \nTherefore, we obtain a massive gauge boson *Z*ʹ12*μ* at the scale *v*12, while *Y*1 + *Y*2 associated to the gauge boson *Y*12*μ* remains unbroken. These results are trivially generalised for the case of more hyperons. The covariant derivative in the new basis is given by\n \n$$\\begin{aligned}\nD\\_{\\mu} & =\\partial\\_{\\mu}-i\\frac{g\\_{1}g\\_{2}}{\\sqrt{g\\_{1}^{2}+g\\_{2}^{2}}}(Y\\_{1}+Y\\_{2})Y\\_{12\\mu}-i\\left(-\\frac{g\\_{1}^{2}}{\\sqrt{g\\_{1}^{2}+g\\_{2}^{2}}}Y\\_{1}+\\frac{g\\_{2}^{2}}{\\sqrt{g\\_{1}^{2}+g^{2}}}Y\\_{2}\\right)Z'\\_{12\\mu}\\label{eq:Covariant\\_Derivative\\_12}\\\\\n & =\\partial\\_{\\mu}-ig\\_{12}(Y\\_{1}+Y\\_{2})Y\\_{12\\mu}-i\\left(-g\\_{1}\\sin\\theta\\_{12}Y\\_{1}+g\\_{2}\\cos\\theta\\_{12}Y\\_{2}\\right)Z'\\_{12\\mu}\\,.\\nonumber \\end{aligned}$$ \nThe fermion couplings in Eq. are readily extracted by expanding the fermion kinetic terms applying Eq. .\nLow scale symmetry breaking[sec:Low-scale-symmetry]\n===================================================\nThe renormalisable Lagrangian of a theory *S**U*(2)*L* × *U*(1)*Y*1 + *Y*2 × *U*(1)*Y*3 with $H(\\mathbf{2})\\_{(0,\\frac{1}{2})}$ and *ϕ*23(**1**)(*q*, − *q*) reads (neglecting fermion content and kinetic mixing, although we will consider the effect of kinetic mixing at the end of the section),\n \n$$\\begin{aligned}\n\\mathcal{L}\\_{\\mathrm{ren}} & =-\\frac{1}{4}F\\_{\\mu\\nu}^{(12)}F^{\\mu\\nu(12)}-\\frac{1}{4}F\\_{\\mu\\nu}^{(3)}F^{\\mu\\nu(3)}-\\frac{1}{4}W\\_{\\mu\\nu}^{a}W\\_{a}^{\\mu\\nu}\\label{eq:23\\_breaking\\_Lagrangian}\\\\\n & +(D\\_{\\mu}H)^{\\dagger}D^{\\mu}H+(D\\_{\\mu}\\phi\\_{23})^{\\*}D^{\\mu}\\phi\\_{23}\\nonumber \\\\\n & -V(H,\\phi\\_{23})\\,,\\nonumber \\end{aligned}$$ \nwhere the covariant derivatives read\n \n$$D\\_{\\mu}H=(\\partial\\_{\\mu}-ig\\_{L}\\frac{\\sigma^{a}}{2}W\\_{\\mu}^{a}-i\\frac{g\\_{3}}{2}B\\_{3\\mu})H\\,,\\label{eq:cov1}$$ \n \n*D**μ**ϕ*23 = (∂*μ* − *i**g*12*q**B*12*μ* + *i**g*3*q**B*3*μ*)*ϕ*23 , \nand *σ**a* with *a* = 1, 2, 3 are the Pauli matrices. The Higgs doublet develops the usual electroweak symmetry breaking VEV as\n \n$$\\left\\langle H\\right\\rangle =\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{c}\n0\\\\\nv\\_{\\mathrm{SM}}\n\\end{array}\\right)\\,,$$ \nwhile the hyperon develops a higher scale VEV as\n \n$$\\left\\langle \\phi\\_{23}\\right\\rangle =\\frac{v\\_{23}}{\\sqrt{2}}\\,,$$ \nwhich spontaneously breaks the group *U*(1)*Y*1 + *Y*2 × *U*(1)*Y*3 down to its diagonal subgroup.\nExpanding the following kinetic terms with the expressions of the covariant derivatives of Eqs. and, we obtain\n \n$$(D\\_{\\mu}H)^{\\dagger}D^{\\mu}H+(D\\_{\\mu}\\phi\\_{23})^{\\*}D^{\\mu}\\phi\\_{23}=\\frac{v\\_{\\mathrm{SM}}^{2}g\\_{L}^{2}}{4}W\\_{\\mu}W^{\\mu\\dagger}+\\frac{q^{2}v\\_{23}^{2}}{2}\\left(\n\\global\\long\\def\\arraystretch{0.7}\\begin{array}{@{}llcc@{}}\n & \\multicolumn{1}{c@{}}{\\phantom{\\!\\,}W\\_{3}^{\\mu}} & \\phantom{\\!\\,}B\\_{12}^{\\mu} & \\phantom{\\!\\,}B\\_{3}^{\\mu}\\\\\n\\cmidrule(l){2-4}\\left.\\,W\\_{3\\mu}\\right| & g\\_{L}^{2}r^{2} & 0 & -g\\_{L}g\\_{3}r^{2}\\\\\n\\left.B\\_{12\\mu}\\right| & 0 & g\\_{12}^{2} & -g\\_{12}g\\_{3}\\\\\n\\left.\\;\\,B\\_{3\\mu}\\right| & -g\\_{L}g\\_{3}r^{2} & -g\\_{12}g\\_{3} & g\\_{3}^{2}+g\\_{3}^{2}r^{2}\n\\end{array}\\right),$$ \nwhere $r=\\frac{v\\_{\\mathrm{SM}}}{2qv\\_{23}}\\ll1$, we have defined $W\\_{\\mu}=(W\\_{\\mu}^{1}+iW\\_{\\mu}^{2})/\\sqrt{2}$, and we denote *M*gauge2 as the off-diagonal matrix above. Given the two different scales in the mass matrix above, we first apply the following transformation\n \n$$\\left(\\begin{array}{c}\nW\\_{3}^{\\mu}\\\\\nY^{\\mu}\\\\\nX^{\\mu}\n\\end{array}\\right)=\\left(\\begin{array}{ccc}\n1 & 0 & 0\\\\\n0 & \\cos\\theta\\_{23} & \\sin\\theta\\_{23}\\\\\n0 & -\\sin\\theta\\_{23} & \\cos\\theta\\_{23}\n\\end{array}\\right)\\left(\\begin{array}{c}\nW\\_{3}^{\\mu}\\\\\nB\\_{12}^{\\mu}\\\\\nB\\_{3}^{\\mu}\n\\end{array}\\right)=\\left(\\begin{array}{c}\nW\\_{3}^{\\mu}\\\\\n\\cos\\theta\\_{23}B\\_{12}^{\\mu}+\\sin\\theta\\_{23}B\\_{3}^{\\mu}\\\\\n-\\sin\\theta\\_{23}B\\_{12}^{\\mu}+\\cos\\theta\\_{23}B\\_{3}^{\\mu}\n\\end{array}\\right)\\,,\\label{eq:C.7}$$ \nwhere\n \n$$\\sin\\theta\\_{23}=\\frac{g\\_{12}}{\\sqrt{g\\_{12}^{2}+g\\_{3}^{2}}}\\,,$$ \nand we denote the rotation in Eq. as *V**θ*23, obtaining\n \n$$V\\_{\\theta\\_{23}}M\\_{\\mathrm{gauge}}^{2}V\\_{\\theta\\_{23}}^{\\dagger}=\\frac{q^{2}v\\_{23}^{2}}{2}\\left(\n\\global\\long\\def\\arraystretch{0.7}\\begin{array}{@{}llcc@{}}\n & \\multicolumn{1}{c@{}}{\\phantom{\\!\\,}W\\_{3}^{\\mu}} & \\phantom{\\!\\,}Y^{\\mu} & \\phantom{\\!\\,}X^{\\mu}\\\\\n\\cmidrule(l){2-4}\\left.W\\_{3\\mu}\\right| & g\\_{L}^{2}r^{2} & -g\\_{L}g\\_{Y}r^{2} & -g\\_{L}g\\_{X}r^{2}\\\\\n\\left.\\;\\,\\,\\,Y\\_{\\mu}\\right| & -g\\_{L}g\\_{Y}r^{2} & g\\_{Y}^{2}r^{2} & g\\_{Y}g\\_{X}r^{2}\\\\\n\\left.\\;\\;X\\_{\\mu}\\right| & -g\\_{L}g\\_{X}r^{2} & g\\_{Y}g\\_{X}r^{2} & g\\_{F}^{2}+g\\_{X}^{2}r^{2}\n\\end{array}\\right)\\,,\\label{eq:MassMatrix\\_MatchLiterature}$$ \nwhere *Y**μ* is the SM hypercharge gauge boson with gauge coupling\n \n$$g\\_{Y}=\\frac{g\\_{12}g\\_{3}}{\\sqrt{g\\_{12}^{2}+g\\_{3}^{2}}}\\simeq0.36\\,,$$ \nwhere the numeric value depicted is evaluated at the electroweak scale, and *X**μ* can be interpreted as an effective gauge boson with effective couplings\n \n$$g\\_{X}=\\frac{g\\_{3}^{2}}{\\sqrt{g\\_{12}^{2}+g\\_{3}^{2}}}\\,,\\qquad\\qquad g\\_{F}=\\sqrt{g\\_{12}^{2}+g\\_{3}^{2}}\\,,$$ \nto the Higgs boson and to *ϕ*23, respectively. In this basis, the covariant derivatives read\n \n$$D\\_{\\mu}H=(\\partial\\_{\\mu}-ig\\_{L}\\frac{\\sigma^{a}}{2}W\\_{\\mu}^{a}-i\\frac{g\\_{Y}}{2}Y\\_{\\mu}-i\\frac{g\\_{X}}{2}X\\_{\\mu})H\\,,$$ \n \n*D**μ**ϕ*23 = (∂*μ* − *i**q**g**F**X**μ*)*ϕ*23 . \nThe mass matrix in this basis can be block-diagonalised by applying the following transformation\n \n$$\\left(\\begin{array}{c}\nA^{\\mu}\\\\\n(Z^{0})^{\\mu}\\\\\nX^{\\mu}\n\\end{array}\\right)=\\left(\\begin{array}{ccc}\n\\sin\\theta\\_{W} & \\cos\\theta\\_{W} & 0\\\\\n\\cos\\theta\\_{W} & -\\sin\\theta\\_{W} & 0\\\\\n0 & 0 & 1\n\\end{array}\\right)\\left(\\begin{array}{c}\nW\\_{3}^{\\mu}\\\\\nY^{\\mu}\\\\\nX^{\\mu}\n\\end{array}\\right)=\\left(\\begin{array}{c}\n\\cos\\theta\\_{W}Y^{\\mu}+\\sin\\theta\\_{W}W\\_{3}^{\\mu}\\\\\n-\\sin\\theta\\_{W}Y^{\\mu}+\\cos\\theta\\_{W}W\\_{3}^{\\mu}\\\\\nX^{\\mu}\n\\end{array}\\right)\\,,\\label{eq:Basis\\_BeforeZ-ZprimeMixing}$$ \nwhere the mixing angle is identified with the usual weak mixing angle as\n \n$$\\sin\\theta\\_{W}=\\frac{g\\_{Y}}{\\sqrt{g\\_{Y}^{2}+g\\_{L}^{2}}}\\,,$$ \nand we denote the matrix in Eq. as *V**θ**W*[11](#fn11), obtaining\n \n$$V\\_{\\theta\\_{W}}V\\_{\\theta\\_{23}}M\\_{\\mathrm{gauge}}^{2}(V\\_{\\theta\\_{W}}V\\_{\\theta\\_{23}})^{\\dagger}=\\frac{q^{2}v\\_{23}^{2}}{2}\\left(\n\\global\\long\\def\\arraystretch{0.7}\\begin{array}{@{}llcc@{}}\n & \\multicolumn{1}{c@{}}{\\phantom{\\!\\,}A^{\\mu}} & \\phantom{\\!\\,}(Z^{0})^{\\mu} & \\phantom{\\!\\,}X^{\\mu}\\\\\n\\cmidrule(l){2-4}\\left.\\quad\\;A\\_{\\mu}\\right| & 0 & 0 & 0\\\\\n\\left.(Z^{0})\\_{\\mu}\\right| & 0 & (g\\_{L}^{2}+g\\_{Y}^{2})r^{2} & -g\\_{X}\\sqrt{g\\_{Y}^{2}+g\\_{L}^{2}}r^{2}\\\\\n\\left.\\quad\\,X\\_{\\mu}\\right| & 0 & -g\\_{X}\\sqrt{g\\_{Y}^{2}+g\\_{L}^{2}}r^{2} & g\\_{F}^{2}+g\\_{X}^{2}r^{2}\n\\end{array}\\right)\\,,$$ \nwhere we have already identified the massless photon. Now we diagonalise the remaining 2 × 2 sub-block in the limit of small *r*2. We obtain\n \n*Z**μ* = cos*θ**Z* − *Z*ʹ23( − sin*θ**W**Y**μ* + cos*θ**W**W*3*μ*) + sin*θ**Z* − *Z*ʹ23*X**μ* , \n \n*Z*ʹ23*μ* = − sin*θ**Z* − *Z*ʹ23( − sin*θ**W**Y**μ* + cos*θ**W**W*3*μ*) + cos*θ**Z* − *Z*ʹ23*X**μ* , \nwhere to leading order in *r*2\n \n$$\\sin\\theta\\_{Z-Z'\\_{23}}\\approx\\frac{\\sqrt{g\\_{Y}^{2}+g\\_{L}^{2}}g\\_{X}}{g\\_{F}^{2}}r^{2}=\\frac{g\\_{3}\\cos\\theta\\_{23}}{\\sqrt{g\\_{Y}^{2}+g\\_{L}^{2}}}\\left(\\frac{M\\_{Z}^{0}}{M\\_{Z'\\_{23}}^{0}}\\right)^{2}\\,.\\label{eq:Z\\_Zp\\_mixing}$$ \nWe can see that the SM *Z* boson carries a small admixture of the *X**μ* boson, which provides a small shift to its mass as\n \n$$M\\_{Z}^{2}\\approx q^{2}v\\_{23}^{2}\\left(g\\_{Y}^{2}+g\\_{L}^{2}\\right)\\left(r^{2}-\\frac{g\\_{X}^{2}}{g\\_{F}^{2}}r^{4}\\right)=(M\\_{Z}^{0})^{2}\\left[1-\\frac{g\\_{3}^{2}\\cos^{2}\\theta\\_{23}}{\\left(g\\_{Y}^{2}+g\\_{L}^{2}\\right)}\\left(\\frac{M\\_{Z}^{0}}{M\\_{Z'\\_{23}}^{0}}\\right)^{2}\\right]$$ \n \n$$M\\_{Z'\\_{23}}^{2}\\approx q^{2}v\\_{23}^{2}g\\_{F}^{2}\\left(1+\\frac{g\\_{X}^{2}}{g\\_{F}^{2}}r^{2}\\right)=(M\\_{Z'\\_{23}}^{0})^{2}\\left[1+\\frac{g\\_{3}^{2}\\cos^{2}\\theta\\_{23}}{\\left(g\\_{Y}^{2}+g\\_{L}^{2}\\right)}\\left(\\frac{M\\_{Z}^{0}}{M\\_{Z'\\_{23}}^{0}}\\right)^{2}\\right]$$ \nwhere\n \n$$M\\_{Z}^{0}=\\frac{v\\_{\\mathrm{SM}}}{2}\\sqrt{g\\_{Y}^{2}+g\\_{L}^{2}}\\,,\\qquad\\qquad M\\_{Z'\\_{23}}^{0}=qv\\_{\\mathrm{23}}\\sqrt{g\\_{12}^{2}+g\\_{3}^{2}}\\,,$$ \nare the masses of the *Z* boson in the SM and the mass of the *Z*ʹ23 boson in absence of *Z* − *Z*ʹ23 mixing, respectively. All these results can be generalised for the case of more hyperons or more Higgs doublets.\nAs expected, the SM *Z* boson mass arises at order *r*2, with a leading correction from *Z* − *Z*ʹ23 mixing arising at order *r*4. Instead, the *Z*ʹ23 boson arises at leading order in the power expansion, with the leading correction from *Z* − *Z*ʹ23 mixing arising at order *r*2. Remarkably, the presence of *Z* − *Z*ʹ23 mixing always shifts the mass of the *Z* boson to smaller values with respect to the SM prediction.\nThe equations obtained match general results in the literature, which consider scenarios where the starting point is a matrix such as Eq. with *g**F* = *g**X*. Our equations match those of these papers when *g**F* = *g**X* (and taking into account that we need to perform an extra rotation *θ*23 to arrive to Eq. ).\nIn the case that a kinetic mixing term sin*χ**F**μ**ν*(12)*F**μ**ν*(3)/2 is included in Eq. , then one can repeat the calculations of this section to finally obtain\n \n$$\\sin\\theta\\_{Z-Z'\\_{23}}=\\frac{g\\_{3}\\cos\\theta\\_{23}}{\\sqrt{g\\_{Y}^{2}+g\\_{L}^{2}}}(1-\\sin\\chi)\\left(\\frac{M\\_{Z}^{0}}{M\\_{Z'\\_{23}}^{0}}\\right)^{2}\\,,\\label{eq:Z\\_Zp\\_mixing\\_kinetic}$$ \nwhere now\n \n$$\\cos\\theta\\_{23}=\\frac{g\\_{3}\\sec^{2}\\chi}{\\sqrt{g\\_{12}^{2}+g\\_{3}^{2}\\left(\\sec\\chi-\\tan\\chi\\right)^{2}}}\\,,\\qquad g\\_{Y}=\\frac{g\\_{12}g\\_{3}\\sec^{2}\\chi}{\\sqrt{g\\_{12}^{2}+g\\_{3}^{2}\\left(\\sec\\chi-\\tan\\chi\\right)^{2}}}\\,,$$ \n \n$$M\\_{Z'\\_{23}}^{0}=qv\\_{\\mathrm{23}}\\sqrt{g\\_{12}^{2}+g\\_{3}^{2}\\left(\\frac{1-\\sin\\chi}{1+\\sin\\chi}\\right)}\\,.$$ \nAssuming that the kinetic mixing parameter sin*χ* is small compared to unity (e.g. absent at tree-level and generated by loop diagrams), then the dominant effect is always the original gauge mixing and kinetic mixing can be neglected. 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The estimation requires knowledge of the mean velocity of the ambient flow and the second-order statistics of the random medium. If these are not known, we show how they may be estimated from additional measurements gathered at the array, using a few known sources. We also show how the transport theory can be used to estimate the mean velocity of the medium. If the array has large aperture and the scattering in the random medium is strong, this estimate does not depend on the knowledge of the statistics of the random medium.\ntime-dependent random medium, Wigner transform, transport, imaging.\n76B15, 35Q99, 60F05.\nIntroduction\n============\nWe study sound wave propagation in a time dependent medium modeled by the wave speed $c(t,\\vx)$ and density $\\rho(t,\\vx)$ that are random perturbations of the constant values *c**o* and *ρ**o*. The medium is moving due to an ambient flow, with velocity $\\vv(t,\\vx)$ that has a constant mean $\\vv\\_o$ and small random fluctuations. The source is at a stationary location and emits a signal in the range direction denoted henceforth by the coordinate *z*, as illustrated in Figure [fig:setup]. The signal is typically a pulse defined by an envelope function of compact support, modulated at frequency $\\om\\_o$. It generates a wave that undergoes scattering as it propagates through the random medium. The goal of the paper is to analyze from first principles the net scattering at long range, and to apply the results to the inverse problem of estimating the source location and medium velocity from measurements of the wave at a remote, stationary array of receivers.\n(0,170)\n(9770,1095)(1568,-1569) (6050,1000)(0,0)[lb] (3650,-1000)(0,0)[lb] (3850,-1180)(0,0)[lb] (2700,-1480)(0,0)[lb]\n[fig:setup]\nVarious models of sound waves in moving media are described in using the linearization of the fluid dynamics equations about an ambient flow, followed by simplifications motivated by scaling assumptions. Here we consider Pierce’s equations derived in for media that vary at longer scales than the central wavelength $\\la\\_o = 2 \\pi c\\_o/\\om\\_o$ of the wave generated by the source. Pierce’s model gives the acoustic pressure\n \n$$p(t,\\vx) = - \\rho(t,\\vx) D\\_t \\phi(t,\\vx), \n\\label{eq:P1}$$ \nin terms of the velocity quasi-potential $\\phi(t,\\vx)$, which satisfies the equation\n \n$$D\\_t \\Big[ \\frac{1}{c^2(t,\\vx)} D\\_t \\phi(t,\\vx)\\Big] -\n\\frac{1}{\\rho(t,\\vx)} \\nabla\\_{\\vx} \\cdot \\Big[ \\rho(t,\\vx) \\nabla\\_{\\vx}\n \\phi(t,\\vx)\\Big] = {\\rm s}(t,\\vx),\n\\label{eq:P2}$$ \nfor spatial variable $\\vx = (\\bx,z) \\in \\RR^{d+1}$ and time $t \\in\n\\RR$, with natural number *d* ≥ 1. Here $\\bx \\in \\RR^d$ lies in the cross-range plane, orthogonal to the range axis *z*. Moreover, $\\nabla\\_{\\vx}$ and $\\nabla\\_{\\vx} \\cdot$ are the gradient and divergence operators in the variable $\\vx$ and \n$$D\\_t = \\partial\\_t + \\vv(t,\\vx) \\cdot \\nabla\\_{\\vx}$$ \n is the material (Lagrangian) derivative, with ∂*t* denoting the partial derivative with respect to time. The source is modeled by the function ${\\rm s}(t,\\vx)$ localized at the origin of range and with compact support. Prior to the source excitation there is no wave\n \n$$\\phi(t,\\vx) \\equiv 0, \\quad t \\ll 0,\n\\label{eq:P3}$$ \nbut the medium is in motion due to the ambient flow.\nSound wave propagation in ambient flows due to wind in the atmosphere or ocean currents arises in applications like the quantification of the effects of temperature fluctuations and wind on the rise time and shape of sonic booms or on radio-acoustic sounding, monitoring noise near airports, acoustic tomography, and so on.\nMoving media also arise in optics, for example in Doppler velocimetry or anemometry which uses lasers to determine the flow velocity $\\vv\\_o$. This has applications in wind tunnel experiments for testing aircraft, in velocity analysis of water flow for ship hull design, in navigation and landing, in medicine and bioengineering. A description of light propagation models used in this context can be found in.\nMuch of the applied literature on waves in moving random media considers either discrete models with Rayleigh or Mie scattering by moving particles or continuum models described by the classic wave equation with wave speed $c(0,\\vx-\\vec{\\bv}t)$. These use Taylor’s hypothesis where the medium is ``frozen\" over the duration of the experiment and simply shifted by the uniform ambient flow. A transport theory in such frozen-in media is obtained for example in and, in the paraxial regime where the waves propagate in a narrow angle cone around the range direction. The formal derivation of this theory assumes that the random fluctuations of the wave speed are Gaussian, and uses the Markov approximation, where the fluctuations are *δ*–correlated in range i.e, at any two distinct ranges, no matter how close, the fluctuations are assumed uncorrelated.\nIn this paper we study the wave equation with coefficients $c(t,\\vx), \\rho(t,\\vx)$ and $\\vv(t,\\vx)$ that have random correlated fluctuations at spatial scale ℓ and temporal scale *T*. These fluctuations are not necessarily Gaussian. We analyze the solution $\\phi(t,\\vx)$ and therefore the acoustic pressure $p(t,\\vx)$ in a forward scattering regime, where the propagation is within a cone (beam) with axis along the range direction *z*. The analysis uses asymptotics in the small parameter $\\ep = \\la\\_o/L \\ll 1$, where *L* is the range scale that quantifies the distance between the source and the array of receivers. Pierce’s equations – are justified for small $\\la\\_o/\\ell \\ll\n1$. By fixing $\\la\\_o/\\ell$ or letting it tend to zero, independent of $\\ep$, and by appropriate scaling of the spatial support of the source ${\\rm s}(t,\\vx)$, we obtain two wave propagation regimes: The first is called the wide beam regime because the cone of propagation has finite opening angle. The second is the paraxial regime, where the cone has very small opening angle. We use the diffusion approximation theory given in and to study both regimes and obtain transport equations that describe the propagation of energy. These equations are simpler in the paraxial case and we use them to study the inverse problem of locating the source. Because the inversion requires knowledge of the mean velocity $\\vv\\_o$ of the ambient flow and the second-order statistics of the random medium, we also discuss their estimation from additional measurements of waves generated by known sources.\nThe paper is organized as follows: We begin in section [sect:form] with the mathematical formulation of the problem. Then we state in section [sect:ResForward] the transport equations. These equations are derived in section [sect:Anal] and we use them for the inverse problem in section [sect:Inverse]. We end with a summary in section [sect:sum].\nFormulation of the problem\n==========================\nWe study the sound wave modeled by the acoustic pressure $p(t,\\vx)$ defined in equation in terms of the velocity quasi-potential $\\phi(t,\\vx)$, the solution of the initial value problem –. The problem is to characterize the acoustic pressure $p(t,\\vx)$ in the scaling regime described in section [sect:scaling] and then use the results for localizing the source and estimating the mean medium velocity $\\vv\\_o$.\nMedium and source\n-----------------\nThe coefficients in equation are random fields, defined by\n \n$$\\begin{aligned}\n\\vv(t,\\vx) &= \\vv\\_o + V \\sigma\\_v \\vnu\n\\Big(\\frac{t}{T},\\frac{\\vx-\\vv\\_o t}{\\ell} \\Big), \\label{eq:F1}\n\\\\ \\rho(t,\\vx) &= \\rho\\_o \\exp \\Big[ \\sigma\\_\\rho\n \\nu\\_\\rho\\Big(\\frac{t}{T},\\frac{\\vx-\\vv\\_o t}{\\ell} \\Big)\n \\Big], \\label{eq:F2} \\\\ c(t,\\vx) &= c\\_o \\Big[ 1 + \\sigma\\_c\n \\nu\\_c\\Big(\\frac{t}{T},\\frac{\\vx-\\vv\\_o t}{\\ell} \\Big) \\Big]^{-1/2},\n \\label{eq:F3}\\end{aligned}$$ \nwhere *c**o*, *ρ**o* are the constant background wave speed and density, $\\vv\\_o$ is the constant mean velocity of the ambient flow, and *V* is a velocity scale (of the order of $|\\vv\\_o|$) that will be specified later. The fluctuations in – are given by the random stationary processes $\\vnu$, *ν**ρ* and *ν**c* of dimensionless arguments and mean zero\n \n$$\\EE\\big[ \\vnu(\\tau,\\vr)\\big] = 0, \\quad \\EE[\\nu\\_\\rho(\\tau,\\vr)] = 0, \\quad \n \\EE[\\nu\\_c(\\tau,\\vr)] = 0.\n\\label{eq:F4}$$ \nWe assume that $\\vnu = (\\nu\\_j)\\_{j=1}^{d+1}$, *ν**ρ* and *ν**c* are twice differentiable, with bounded derivatives almost surely, have ergodic properties in the *z* direction, and are correlated, with covariance entries\n \n$$\\EE\\big[ \\nu\\_\\alpha (\\tau, \\vr) \\nu\\_\\beta (\\tau',\\vr') \\big] =\n\\cR\\_{\\alpha \\beta} (\\tau-\\tau',\\vr-\\vr').\n\\label{eq:F5}$$ \nHere the indices *α* and *β* are either 1, …, *d* + 1, or *ρ*, or *c*. The covariance is an even and integrable symmetric matrix valued function, which is four times differentiable and satisfies the normalization conditions\n \n$$\\cR\\_{\\alpha \\alpha}(0,0) = 1 \\mbox{ or } O(1), \\quad \\quad \\int\\_{\\RR} {\\rm d} \\tau\n\\int\\_{\\RR^{d+1}} {\\rm d} \\vr \\, \\cR\\_{\\alpha \\alpha}(\\tau,\\vr) = 1 ~\n\\mbox{ or } O(1).\n\\label{eq:F6}$$ \nThe scale *T* in definitions – is the correlation time, the typical lifespan of a spatial realization of the fluctuations, and ℓ is the correlation length, the typical length scale of the fluctuations. The dimensionless positive numbers *σ**v*, *σ**ρ* and *σ**c* quantify the standard deviation of the fluctuations. They are of the same order and small, so definitions and can be approximated by\n \n$$\\begin{aligned}\n\\rho(t,\\vx) & \\approx \\rho\\_o \\Big[ 1 + \\sigma\\_\\rho\n \\nu\\_\\rho\\Big(\\frac{t}{T},\\frac{\\vx-\\vv\\_o t}{\\ell} \\Big) \\Big],\n\\quad c(t,\\vx) \\approx c\\_o \\Big[ 1 - \\frac{\\sigma\\_c}{2}\n \\nu\\_c\\Big(\\frac{t}{T},\\frac{\\vx-\\vv\\_o t}{\\ell} \\Big) \\Big],\\end{aligned}$$ \nwith *c**o* and *ρ**o* close to the mean wave speed and density. The exponential in and the inverse of the square root in are used for convenience because some important effective properties of the medium are defined in terms of $\\EE[\\log \\rho]$ and $\\EE[c^{-2}]$, which are equal to log*ρ**o* and *c**o*− 2.\nThe origin of the coordinates is at the center of the source location, modeled by\n \n$${\\rm s}(t,\\vx) = \\sigma\\_s e^{-i \\om\\_o t} S\n\\Big(\\frac{t}{T\\_s},\\frac{\\bx}{\\ell\\_s} \\Big) \\delta(z),\n\\label{eq:F7}$$ \nfor $\\vx = (\\bx,z)$, using the continuous function *S* of dimensionless arguments and compact support. The length scale ℓ*s* is the radius of the support of ${\\rm s}(t,\\vx)$ in cross-range and the time scale *T**s* is the duration of the emitted signal. Note that ${\\rm\n s}(t,\\vx)$ is modulated by the oscillatory exponential at the frequency $\\om\\_o$. We call it the central frequency because the Fourier transform of ${\\rm s}(t,\\vx)$ with respect to time is supported in the frequency interval $|\\om-\\om\\_o| \\le O(1/T\\_s)$. The solution $\\phi(t,\\vx)$ of depends linearly on the source, so we use *σ**s* to control its amplitude.\nTo be able to set radiation conditions for the wave field resolved over frequencies, we make the mathematical assumption that the random fluctuations of $\\vv(t,\\vx)$, $\\rho(t,\\vx)$ and $c(t,\\vx)$ are supported in a domain of finite range that is much larger than *L*. In practice this assumption does not hold, but the wave equation is causal and with finite speed of propagation, so the truncation of the support of the fluctuations does not affect the wave measured at the array up to time *O*(*L*/*c**o*).\nScaling regime\n--------------\nBecause the fluctuations of the coefficients – are small, they have negligible effect on the wave at short range, meaning that $\\phi(t,\\vx) \\approx\n\\phi\\_o(t,\\vx)$, the solution of – with constant wave speed *c**o*, density *ρ**o* and velocity $\\vv\\_o$. We are interested in a long range *L*, where the wave undergoes many scattering events in the random medium and $\\phi(t,\\vx)$ is quite different from $\\phi\\_o(t,\\vx)$. We model this long range regime with the small and positive, dimensionless parameter\n \n$$\\ep = \\frac{\\la\\_o}{L} \\ll 1,\n\\label{eq:F8}$$ \nand use asymptotics in the limit $\\ep \\to 0$ to study the random field $\\phi(t,\\vx)$.\nThe relation between the wavelength, the correlation length and the cross-range support of the source is described by the positive, dimensionless parameters\n \n$$\\gamma = \\frac{\\la\\_o}{\\ell}, \\quad \\gamma\\_s = \\frac{\\la\\_o}{\\ell\\_s},\n\\label{eq:F9}$$ \nwhich are small, but independent of $\\ep$. The positive, dimensionless parameter\n \n$$\\eta = \\frac{T}{T\\_L}, \n\\label{eq:F10}$$ \ndetermines how fast the medium changes on the scale of the travel time *T**L* = *L*/*c**o*.\nThe duration of the source signal is modeled by the positive, dimensionless parameter\n \n$$\\eta\\_s = \\frac{T\\_s}{T\\_L},\n\\label{eq:F11}$$ \nwhich is independent of $\\ep$. The Fourier transform of this signal is supported in the frequency interval centered at $\\om\\_o$ and of length (bandwidth) *O*(1/*T**s*), where\n \n$$\\label{eq:b2}\n \\frac{1}{T\\_s} = \\frac{1}{\\eta\\_s T\\_L} \\ll \\frac{1}{\\ep T\\_L} \n = \\frac{c\\_o}{\\ep L} = \\frac{c\\_o}{\\lambda\\_o} = O(\\omega\\_o).$$ \nThus, the source has a small bandwidth in the $\\ep \\to 0$ limit.\nOur asymptotic analysis assumes the order relation\n \n$$\\ep \\ll \\min \\{\\gamma, \\gamma\\_s, \\eta, \\eta\\_s\\},$$ \nmeaning that we take the limit $\\ep \\to 0$ for fixed *γ*, *γ**s*, *η*, *η**s*. The standard deviations of the fluctuations are scaled as\n \n$$\\sigma\\_c = \\sqrt{\\ep \\gamma} \\bar\\sigma\\_c, \\quad\\sigma\\_\\rho = \\sqrt{\\ep \\gamma} \\bar\\sigma\\_\\rho, \\quad\n\\sigma\\_v = \\sqrt{\\ep \\gamma} \\bar\\sigma\\_v,\n\\label{eq:F12}$$ \nwith $\\bar\\sigma\\_c,\\bar\\sigma\\_\\rho,\\bar\\sigma\\_v=O(1)$ to obtain a *O*(1) net scattering effect.\nThe ambient flow, due for example to wind, has much smaller velocity than the reference sound speed *c**o*. We model this assumption with the scaling relation\n \n$$|\\vv\\_o|/V = O(1), \\quad \\mbox{where } V = \\ep c\\_o.\n\\label{eq:F13}$$ \nAlthough *V* ≪ *c**o*, the medium moves on the scale of the wavelength over the duration of the propagation\n \n$$\\label{eq:b1}\n V T\\_L = V \\frac{L}{c\\_o} = \\ep L =\n \\lambda\\_o < \\ell,$$ \nso the motion has a *O*(1) net scattering effect. Slower motion is negligible, whereas faster motion gives different phenomena than those analyzed in this paper.\nWe scale the amplitude of the source as\n \n$$\\sigma\\_s = \\frac{1}{\\ep \\eta\\_s L} \\Big(\\frac{\\gamma\\_s\n}{\\ep}\\Big)^{d},\n\\label{eq:F14}$$ \nto obtain $\\phi(t,\\vx) = O(1)$ in the limit $\\ep \\to 0$. Since equation is linear, any other source amplitude can be taken into account by multiplication of our wave field with that given amplitude.\nNote that in section [sect:resPar] we consider the secondary scaling relation\n \n*γ* ∼ *γ**S* ≪ 1, \ncorresponding to the paraxial regime, where the symbol “ ∼ ” means of the same order. Moreover, in section [sect:Inv2] we assume *η*/*η**s* ≪ 1 corresponding to a regime of statistical stability. In this secondary scaling regime we let |o| = O(), to obtain the distinguished limit in which the medium velocity impacts the quantities of interest.\nResults of the analysis of the wave field\n=========================================\nWe show in section [sect:Anal] that in the scaling regime described in equations –, the pressure is given by\n \n$$\\begin{aligned}\np(t,\\vx) \\approx i \\om\\_o {\\rho\\_o} \\int\\_{\\mathcal{O}} \\frac{{\\rm d}\\om\n {\\rm d} \\bk}{(2 \\pi)^{d+1}} \\frac{ a(\\om,\\bk,z) }{\\sqrt{\\beta(\\bk)}}\ne^{-i (\\om\\_o + \\om)t +i \\vec{\\bk} \\cdot \\vx},\n\\label{eq:B3}\\end{aligned}$$ \nfor $\\vx = (\\bx,z)$ and $\\mathcal{O} = \\{ \\om \\in \\RR \\} \\times \\{ \\bk\n\\in \\RR^d, ~ |\\bk| < k\\_o\\}, $ where the approximation error vanishes in the limit $\\ep \\to 0$. This expression is a Fourier synthesis of forward propagating time-harmonic plane waves (modes) at frequency $\\om\\_o + \\om$, with wave vectors $\\vec{\\bk}$ defined by\n \n$$\\vec{\\bk} = \\big(\\bk,\\beta(\\bk)\\big), \\quad \\beta(\\bk) = \\sqrt{k\\_o^2 -\n |\\bk|^2}, \\quad k\\_o = 2 \\pi/\\la\\_o.\n\\label{eq:R1}$$ \nThe scattering effects in the random medium are captured by the mode amplitudes, which form a Markov process $\\big(a(\\om,\\bk,z)\\big)\\_{(\\om,\\bk)\\in \\mathcal{O}}$ that evolves in *z*, starting from\n \n$$a(\\om,\\bk,0) = a\\_o(\\om,\\bk) = \\frac{i \\sigma\\_s T\\_s\\ell\\_s^d }{2\n \\sqrt{\\beta(\\bk)}} \\hat S(\\om T\\_s, \\ell\\_s \\bk).\n\\label{eq:R2}$$ \nThis process satisfies the conservation relation\n \n$$\\int\\_{\\mathcal{O}} {\\rm d}\\om {\\rm d} \\bk\\, \\big|a(\\om,\\bk,z)\\big|^2 =\n\\int\\_{\\mathcal{O}} {\\rm d}\\om {\\rm d} \\bk\\, \\big|a\\_o(\\om,\\bk)\\big|^2,\n\\quad \\forall z > 0.\n\\label{eq:R3}$$ \nThe statistical moments of $\\big(a(\\om,\\bk,z)\\big)\\_{(\\om,\\bk)\\in \\mathcal{O}}$ are characterized explicitly in the limit $\\ep \\to 0$, as explained in section [sect:MarkLim] and Appendix [ap:DL]. Here we describe the expectation of the amplitudes, which defines the coherent wave, and the second moments that define the mean Wigner transform of the wave i.e., the energy resolved over frequencies and direction of propagation.\nThe coherent wave\n-----------------\nThe expectation of the acoustic pressure (the coherent wave) is obtained from using the mean amplitudes\n \n$$\\EE[ a(\\om,\\bk,z)] = a\\_o(\\om,\\bk) \\exp \\left[ i \\theta(\\om,\\bk) z +\n D( \\bk) z\\right].\n\\label{eq:R4}$$ \nThese are derived in Section [sect:mean], with $a\\_o(\\om,\\bk)$ given in. The exponential describes the effect of the random medium, as follows:\nThe first term in the exponent is the phase\n \n$$\\theta(\\omega,\\bk) = \\frac{k\\_o}{\\beta(\\bk)} \\Big(\\frac{\\om}{c\\_o} -\n\\frac{\\bv\\_o}{c\\_o} \\cdot \\bk\\Big) + \n\\frac{ \\sigma\\_\\rho^2}{8 \\beta(\\bk) \\ell^2} \n \\Delta\\_{\\vr} \\cR\\_{\\rho \\rho}(0,\\vr)|\\_{\\vr =\n 0},\n\\label{eq:R5}$$ \nand consists of two parts: The first part models the Doppler frequency shift and depends on the cross-range component $\\bv\\_o$ of the mean velocity $\\vec{\\bv}\\_o=(\\bv\\_o, v\\_{oz})$. It comes from the expansion of the mode wavenumber \n$$\\sqrt{\\Big(k\\_o + \\frac{\\om- \\bv\\_o \\cdot \\bk}{c\\_o}\\Big)^2 - |\\bk|^2}\n\\approx \\beta(\\bk) + \\frac{k\\_o (\\om- \\bv\\_o \\cdot \\bk)}{c\\_o \\beta(\\bk)}\n,$$ \n in the limit $\\ep \\to 0$, using the scaling relation ([eq:F13]) and $\\om \\ll \\om\\_o$ obtained from. The second part is due to the random medium and it is small when *γ* ≪ 1, i.e., $\\la\\_o \\ll \\ell$.\nThe second term in the exponent in is\n \n$$\\begin{aligned}\nD(\\bk) = - \\frac{k\\_o^4 \\ell^{d+1}}{4} \\int\\_{|\\bk'| < k\\_o}\n\\frac{{\\rm d}\\bk'}{(2\\pi)^d} \\, \\frac{1}{\\beta(\\bk) \\beta(\\bk')}\n\\int\\_{\\RR^d} {\\rm d}\\br \\int\\_0^\\infty {\\rm d}r\\_z \\, e^{- i \\ell\n (\\vec{\\bk}-\\vec{\\bk}') \\cdot \\vr} \\nonumber \\\\\n \\times \\Big[\n\\sigma\\_c^2 \\cR\\_{cc}(0,\\vr) + \\frac{\\sigma\\_{\\rho}^2}{4 (k\\_o \\ell)^4}\n \\Delta\\_{\\vr}^2 \\cR\\_{\\rho \\rho}(0,\\vr) - \\frac{\\sigma\\_{\\rho}\\sigma\\_c}{(k\\_o\n \\ell)^2} \\Delta\\_{\\vr} \\cR\\_{c \\rho}(0,\\vr) \\Big], \\label{eq:R6}\\end{aligned}$$ \nwhere we used the notation $\\vr = (\\br,r\\_z)$ and definition. This complex exponent accounts for the significant effect of the random medium, seen especially in the term proportional to $\\cR\\_{cc}$ which dominates the other ones in the *γ* ≪ 1 regime. Because the covariance is even, the real part of $D(\\bk)$ derives from\n \n$$\\int\\_{\\RR^{d+1}} {\\rm d} \\vr \\, \\cR\\_{cc}(0,\\vr) e^{-i\n \\ell(\\vec{\\bk}-\\vec{\\bk'}) \\cdot \\vr} = \\int\\_{\\RR} \\frac{{\\rm d}\n \\Omega}{2 \\pi} \\,\n\\widetilde\\cR\\_{cc}\\big(\\Omega,\\ell(\\vec{\\bk}-\\vec{\\bk'})\\big),\n\\label{eq:TR4p}$$ \nwhere\n \n$$\\widetilde \\cR\\_{cc}(\\Omega,\\vq) = \\int\\_{\\RR} {\\rm d} \\tau\n\\int\\_{\\RR^{d+1}} {\\rm d} \\vr \\, \\cR\\_{cc}(\\tau,\\vr) e^{i \\Omega \\tau -\n i \\vq \\cdot \\vr} \\ge 0,\n\\label{eq:TR4}$$ \nis the power spectral density of *ν**c*. This is non-negative by Bochner’s theorem, so $\\mbox{Re} \\big[ D(\\bk)\\big] < 0$ and the mean amplitudes decay exponentially in *z*, on the length scale\n \n$$\\mathscr{S}(\\bk) = - \\frac{1}{\\mbox{Re} \\big[ D(\\bk)\\big]},\n\\label{eq:R8}$$ \ncalled the scattering mean free path. Note that $|\\bk|, |\\bk'| =\nO(1/\\ell)$ in the support of $\\widetilde \\cR\\_{cc}$ in and that by choosing the standard deviation *σ**c* as in, we obtain from – that $\\mathscr{S}(\\bk) = O(L)$ in the $\\ep \\to 0$ followed by the *γ* → 0 limit. This shows that the decay of the mean amplitudes in *z* is significant in our regime. It is the manifestation of the randomization of the wave due to scattering in the medium.\nThe Wigner transform\n--------------------\nThe strength of the random fluctuations of the mode amplitudes is described by the Wigner transform (energy density)\n \n$$\\begin{aligned}\nW(\\omega,\\bk,\\bx,z) = \\int \\frac{{\\rm d} {\\itbf q}}{(2 \\pi)^d} \\, e^{i\n \\itbf q \\cdot (\\nabla \\beta(\\bk) z + \\bx)}\n \\EE \\Big[ a \\Big(\\omega,\\bk + \\frac{{\\itbf q}}{2},z\\Big) \\overline{\n a\\Big(\\omega,\\bk-\\frac{ {\\itbf q}}{2},z\\Big)}\\Big],\n \\label{eq:T6:0}\\end{aligned}$$ \nwhere the bar denotes complex conjugate and the integral is over all $\\bq \\in \\RR^d$ such that $|\\bk \\pm \\bq/2|<k\\_o$. The Wigner transform satisfies the equation\n \n$$\\begin{aligned}\n\\big[ \\partial\\_z - \\nabla \\beta(\\bk) \\cdot \\nabla\\_\\bx \\big] W\n(\\omega,\\bk,\\bx,z) = \\int\\_{\\mathcal{O}}\\frac{{\\rm d}\\omega'{\\rm d}\n \\bk'}{(2 \\pi)^{d+1}}\\, {\\mathcal Q}(\\omega,\\omega',\\bk,\\bk') \\big[\n W(\\omega',\\bk',\\bx,z) \\nonumber \\\\ -W(\\omega,\\bk,\\bx,z) \\big],\n\\label{eq:TRresult}\\end{aligned}$$ \nfor *z* > 0, with initial condition\n \n$$\\label{eq:TRinitial}\nW (\\omega,\\bk,\\bx,0) = |a\\_o(\\om,\\bk)|^2 \\delta(\\bx).$$ \nThe integral kernel in is called the differential scattering cross-section. It is defined by\n \n$$\\begin{aligned}\n&{\\mathcal Q}(\\omega,\\omega',\\bk,\\bk') = \\frac{k\\_o^4 \\ell^{d+1} T}{4 \\beta(\\bk)\\beta(\\bk') } \\Big[\n\\sigma\\_c^2 \\widetilde\n\\cR\\_{cc} + \\frac{\\sigma\\_{\\rho}^2}{4(k\\_o \\ell)^4}\n\\widetilde{ \\Delta\\_{\\scriptsize{\\overrightarrow{\\br}}}^2 {\\cR}\\_{\\rho\n \\rho}}- \\frac{\\sigma\\_c \\sigma\\_{\\rho}}{(k\\_o \\ell)^2}\n \\widetilde{ \\Delta\\_{\\scriptsize{\\overrightarrow{\\br}}} {\\cR}\\_{c \\rho} }\n \\Big],\n \\label{eq:DCS}\\end{aligned}$$ \nwhere the power spectral densities in the square bracket are evaluated as [eq:DCS2] cc = cc(T (-’ - (-’) o), (-)), and similar for the other two terms, which are proportional to the Fourier transform of $\\Delta\\_{\\vr}^2\n\\cR\\_{\\rho \\rho}$ and $\\Delta\\_{\\vr} \\cR\\_{c \\rho}$. The total scattering cross section is defined by the integral of and satisfies\n \n$$\\begin{aligned}\n\\Sigma(\\bk) = \\int\\_{\\mathcal{O}}\\frac{{\\rm d} \\omega'{\\rm d} \\bk'}{(2\n \\pi)^{d+1}}\\, {\\mathcal Q}(\\omega,\\omega',\\bk,\\bk') =\n \\frac{2}{{\\mathscr{S}(\\bk)}}.\\end{aligned}$$ \nNote that the last two terms in the square bracket in are small in the *γ* ≪ 1 regime, because 1/(*k**o*ℓ) = *γ*/(2*π*) ≪ 1 and *σ**ρ*/*σ**c* = *O*(1). If *σ**ρ*/*σ**c* were large, of the order *γ*− 2, then these terms would contribute. However, this would only change the interpretation of the differential scattering cross section and not its qualitative form.\n### The radiative transfer equation\nThe evolution equation for the Wigner transform is related to the radiative transfer equation. Indeed, we show in Appendix [app:tranproof] that $W(\\omega,\\bk,\\bx,z)$ is the solution of - if and only if\n \n$$\\label{eq:T}\nV(\\omega,\\vec\\bk,\\vec\\bx) = \\frac{1}{\\beta(\\bk)} W(\\omega,\\bk,\\bx,z)\n\\delta\\big( k\\_z - \\beta(\\bk)\\big), \\quad \\vec \\bk = (\\bk,k\\_z),$$ \nsolves the radiative transfer equation\n \n$$\\begin{aligned}\n\\nonumber\n\\nabla\\_{\\vec\\bk} \\Omega(\\vec\\bk) \\cdot \\nabla\\_{\\vec\\bx}\nV(\\omega,\\vec\\bk,\\vec\\bx) = \\int\\_{\\RR^{d+1}} \\frac{{\\rm d}\n \\vec\\bk'}{(2\\pi)^{d+1}} \\int\\frac{{\\rm d}\\omega'}{2\\pi}\n\\mathfrak{S}\\big(\\omega,\\omega',\\vec\\bk,\\vec\\bk'\\big) \\big[\n V(\\omega',\\vec\\bk',\\vec\\bx) \\\\- V(\\omega,\\vec\\bk,\\vec\\bx) \\big],\n\\label{eq:rte3d}\\end{aligned}$$ \nwith $\\Omega(\\vec\\bk)=c\\_o |\\vec\\bk|$ and the scattering kernel\n \n$$\\mathfrak{S}\\big(\\omega,\\omega',\\vec\\bk,\\vec\\bk'\\big) = \\frac{2 \\pi\n c\\_o^2}{k\\_o^2} \\beta(\\bk) \\beta(\\bk') \\mathcal{Q}(\\om,\\om',\\bk,\\bk')\n\\delta\\big( \\Omega(\\vec\\bk)-\\Omega(\\vec\\bk')\\big).$$ \nThe initial condition is specified at $\\vx = (\\bx,0)$ by\n \n$$V(\\om,\\vec\\bk,(\\bx,0)) = |a\\_o(\\om,\\bk)|^2 \\delta(\\bx)\n\\delta(k\\_z-\\beta(\\bk)),\n\\label{eq:trIni}$$ \nwith $a\\_o(\\om,\\bk)$ defined in.\nThis result shows that the generalized (singular) phase space energy evolves as in the standard 3D radiative transfer equation, but it is supported on the phase vectors with range component\n \n$$\\label{eq:hs} \nk\\_z=\\beta(\\bk), \\quad |\\bk| < k\\_o.$$ \nIndeed, if $\\vec\\bk' = (\\bk',\\beta(\\bk'))$ and $\\vec \\bk= (\\bk,k\\_z)$, then \n$$\\delta \\big( \\Omega(\\vec\\bk)- \\Omega(\\vec\\bk') \\big) = \\frac{1}{c\\_o}\n\\delta \\big(|\\vec \\bk|- k\\_o\\big) = \\frac{k\\_o}{c\\_o \\beta(\\bk)}\n\\delta\\big(k\\_z -\\beta(\\bk)\\big),$$ \n so the evolution of $V(\\om,\\vec\\bk,\\vx)$ is confined to the hypersurface in equation ([eq:hs]). Physically, this means that the wave energy is traveling with constant speed in a cone of directions centered at the range axis *z*.\n### Paraxial approximation\nThe paraxial approximation of the Wigner transform is obtained from - in the limit \n$$\\gamma = \\la\\_o/\\ell \\to 0, \\quad \\mbox{so that} ~ \n\\gamma/\\gamma\\_s = {\\rm finite},$$ \n as explained in section [sect:Anal7]. In this case the phase space decomposition of the initial wave energy given by and is supported in a narrow cone around the range axis *z*, with opening angle scaling as = s 1. Moreover, from the expression of the differential scattering cross-section and we see that the energy coupling takes place in a small cone of differential directions whose opening angle is = 1.\nIn the paraxial regime equation simplifies to\n \n$$\\begin{aligned}\n\\left[\\partial\\_{z} + \\frac{\\bk}{k\\_o} \\cdot \\nabla\\_{\\bx} \\right]\n W(\\omega,\\bk,\\bx,{z}) = \\int\\_{\\mathbb{R}^d} \\frac{{\\rm d} \\bk'}{(2\n \\pi)^d} \\int\\_{\\mathbb{R}}\\frac{{\\rm d} \\omega'}{2 \\pi} \\,{\\mathcal\n Q}\\_{\\rm par}(\\omega',\\bk') \\nonumber \\\\ \\times W\\left( \\omega -\n \\omega' - \\bk' \\cdot \\bv\\_o, \\bk-\\bk',\\bx,{z}\\right) -\\Sigma\\_{{\\rm\n par}} W(\\omega,\\bk,\\bx,{z}), \\label{eq:TRresultP}\\end{aligned}$$ \nwhere we obtained from definition and the scaling relations, that in the limit *γ* → 0, () ko, | () -(’) | 0, T | voz ( () -(’) )| 0. The differential scattering cross-section becomes\n \n$$\\begin{aligned}\n{\\mathcal Q}\\_{\\rm par}(\\omega, \\bk ) = \\frac{k\\_o^2 \\sigma\\_c^2\n \\ell^{d+1} T}{4 } \\widetilde \\cR\\_{cc} \\big(T \\omega, \\ell\\bk,\n0\\big), \\label{eq:TrP2}\\end{aligned}$$ \nand the total scattering cross-section is\n \n$$\\begin{aligned}\n\\Sigma\\_{{\\rm par}} = \\int\\_{\\mathbb{R}^d} \\frac{{\\rm d} \\bk'}{(2\n \\pi)^d} \\int\\_{\\mathbb{R}}\\frac{{\\rm d} \\omega'}{2 \\pi} \\,{\\mathcal\n Q}\\_{\\rm par}(\\omega',\\bk') = \\frac{ \\sigma\\_c^2 \\ell k\\_o^2\n }{4 } \\cR(0,{\\bf 0})= \\frac{2}{\\mathscr{S}\\_{\\rm par}}\n,\\label{eq:TrP3}\\end{aligned}$$ \nwhere $\\mathscr{S}\\_{\\rm par}$ is the scattering mean free path in the paraxial regime and\n \n$$\\cR(\\tau,\\br) = \\int\\_{\\RR} {\\rm d} r\\_z \\, \\cR\\_{cc}(\\tau,\\vr), \\quad \\vr = (\\br,r\\_z).\n\\label{eq:Sol01}$$ \nThe initial condition is as in, with *a**o* defined in,\n \n$$W(\\om,\\bk,\\bx,0) = \\frac{\\sigma\\_s^2 T\\_s^2 \\ell\\_s^{2d}}{4 k\\_o}\n\\big|\\hat S(T\\_s \\om,\\ell\\_s \\bk)\\big|^2 \\delta(\\bx).\n\\label{eq:paraxini}$$ \nNote that the right-hand side of equation is a convolution, so we can write the Wigner transform explicitly using Fourier transforms, as explained in Appendix [ap:SolPar]. The result is\n \n$$\\begin{aligned}\nW(\\om,\\bk,&\\bx,z) = \\frac{\\sigma\\_s^2 T\\_s \\ell\\_s^d}{4 k\\_o} \\int\\_{\\RR}\n\\frac{{\\rm d} \\Omega}{2 \\pi} \\int\\_{\\RR^d} \\frac{{\\rm d} \\bK}{(2\n \\pi)^d} \\, |\\hat S(\\Omega, \\bK)|^2 \\int\\_{\\RR}{\\rm d} t \\int\\_{\\RR^d}\n {\\rm d} \\by \\nonumber \\\\&\\times \\int\\_{\\RR^d} \\frac{{\\rm d}\n \\bq}{(2 \\pi)^d}\\, \\exp \\left\\{i\\Big(\\om-\\frac{\\Omega}{T\\_s}\n \\Big)t -i \\by \\cdot \\Big(\\bk - \\frac{\\bK}{\\ell\\_s}\\Big)+i \\bq\n \\cdot \\Big(\\bx -\\frac{\\bK}{k\\_o} \\frac{z}{\\ell\\_s} \\Big)\\right.\n \\nonumber \\\\ & \\left. \\hspace{0.3in}+ \\frac{\\sigma\\_c^2 \\ell\n k\\_o^2}{4} \\int\\_0^z {\\rm d} z' \\, \\Big[\n \\cR\\Big(\\frac{t}{T},\\frac{\\by - \\frac{\\bq}{k\\_o}(z- z') -\n \\bv\\_o t}{\\ell} \\Big) - \\cR(0,{\\bf 0})\\Big]\n \\right\\},\\label{eq:Sol0}\\end{aligned}$$ \nand we use it next in the inverse problem of estimating the source location and the mean flow velocity $\\vv\\_o$.\nApplication to imaging\n======================\nIn this section we use the transport theory in the paraxial regime, stated in section [sect:resPar], to localize a stationary in space time-harmonic source in a moving random medium with smooth and isotropic random fluctuations, from measurements at a stationary array of receivers. The case of a time-harmonic source is interesting because it shows the beneficial effect of the motion of the random medium for imaging. In the absence of this motion, the wave received at the array is time-harmonic, it oscillates at the frequency $\\om\\_o$, and it is impossible to determine from it the range of the source. The random motion of the medium causes broadening of the frequency support of the wave field, which makes the range estimation possible.\nWe consider a strongly scattering regime, where the wave received at the array is incoherent. This means explicitly that the range *L* is much larger than the scattering mean free path $\\mathscr{S}\\_{\\rm par}$ or, equivalently, from,\n \n$$\\frac{\\sigma\\_c^2 \\ell k\\_o^2 L}{4} \\cR(0,{\\bf 0}) \\gg 1.\n\\label{eq:StrScat}$$ \nWe also suppose that\n \n$$\\frac{\\eta}{\\eta\\_s} = \\frac{T}{T\\_s} \\ll 1,\n\\label{eq:stability}$$ \nto ensure that the imaging functions are statistically stable with respect to the realizations of the random medium. We begin in section [sect:Inv1] with the approximation of the Wigner transform for a time-harmonic source, in the strongly scattering regime. This Wigner transform quantifies the time-space coherence properties of the wave, as described in section [sect:TXcoh]. Then, we explain in section [sect:Inv2] how we can estimate the Wigner transform from the measurements at the array. The source localization problem is discussed in section [sect:Inv3] and the estimation of the mean medium velocity is discussed in section [sect:Inv4].\nWigner transform for time-harmonic source and strong scattering\n---------------------------------------------------------------\nTo derive the Wigner transform for a time-harmonic source, we take the limit *T**s* → ∞ in, after rescaling the source amplitude as\n \n$$\\sigma\\_s = \\sigma/\\sqrt{T\\_s}, \\quad \\sigma = O(1).\n\\label{eq:I1}$$ \nWe assume for convenience[1](#fn1) that the source has a Gaussian profile,\n \n$$\\int\\_{\\RR} {\\rm d} \\Omega \\, \\big|\\hat S(\\Omega,\\bK)\\big|^2 = (2\n\\pi)^{d} e^{-|\\bK|^2},\n\\label{eq:I2}$$ \nso we can calculate explicitly the integral over $\\bK$ in. We obtain after the change of variables $ \\by = \\bxi\n+ ({\\bq}/{k\\_o}) z,$ that\n \n$$\\begin{aligned}\nW(\\om,\\bk,\\bx,z) =& \\frac{\\sigma^2\\ell\\_s^d \\pi^{d/2}}{4 k\\_o(2 \\pi)^d}\n\\int\\_{\\RR} {\\rm d} t \\, \\int\\_{\\RR^d} {\\rm d} \\bxi \\int\\_{\\RR^d} {\\rm d}\n\\bq \\, \\exp \\left\\{ i \\om t - \\frac{|\\bxi|^2}{4 \\ell\\_s^2} - i \\bxi\n\\cdot \\bk + i \\bq \\cdot \\Big(\\bx -\\frac{\\bk}{k\\_o} z\n\\Big)\\right. \\nonumber \\\\ &\\hspace{0.3in}\\left. + \\frac{\\sigma\\_c^2\n \\ell k\\_o^2}{4} \\int\\_0^z {\\rm d} z' \\, \\Big[\n \\cR\\Big(\\frac{t}{T},\\frac{\\bxi+\\frac{\\bq}{k\\_o}z' - \\bv\\_o t}{\\ell}\n \\Big) - \\cR(0,{\\bf 0})\\Big] \\right\\}.\\label{eq:I3}\\end{aligned}$$ \nNote that the last term in the exponent in is negative, because $\\cR$ is maximal at the origin. Moreover, the relation that defines the strongly scattering regime implies that the integrand in is negligible for *t*/*T* ≥ 1 and $|\\bxi+\\bq/k\\_o z' - \\bv\\_o t| /\\ell \\ge 1$. Thus, we can restrict the integral in to the set \n$$\\Big\\{(t,\\bxi,\\bq) \\in \\RR^{2d+1}: ~ ~ |t| \\ll T,~\n~ \\big|\\bxi +\\frac{\\bq}{k\\_o}z' - \\bv\\_o t \\big | \\ll \\ell \\Big\\},$$ \n and approximate\n \n$$\\cR(\\tau,\\br) \\approx \\cR(0,{\\bf 0}) - \\frac{\\alpha\\_o}{2} \\tau^2\n-\\frac{\\vartheta\\_o}{2} |\\br|^2,\n\\label{eq:Taylor}$$ \nwith *α**o*, *ϑ**o* > 0. Here we used that the Hessian of $\\cR$ evaluated at the origin is negative definite and because the medium is statistically isotropic, it is also diagonal, with the entries − *α**o* and − *ϑ**o*. We obtain that\n \n$$\\frac{\\sigma\\_c^2 \\ell k\\_o^2}{4}\n\\left[\\cR\\Big(\\frac{t}{T},\\frac{\\bxi+\\frac{\\bq}{k\\_o}z' - \\bv\\_o\n t}{\\ell} \\Big) - \\cR(0,{\\bf 0})\\right] \\approx -\\frac{\\alpha}{2}\n\\Big(\\frac{t}{T}\\Big)^2 - \\frac{\\vartheta}{2}\n\\bigg(\\frac{|\\bxi+\\frac{\\bq}{k\\_o}z' - \\bv\\_o t|}{\\ell} \\bigg)^2$$ \nwith the positive parameters\n \n$$\\label{eq:pn}\n \\alpha = \\alpha\\_o \\frac {\\sigma\\_c^2 \\ell k\\_o^2}{4}, \\quad \\quad\n \\vartheta = \\vartheta\\_o \\frac{\\sigma\\_c^2 \\ell k\\_o^2}{4}.$$ \nSubstituting in and integrating in *z*ʹ we obtain\n \n$$\\begin{aligned}\nW(\\om,\\bk,\\bx,z) &\\approx \\frac{\\sigma^2 \\ell\\_s^d \\pi^{d/2}}{4 k\\_o(2\n \\pi)^d} \\int\\_{\\RR} {\\rm d} t \\, \\int\\_{\\RR^d} {\\rm d} \\bxi\n\\int\\_{\\RR^d} {\\rm d} \\bq \\, \\exp \\left\\{ i \\om t - \\frac{\\alpha z}{2}\n\\Big(\\frac{t}{T}\\Big)^2 -\\frac{\\vartheta z}{2 \\ell^2} \\big|\\bxi- \\bv\\_o\nt\\big|^2 \\right. \\nonumber \\\\ &\\hspace{-0.2in}\\left. -\n\\frac{|\\bxi|^2}{4 \\ell\\_s^2}- i \\bxi \\cdot \\bk - \\frac{\\vartheta z^2}{2\n \\ell^2}(\\bxi - \\bv\\_o t)\\cdot \\frac{\\bq}{k\\_o} - \\frac{\\vartheta\n z^3}{6 \\ell^2} \\Big| \\frac{\\bq}{k\\_o}\\Big|^2+i \\bq \\cdot \\Big(\\bx\n-\\frac{\\bk}{k\\_o} z\\Big) \\right\\}.\\label{eq:IM3}\\end{aligned}$$ \nThe imaging results are based on this expression. Before we present them, we study the coherence properties of the transmitted wave and define the coherence parameters which affect the performance of the imaging techniques.\nTime-space coherence\n--------------------\nLet us define the time-space coherence function\n \n$$\\label{eq:Coh1}\n {C}(\\Delta t, \\Delta \\bx,\\bx,z) = \\frac{\\lambda\\_o}{2\\pi (c\\_o\n \\rho\\_o)^2}\\int\\_\\RR {\\rm d}t \\, p(t+\\Delta t,\\bx+\\Delta\\bx/2,z)\n \\overline{ p(t,\\bx-\\Delta\\bx/2,z) } e^{i \\omega\\_o \\Delta t},$$ \nand obtain from that in the paraxial regime\n \n$$\\begin{aligned}\nC(\\Delta t, \\Delta \\bx,\\bx,z) \\approx& \\int\\_{\\RR } \\frac{{\\rm d}\n \\omega}{2\\pi} \\int\\_{\\RR^{d }} \\frac{ {\\rm d}\\bk}{(2\\pi)^d}\n\\int\\_{\\RR^{d }} \\frac{{\\rm d}\\bq}{(2\\pi)^d} \\, a(\\omega,\\bk+\\bq/2,z)\n\\overline{a(\\omega,\\bk-\\bq/2,z)} \\nonumber \\\\ &\\times\n\\exp\\Big\\{i \\bq \\cdot [z \\nabla \\beta(\\bk) + \\bx] +i\n\\Delta \\bx\\cdot \\bk -i \\omega \\Delta t\\Big\\}. \\label{eq:Coh2}\\end{aligned}$$ \nMoreover, in view of and the fact that we average in time so that the statistical fluctuations of *C* are small (see Remark [rem:stability]) we have\n \n$$\\begin{aligned}\nC(\\Delta t, \\Delta \\bx,\\bx,z) &\\approx \\EE\\Big[{C}(\\Delta t, \\Delta\n \\bx,\\bx,z)\\Big] \\nonumber \\\\&\\approx \\int\\_{\\RR } \\frac{{\\rm d} \\omega}{2\\pi}\n\\int\\_{\\RR^{d }} \\frac{ {\\rm d}\\bk}{(2\\pi)^d} W(\\omega, \\bk,\\bx,z)\ne^{-i \\omega \\Delta t+i \\Delta \\bx\\cdot \\bk }. \n\\label{eq:Coh3}\\end{aligned}$$ \nThis shows formally that we can characterize the Wigner transform as the Fourier transform of the coherence function\n \n$$W(\\omega, \\bk,\\bx,z) \\approx \\int\\_{\\RR } {\\rm d}\\Delta t\n \\int\\_{\\RR^{d }} {\\rm d}\\Delta\\bx \\, {C}(\\Delta t, \\Delta\n \\bx,\\bx,z) e^{i \\omega \\Delta t-i \\Delta \\bx\\cdot \\bk }.\n\\label{eq:Coh4}$$ \nUsing the expression of the Wigner transform in we find after evaluating the integrals that\n \n$$\\begin{aligned}\nC(\\Delta t, \\Delta \\bx,\\bx, z) &\\approx\n \\frac{\\sigma^2 \\ell\\_s^d }{2^{2+d/2} k\\_o\n R\\_z^d } \\, \\exp \\big[i \\varphi( \\Delta t, \\Delta \\bx,\\bx, z)]\n\\nonumber \\\\ & \\times \\exp\\bigg[ - \\frac{\\Delta t^2}{2{\\cal T}\\_z^2} -\n \\frac{|\\bx|^2}{2 R\\_z^2} - \\frac{|\\Delta \\bx|^2}{2 {\\mathcal\n D}\\_{1z}^2} - \\frac{|H\\_z \\Delta \\bx - \\bv\\_o \\Delta t |^2}{2\n {\\cal D}\\_{2z}^2} \\bigg], \\label{eq:Cbar}\n \\end{aligned}$$ \nwith phase\n \n$$\\varphi( \\Delta t, \\Delta \\bx,\\bx, z) = \\frac{k\\_o \\bx\\cdot \\Big[\n \\Big(1 + \\vartheta z\\Big(\\frac{\\ell\\_s}{\\ell}\\Big)^2\\Big) \\Delta\n \\bx - \\vartheta z\\Big(\\frac{\\ell\\_s}{\\ell}\\Big)^2 \\bv\\_o \\Delta t\n \\Big]}{z \\Big[ 1 + \\frac{2}{3}\\vartheta\n z\\Big(\\frac{\\ell\\_s}{\\ell}\\Big)^2\\Big] },$$ \nand coefficients\n \n$$\\begin{aligned}\n{\\cal T}\\_z =& \\frac{T}{\\sqrt{\\alpha z}}, \\quad \\quad R\\_z =\n\\frac{z}{\\sqrt{2}\\ell\\_s k\\_o} \\left( 1 + \\frac{2\\ell\\_s^2}{3{\\cal D}\\_z^2}\n\\right)^{1/2}, \n \\quad \\quad\n{\\cal D}\\_z = \\frac{\\ell}{\\sqrt{\\vartheta z}}, \n\\label{eq:Adef}\n\\\\\n{\\cal D}\\_{1z} =& 2 {\\cal D}\\_z \\left[3 \\Big(1+\n \\frac{\\ell\\_s^2}{6{\\cal D}\\_z^2}\\Big)\\right]^{1/2} \n,\\label{eq:Coeff1} \\quad \n{\\cal D}\\_{2z} = {\\cal D}\\_z\\left(\\frac{1+ \\frac{2\\ell\\_s^2}{3{\\cal D}\\_z^2}}{1+\n \\frac{\\ell\\_s^2}{6{\\cal D}\\_z^2}}\\right)^{1/2}, \\quad \n H\\_z = 1 - \\frac{1}{2\n \\Big(1+ \\frac{\\ell\\_s^2}{6 {\\cal D}\\_z^2}\\Big)}. \n \\end{aligned}$$ \nThe decay of the coherence function in Δ*x* models the spatial decorrelation of the wave on the length scale corresponding to the characteristic speckle size. This is quantified by the length scales ${\\cal D}\\_{1z}$ and ${\\cal D}\\_{2z}$, which are of the order of ${\\cal\n D}\\_z$. We call ${\\cal D}\\_z$ the decoherence length and obtain from and that it is of the order of the typical size ℓ of the random fluctuations of the medium,\n \n$$\\label{eq:defDz}\n{\\cal D}\\_z = \\frac{\\ell}{\\sqrt{\\vartheta z}} =\n\\frac{\\ell}{\\pi\\sqrt{\\vartheta\\_o}} \\sqrt{\\frac{L}{z}} = O(\\ell).$$ \nThe decay of the coherence function in Δ*t* models the temporal decorrelation of the wave, on the time scale\n \n$$\\mathcal{T}\\_z = \\frac{T}{\\sqrt{\\alpha z}} = \\frac{T}{\\pi\n \\sqrt{\\alpha\\_o}}\\sqrt{\\frac{L}{z}} = O(T),\n\\label{eq:defTz}$$ \nwhere we used definitions and. We call ${\\cal T}\\_z$ the decoherence time and note that it is of the order of the life span *T* of the random fluctuations of the medium.\nThe decay of the coherence function in $|\\bx|$ means that the waves propagate in a beam with radius *R**z*, which evolves in *z* as described in and satisfies\n \n$$R\\_z \\approx \\left\\{ \\begin{array}{lll}\n\\displaystyle \\frac{z}{\\sqrt{2} \\ell\\_s k\\_o} &\n \\hbox{for} & \\ell\\_s \\ll {\\cal D}\\_z, \\\\\n\\displaystyle \\sqrt{\\frac{\\vartheta}{3}}\n \\frac{z^{3/2}}{k\\_o \\ell} & \\hbox{for} & \\ell\\_s \\gg {\\cal D}\\_z.\n \\end{array} \n \\right.\n\\label{eq:defRz}$$ \nThis shows that the transition from diffraction based beam spreading to scattering based beam spreading happens around the critical propagation distance\n \n$$z^\\* = \\frac{1}{\\vartheta} \\Big(\\frac{\\ell}{\\ell\\_s} \\Big)^2 = L\n \\Big( \\frac{\\gamma\\_s}{\\gamma} \\Big)^2 \\frac{1}{\\pi^2 \\vartheta\\_o\n }. \\label{eq:defzstar}$$ \nThis expression is derived from equation $\\ell\\_s = {\\cal D}\\_{z^\\star}$ and definitions and, and it shows that *z*⋆/*L* is finite in our regime[2](#fn2).\nNote that when *z* ≫ *z*\\* i.e., ${\\cal D}\\_z \\ll \\ell\\_s$, the coefficients become\n \n$$\\begin{aligned}\n\\label{eq:Coeff1Z}\n {\\cal D}\\_{1z} \\approx & \\sqrt{2} \\ell\\_s, \\quad {\\cal D}\\_{2z} \\approx\n 2 {\\cal D}\\_z, \\quad H\\_z \\approx 1,\\end{aligned}$$ \nand the coherence function satisfies\n \n$$\\frac{|C(\\Delta t, \\Delta \\bx,\\bx, z) |}{|C(0,{\\bf 0},{\\bf 0},z)|}\n \\approx \\exp\\bigg( - \\frac{\\Delta t^2}{2{\\cal T}\\_z^2} -\n \\frac{|\\bx|^2}{2 R\\_z^2 } - \\frac{|\\Delta \\bx|^2}{4\n \\ell\\_s^2} - \\frac{| \\Delta \\bx - \\bv\\_o \\Delta t |^2}{8{\\cal D}\\_z^2 }\n \\bigg).$$ \nThus, the spatial spreading and decorrelation of the wave field for *z* ≫ *z*\\* are governed by the parameters *R**z*, ℓ*s* and ${\\cal D}\\_z$, with *R**z* given by the second case in and ${\\cal D}\\_z$ given in. These parameters scale with the propagation distance *z* < *L* as *R**z* ∼ *z*3/2 and ${\\cal D}\\_z \\sim\n z^{-1/2}$. The temporal decorrelation is on the scale ${\\cal T}\\_z \\sim z^{-1/2}$.\nEstimation of the Wigner transform\n----------------------------------\nSuppose that we have a receiver array centered at $(\\bx\\_o,z)$, with aperture in the cross-range plane modeled by the appodization function\n \n$$\\mathscr{A}(\\bx) = \\exp\\left(-\\frac{|\\bx-\\bx\\_o|^2}{2\n (\\varkappa/k\\_o)^2}\\right).\n\\label{eq:EW1}$$ \nThe linear size of the array is modeled by the standard deviation $\n\\varkappa /k\\_o $, with dimensionless $\\varkappa >0$ defining the diameter of the array expressed in units of $\\la\\_o$.\nRecalling the wave decomposition and that $\\beta(\\bk)\n\\sim k\\_o$ in the paraxial regime, we define the estimated mode amplitudes by\n \n$$\\begin{aligned}\na\\_{\\rm est}(\\om,\\bk,z) &= \\frac{k\\_o e^{-i \\beta(\\bk) z}}{i \\om\\_o\n {\\rho\\_o} } \\int\\_\\RR dt \\int\\_{\\RR^d} {\\rm d} \\bx \\, \\mathscr{A}(\\bx)\np(t,\\bx,z) e^{i(\\om+\\om\\_o)t} e^{-i \\bk \\cdot \\bx} \\nonumber \\\\ &=\n\\left( \\frac{\\varkappa^2}{2 \\pi k\\_o^2}\\right)^{d/2} \\int\\_{\\RR^d} {\\rm\n d} {\\widetilde}\\bk \\, a(\\om,\\bk+{\\widetilde}\\bk,z) e^{ i [\\beta(\\bk+{\\widetilde}\\bk) - \\beta(\\bk)] z + i {\\widetilde}\\bk \\cdot \\bx\\_o -\n \\frac{\\varkappa^2 |{\\widetilde}\\bk|^2}{2 k\\_o^2}}.\n\\label{eq:EW2}\\end{aligned}$$ \nWith these amplitudes we calculate the estimated Wigner transform\n \n$$\\begin{aligned}\nW\\_{\\rm est}(\\om,\\bk,\\bx,z) &= \\int\\_{\\RR^d} \\frac{{\\rm d} \\bq}{(2\n \\pi)^d} e^{i \\bq \\cdot (\\nabla \\beta(\\bk) z + \\bx)} a\\_{\\rm\n est}\\Big(\\om,\\bk + \\frac{\\bq}{2},z\\Big) \\overline{ a\\_{\\rm\n est}\\Big(\\om,\\bk - \\frac{\\bq}{2},z\\Big)} \\end{aligned}$$ \nand obtain after carrying out the integrals and using the approximation\n \n$$\\Big[ \\beta\\Big(\\bk + \\frac{\\bq}{2}\\Big) - \\beta\\Big(\\bk-\n \\frac{\\bq}{2}\\Big)\\Big]z \\approx \\bq \\cdot \\nabla \\beta(\\bk) z\n,$$ \nthat\n \n$$W\\_{\\rm est}(\\om,\\bk,\\bx,z) \\approx\n\\Big(\\frac{\\varkappa^2}{\\pi k\\_o^2}\\Big)^{d/2}\\hspace{-0.03in}\ne^{-\\frac{k\\_o^2|\\bx-\\bx\\_o|^2}{\\varkappa^2}}\n\\int\\_{\\RR^d}\\hspace{-0.03in} {\\rm d} \\bK \\, e^{-\\frac{\\varkappa^2\n |\\bK|^2}{k\\_o^2}} W(\\om,\\bk + \\bK,\\bx,z).\n\\label{eq:EW4}$$ \nWe can now use the expression in this equation, to obtain an explicit approximation for $W\\_{\\rm est}$. Equivalently, we can substitute in and obtain after integrating in $\\bK$ that\n \n$$\\begin{aligned}\nW\\_{\\rm est}(\\om,\\bk,\\bx,z) \\approx e^{\n -\\frac{k\\_o^2|\\bx-\\bx\\_o|^2}{\\varkappa^2}} \\hspace{-0.03in}\\int\\_{\\RR }\n{\\rm d}\\Delta t \\int\\_{\\RR^{d }} \\hspace{-0.03in} {\\rm d}\\Delta\\bx \\,\nC(\\Delta t, \\Delta \\bx,\\bx,z) e^{i \\omega \\Delta t -i \\Delta \\bx\\cdot\n \\bk -\\frac{k\\_o^2|\\Delta \\bx |^2}{4 \\varkappa^2}},\n\\label{eq:EW5}\\end{aligned}$$ \nwith *C* given in.\n[rem:stability] Note from that the time integration that defines the coherence function is over a time interval determined by the pulse duration *T**s*, which is larger than the coherence time *T* of the medium by assumption. If we interpret the wave as a train of *T**s*/*T* pulses of total duration *T*, each individual pulse travels through uncorrelated layers of medium because the correlation radius of the medium ℓ is much smaller than *c**o**T*. This follows from the fact that $\\ell/(c\\_oT) = \\eps/(\\eta\\gamma)$ and $\\eps \\ll \\gamma\\eta$. Thus, $C(\\Delta t,\n\\Delta \\bx,\\bx,z)$ is the superposition of approximately *T**s*/*T* uncorrelated components and its statistical fluctuations are small by the law of large numbers. Moreover, we conclude from that the estimated Wigner transform is approximately equal to its expectation, up to fluctuations of relative standard deviation that is smaller than $\\sqrt{T/T\\_s}$.\nSource localization\n-------------------\nWe now show how we can use the estimated Wigner transform to localize the source. Recall that we use the system of coordinates with origin at the center of the source. Thus, the location $(\\bx\\_o,z)$ of the center of the array relative to the source is unknown and the goal of imaging is to estimate it. We begin in section [sect:imagCR] with the estimation of the direction of arrival of the waves at the array, and then describe the localization in range in section [sect:imagR]. These two estimates determine the source location in the cross-range plane, as well.\n### Direction of arrival estimation\nWe can estimate the direction of arrival of the waves from the peak (maximum) in $\\bk$ of the imaging function\n \n$$\\begin{aligned}\n\\mathcal{O}\\_{\\rm DoA}(\\bk,z) &= \\int\\_{\\RR} \\frac{{\\rm d} \\om }{2\\pi}\nW\\_{\\rm est}(\\om,\\bk,\\bx\\_o,z), \\label{eq:DoA1}\\end{aligned}$$ \ndetermined by the estimated Wigner transform at the center of the array of receivers. If the medium were homogeneous, the maximum of $\\bk \\mapsto \\mathcal{O}\\_{\\rm DoA}(\\bk,z)$ would be at the cross-range wave vector $\\bk^\\* = k\\_o \\frac{\\bx\\_o}{z},$ and the width of the peak (the resolution) would be $1/(\\sqrt{2} \\varkappa)$. However, cumulative scattering in the random medium gives a different result, as we now explain:\nSubstituting in and using the expression, we obtain after evaluating the integrals that\n \n$$\\begin{aligned}\n\\frac{ \\mathcal{O}\\_{\\rm DoA}(\\bk,z) } { \\max\\_{\\bk'} \\mathcal{O}\\_{\\rm\n DoA}(\\bk',z) } = \\exp \\left\\{- \\frac{1}{2 \\vartheta\\_{\\rm\n DoA}^2(z)} \\left| \\frac{\\bk - \\bk(z)}{k\\_o}\n\\right|^2\\right\\},\n \\label{eq:DoA5}\\end{aligned}$$ \nwith\n \n$$\\vartheta\\_{\\rm DoA}(z) = \\left\\{ \\frac{1}{3 {\\cal D}\\_z^2 k\\_o^2} \\left(\\frac{1\n + \\frac{\\ell\\_s^2}{2 {\\cal D}\\_z^2}}{ 1 + \\frac{2 \\ell\\_s^2}{3 {\\cal D}\\_z^2}}\n \\right) + \\frac{1}{2 \\varkappa^2} \\right\\}^{1/2},\n \\quad \\quad\n \\bk(z) = k\\_o \\frac{\\bx\\_o}{z} \\left(\\frac{1 + \\frac{\\ell\\_s^2}{{\\cal D}\\_z^2}}{\n 1 + \\frac{2\\ell\\_s^2}{3{\\cal D}\\_z^2} }\\right).\n\\label{eq:DoA6}$$ \nTherefore, the maximum of $\\bk \\mapsto \\mathcal{O}\\_{\\rm DoA}(\\bk,z)$ is at the cross-range wave vector $\\bk(z)$ and the width of the peak (the resolution) is determined by $\\vartheta\\_{\\rm DoA}(z)$. This resolution improves for larger array aperture (i.e., $\\varkappa$) and deteriorates as *z* increases. Depending on the magnitude of *z* relative to the critical range *z*\\* defined in, we distinguish three cases:\n1. In the case *z* ≪ *z*\\* i.e., $\\ell\\_s \\ll {\\cal D}\\_z$, the intensity travels along the deterministic characteristic, meaning that $\\mathcal{O}\\_{\\rm DoA}(\\bk,z)$ peaks at\n \n$$\\bk(z) \\approx k\\_o \\frac{\\bx\\_o}{z}.$$ \nHowever, the resolution is worse than in the homogeneous medium,\n \n$$\\vartheta\\_{\\rm DoA}(z) \\approx \\left\\{ \\frac{1 }{3 {\\cal D}\\_z^2 k\\_o^2} +\n\\frac{1}{2\\varkappa^2} \\right\\}^{1/2},\n\\label{eq:DoA6a}$$ \nwith ${\\cal D}\\_z$ defined in.\n2. In the case *z* ≫ *z*\\*, i.e., $\\ell\\_s \\gg {\\cal D}\\_z$, the peak of $\\mathcal{O}\\_{\\rm DoA}(\\bk,z) $ is at the cross-range wave vector\n \n$$\\bk(z) \\approx \\frac{3}{2} k\\_o \\frac{\\bx\\_o}{z},$$ \nand the resolution is\n \n$$\\vartheta\\_{\\rm DoA}(z) \\approx \\left\\{ \\frac{1 }{4 {\\cal D}\\_z^2 k\\_o^2} +\n\\frac{1}{2\\varkappa^2} \\right\\}^{1/2}.\n\\label{eq:DoA6b}$$ \nHere the peak corresponds to a straight line characteristic, but with a different slope than in the homogeneous medium. The resolution is also worse than in the homogeneous medium.\n3. In the case *z* = *O*(*z*\\*), the characteristic can no longer be approximated by a straight line, as seen from. Nevertheless, we can still estimate the source position from the observed peak $\\bk(z)$, provided that we have an estimate of the range *z*. The resolution of the estimate of $\\bk(z)$ is $\\vartheta\\_{\\rm DoA}(z) $ given by ([eq:DoA6]) that is bounded from below by ([eq:DoA6b]) and from above by ([eq:DoA6a]).\n[rem:DOA] Note that ([eq:DoA6]) is a decreasing function of the array diameter $\\varkappa /k\\_o$, as long as this satisfies $\\varkappa /k\\_o\n\\le \\sqrt{2} {\\cal D}\\_z$. Thus, increasing the aperture size beyond the critical value $\\sqrt{2} {\\cal D}\\_z$ does not bring any resolution improvement.\n### Range estimation\nThe results of the previous section show that the direction of arrival estimation is coupled with the estimation of the range *z* in general, with the exception of the two extreme cases 1. and 2. outlined above.\nTo estimate the range *z*, we use the imaging function\n \n$$\\begin{aligned}\n\\mathcal{O}\\_{\\rm range}(t,z) &= \\int\\_{\\RR} \\frac{{\\rm d} \\om }{2\\pi}\ne^{-i \\om t} \\int\\_{\\RR^d} \\frac{{\\rm d} \\bk}{(2\\pi)^d} W\\_{\\rm\n est}(\\om,\\bk,\\bx\\_o,z) \\approx {C}(t,{\\bf 0},\\bx\\_o,z),\n\\label{eq:Ra1}\\end{aligned}$$ \nderived from. Substituting the expression of the coherence function in this equation we obtain\n \n$$\\begin{aligned}\n\\frac{ |\\mathcal{O}\\_{\\rm range}(t,z)| } { \\max\\_{t'} |\\mathcal{O}\\_{\\rm\n range}(t',z)| } &= \\exp \\left\\{- \\frac{t^2}{2 \\vartheta\\_{\\rm\n range}^2(z)} \\right\\},\n\\label{eq:Ra2}\\end{aligned}$$ \nwith\n \n$$\\begin{aligned}\n\\vartheta\\_{\\rm range}(z) = {\\cal T}\\_z \\left\\{ 1 + \\frac{|\\bv\\_o|^2\n {\\cal T}\\_z^2}{{\\cal D}\\_z^2} \\left( \\frac{1 + \\frac{\\ell\\_s^2}{6 {\\cal\n D}\\_z^2}}{1 + \\frac{2 \\ell\\_s^2}{3 {\\cal\n D}\\_z^2}}\\right)\\right\\}^{-1/2}.\n\\label{eq:Ra3}\\end{aligned}$$ \nAs a function of *t*, this peaks at *t* = 0 and its absolute value decays as a Gaussian, with standard deviation $\\vartheta\\_{\\rm\n range}(z)$. If we know the statistics of the medium (the decoherence time ${\\cal T}\\_z$ and length ${\\cal D}\\_z$) and the magnitude of the cross-range velocity $|\\bv\\_o|$, then we can determine the range *z* by estimating the rate of decay of $\\mathcal{O}\\_{\\rm\n range}(t,z)$. Note that the array dimameter $\\varkappa /k\\_o$ plays no role for the range estimation.\n[rem:range2] We can also estimate the mean velocity $\\vv\\_o = (\\bv\\_o,v\\_{oz})$ from, by considering different beam orientations in the case that the sources and also the medium statistics (the decoherence time ${\\cal T}\\_z$ and length ${\\cal D}\\_z$) are known. That is to say, with three known beams we can get the vector $\\vv\\_o$, and then we can use it to localize the unknown source using the direction of arrival and range estimation described above. See also section [sect:Inv4] for a more detailed analysis of the velocity estimation.\n[rem:range3] If the decoherence time ${\\cal T}\\_z$ and length ${\\cal D}\\_z$ are not known, they can also be estimated using additional known sources. Definitions – show that ${\\cal D}\\_z z^{1/2}$ and ${\\cal T}\\_z z^{1/2}$ are constant with respect to *z*. Once estimated, these constants can be used in the imaging of the unknown source.\nSingle beam lateral velocity estimation\n---------------------------------------\nWe observe from and – that the source localization depends only on the Euclidian norm $|\\bv\\_o|$ of the cross-range component of the mean velocity of the medium. We show here that $\\bv\\_o$ can be obtained with only one beam and, when the receiver array is large and *z* ≫ *z*\\* i.e., ℓ*s* ≫ *D**z*, the velocity estimate is independent of the medium statistics and the source location.\nThe estimation of $\\bv\\_o$ is based on the imaging function\n \n$$\\begin{aligned}\n \\mathcal{O}\\_{\\rm v}(\\by,t,z) &= \\int\\_{\\RR}\\frac{ {\\rm d} \\om}{2\\pi} e^{-i\n \\om t} \\int\\_{\\RR^d} \\frac{{\\rm d} \\bk }{(2\\pi)^d} e^{ i \\bk \\cdot\n \\by} \\int\\_{\\RR^d} {\\rm d} \\bx \\, W\\_{\\rm est}(\\om,\\bk,\\bx,z)\n \\nonumber \\\\ &\\approx \\exp\\Big( - \\frac{k\\_o^2 |\\by|^2}{4 \\varkappa^2}\n \\Big) \\int\\_{\\RR^d} {\\rm d} \\bx \\, \\exp\\Big(\n -\\frac{k\\_o^2|\\bx-\\bx\\_o|^2}{\\varkappa^2}\\Big) \\, {C}(t,\\by,\\bx, z).\\end{aligned}$$ \nSubstituting the expression of the coherence function and carrying out the integrals we obtain that\n \n$$\\begin{aligned}\n |\\mathcal{O}\\_{\\rm v}(\\by,t,z)| \\approx & \\frac{\\sigma^2 \\pi^{d/2 } (\\varkappa\\ell\\_s)^d }{2^{2+d} k\\_o^{d+1} {\\cal A}^{d}\\_z } \\exp\\left\\{ - \\frac{t^2}{2 {\\cal\n T}\\_z^2} -\\frac{|\\by- s\\_z t \\bv\\_o|^2}{2 \\mathfrak{m}^2\\_z{\\cal A}^2\\_z} -\n\\frac{|t \\bv\\_o|^2}{ \\mathfrak{n}^2\\_z {\\cal A}\\_z^2} - \\frac{| \\bx\\_o|^2}{4\n {\\cal A}^2\\_z} \\right\\}, \n\\label{eq:Ovy}\\end{aligned}$$ \nwith the effective apperture\n \n$$\\begin{aligned}\n {\\cal A}^2\\_z = \\frac{1}{4} \\Big[\\Big(\\frac{\\varkappa}{k\\_o}\\Big)^2 +\n \\Big(\\frac{z}{k\\_o \\ell\\_s}\\Big)^2 \\Big(1 + \\frac{2 \\ell\\_s^2}{3\n D\\_z^2}\\Big)\\Big],\n\\label{eq:defAz}\\end{aligned}$$ \nand dimensionless parameters\n \n$$\\begin{aligned}\n {\\mathfrak m}\\_z^2 =& \\frac{8}{1 +\n \\frac{2}{3 {\\cal D}\\_z^2} \\big(\\frac{z}{k\\_o \\ell\\_s}\\big)^2\\Big(1 +\n \\frac{\\ell\\_s^2}{2 {\\cal D}\\_z^2}\\Big) + \\big(\\frac{\\varkappa}{k\\_o\n \\ell\\_s}\\big)^2 \\Big(1 + \\frac{2 \\ell\\_s^2}{{\\cal D}\\_z^2}\\Big) +\n \\big(\\frac{k\\_o}{\\varkappa}\\big)^2 \\big(\\frac{z}{k\\_o \\ell\\_s}\\big)^2\n \\Big( 1 + \\frac{2 \\ell\\_s^2}{3 {\\cal D}\\_z^2}\\Big)}, \\\\ \\mathfrak{n}\\_z^2 =\n & \\frac{\\mathfrak{m}\\_z^2}{s\\_z(q\\_z - s\\_z/2)},\\\\\ns\\_z =& \\frac{\\mathfrak{m}\\_z^2}{4{\\cal D}\\_z^2} \\Big[\n \\Big(\\frac{\\varkappa}{k\\_o}\\Big)^2 + \\frac{1}{2} \\Big(\\frac{z}{k\\_o\n \\ell\\_s}\\Big)^2 \\Big(1 + \\frac{\\ell\\_s^2}{3 {\\cal D}\\_z^2} \\Big)\\Big],\n\\quad \\quad q\\_z = \\frac{\\Big(\\frac{\\varkappa}{k\\_o}\\Big)^2 + \n \\Big(\\frac{z}{k\\_o\n \\ell\\_s}\\Big)^2 \\Big(1 + \\frac{\\ell\\_s^2}{6 {\\cal D}\\_z^2} \\Big)}{\n2\\Big(\\frac{\\varkappa}{k\\_o}\\Big)^2+ \n\\Big(\\frac{z}{k\\_o\n \\ell\\_s}\\Big)^2 \\Big(1 + \\frac{\\ell\\_s^2}{3 {\\cal D}\\_z^2} \\Big)}.\n \\end{aligned}$$ \nThese depend on the radii $\\varkappa/k\\_o$ of the array and ℓ*s* of the source, the decoherence length ${\\cal D}\\_z$ and the ratio *z*/(*k**o*ℓ*s*) that quantifies the cross-range resolution of focusing of a wave using time delay beamforming at a source of radius ℓ*s*.\nTo estimate $\\bv\\_o$ we can proceed as follows: First, we estimate for each time *t* the position $\\by\\_{\\rm max}(t)$ that maximizes $\n\\by\\mapsto \\mathcal{O}\\_{\\rm v}(\\by;t,z)$. Second, we note from ([eq:Ovy]) that $\\by\\_{\\rm max}(t)$ should be a linear function in *t*, of the form $\\by\\_{\\rm max}(t) = s\\_z \\bv\\_o t$. Therefore, we can estimate $ s\\_z \\bv\\_o$ with a weighted linear least squares regression of $\\by\\_{\\rm max}(t) $ with respect to *t*. In practice *s**z* is likely unknown. However, in the case of a large receiver array with radius satisfying\n \n$$\\frac{\\varkappa}{k\\_o} \\gg \\max\\left\\{\\frac{z}{k\\_o \\ell\\_s},\n\\frac{z}{k\\_o {\\cal D}\\_z} \\right\\},\n\\label{eq:LA1}$$ \nand for *z* ≫ *z*⋆, so that $\\ell\\_s \\gg {\\cal D}\\_z$, we obtain that *s**z* ≈ 1. Thus, the least squares regression gives an unbiased estimate of $\\bv\\_o$.\nIn view of ([eq:Ovy]), the least squares regression can be carried out over a time interval with length of the order of $\\min(\n{\\cal T}\\_z, \\mathfrak{n}\\_z {\\cal A}\\_z/|{\\itbf v}\\_o|) $. Beyond this critical time the function $ \\mathcal{O}\\_{\\rm v}$ vanishes. Therefore, as long as $|{\\itbf v}\\_o|< {{\\mathfrak n}\\_z {\\cal A}\\_z}/{{\\cal T}\\_z} $, the velocity resolution is\n \n$$\\begin{aligned}\n \\label{eq:resv1}\n \\mbox{res}\\_{\\rm v}\n = \\frac{{\\mathfrak m}\\_z {\\cal A}\\_z}{s\\_z {\\cal T}\\_z } \\approx\n \\frac{{\\cal D}\\_z}{{\\cal T}\\_z},\\end{aligned}$$ \nwhere the approximation is for a large array and $\\ell\\_s \\gg {\\cal\n D}\\_z$. If $|{\\itbf v}\\_o|$ is larger than $ {{\\mathfrak n}\\_z {\\cal A}\\_z}/{{\\cal T}\\_z} $, then the resolution is reduced to\n \n$$\\begin{aligned}\n\\label{eq:resv2}\n\\mbox{res}\\_{\\rm v} = \n\\frac{{\\mathfrak m}\\_z {\\cal A}\\_z}{s\\_z {\\mathfrak n}\\_z {\\cal\n A}\\_z/|\\bv\\_o| } \\approx \\frac{{\\cal D}\\_z}{{\\cal T}\\_z} \\frac{|\\bv\\_o|\n {\\cal T}\\_z}{\\mathfrak{n}\\_z {\\cal A}\\_z}.\\end{aligned}$$ \nAnalysis of the wave field\n==========================\nTo derive the results stated in section [sect:ResForward], we begin in section [sect:Anal1] with a slight reformulation, which transforms equation into a form that is more convenient for the analysis. We scale the resulting equation in section [sect:Anal2], in the regime defined in section [sect:scaling], and then we change coordinates to a moving frame in section [sect:Anal3]. In this frame we write the wave as a superposition of time-harmonic, plane waves with random amplitudes that model the net scattering in the random medium, as described in section [sect:Anal4]. We explain in section [sect:Anal5] that the backward going waves are negligible, and use the diffusion approximation theory in section [sect:Anal6] to analyze the amplitudes of the forward going waves, in the limit $\\ep \\to 0$. We end in section [sect:Anal7] with the paraxial limit.\nTransformation of the wave equation\n-----------------------------------\nLet us define the new potential\n \n$$\\psi(t,\\vx) = \\frac{\\sqrt{\\rho(t,\\vx)}}{\\sqrt{\\rho\\_o}} \\phi(t,\\vx),\n\\label{eq:An1}$$ \nand substitute it in to obtain the wave equation\n \n$$\\begin{aligned}\nD\\_t \\left[ \\frac{1}{c^2(t,\\vx)} D\\_t \\psi(t,\\vx)\\right] - \\frac{D\\_t\n \\psi(t,\\vx)D\\_t \\ln \\rho(t,\\vx)}{c^2(t,\\vx)}\n-\\Delta\\_{\\vx} \\psi(t,\\vx) +q(t,\\vx) \\psi(t,\\vx) \\nonumber \\\\ \n=\\sigma\\_s \\frac{\\sqrt{\\rho(t,\\vx)}}{\\sqrt{\\rho\\_o}} e^{-i \\om\\_o t} S\n\\Big(\\frac{t}{T\\_s},\\frac{\\bx}{\\ell\\_s}\\Big) \\delta(z),\n\\label{eq:An3}\\end{aligned}$$ \nfor $t \\in \\RR$ and $\\vx = (\\bx,z) \\in \\RR^{d+1}$, where $\\Delta\\_{\\vx}$ is the Laplacian operator and\n \n$$\\begin{aligned}\nq(t,\\vx) = \\frac{\\Delta\\_{\\vx} \\sqrt{\\rho(t,\\vx)}}{\\sqrt{\\rho(t,\\vx)}}\n- \\frac{1}{c^2(t,\\vx)} \\left\\{ \\frac{D^2\\_t\n \\sqrt{\\rho(t,\\vx)}}{\\sqrt{\\rho(t,\\vx)}} - \\frac{1}{2}\\left[ D\\_t \\ln\n \\rho(t,\\vx) \\right]^2 \\right\\} \\nonumber \\\\-\\frac{1}{2}D\\_t\nc^{-2}(t,\\vx) D\\_t \\ln{\\rho(t,\\vx)}.\n\\label{eq:An4}\\end{aligned}$$ \nThe initial condition becomes\n \n$$\\psi(t,\\vx) \\equiv 0, \\quad t \\ll -T\\_s.\n\\label{eq:An4p}$$ \nScaled wave equation\n--------------------\nWe use the scaling regime defined in section [sect:scaling] and denote with primes the dimensionless, order one variables\n \n$$\\vx = L \\vx', \\quad t = T\\_L t'.\n\\label{eq:An6}$$ \nWe also let\n \n$$\\vv\\_o = V \\vv\\_o', \\quad c\\_o = c\\_o c\\_o', \\quad \\om\\_o = \\om\\_o\n\\frac{\\om\\_o'}{2 \\pi},\n\\label{eq:An7}$$ \nwhere the constants *c**o*ʹ = 1 and $\\om\\_o' = 2 \\pi$ are introduced so that the scaled equation is easier to interpret.\nIn the scaled variables, and using the source amplitude, the right-hand side in becomes\n \n$$\\begin{aligned}\n\\hspace{-0.1in}\\sigma\\_s \\frac{\\sqrt{\\rho(t,\\vx)}}{\\sqrt{\\rho\\_o}} e^{-i \\om\\_o t} S\n\\Big(\\frac{t}{T\\_s},\\frac{\\bx}{\\ell\\_s}\\Big) \\delta(z) = \\frac{\\big[1+\n O(\\sqrt{\\ep})\\big]}{\\ep \\eta\\_s L^2} \\Big(\\frac{\\gamma\\_s}{\\ep}\n\\Big)^d e^{-i \\frac{\\om\\_o'}{\\ep} t'} S\n\\Big(\\frac{t'}{\\eta\\_s},\\frac{\\gamma\\_s \\bx'}{\\ep} \\Big) \\delta (z').\n\\label{eq:An7p}\\end{aligned}$$ \nWe also have from definitions – that the random coefficients take the form\n \n$$\\begin{aligned}\n\\frac{\\vv(t,\\vx)}{V} &= \\vv'(t',\\vx') = \\vv\\_o' + \\sqrt{\\ep \\gamma}\n\\,\\bar\\sigma\\_{v} \\,\\vnu \\Big(\\frac{t'}{\\eta}, \\frac{\\vx'-\\ep \\vv\\_o'\n t'}{\\ep/\\gamma}\\Big),\n\\label{eq:An9} \\\\\n\\frac{\\rho(t,\\vx)}{\\rho\\_o} &= \\exp \\left[ \\sqrt{\\ep \\gamma}\\, \n \\bar\\sigma\\_{\\rho} \\, \\nu\\_\\rho \\Big(\\frac{t'}{\\eta}, \\frac{\\vx'-\\ep\n \\vv\\_o't'}{\\ep/\\gamma} \\Big) \\right], \\label{eq:An9p}\n\\\\ \\frac{c\\_o^2}{c^2(t,\\vx)} &= \\frac{1}{(c\\_o')^2} \\left[ 1 + \\sqrt{\\ep\n \\gamma}\\, \\bar{\\sigma}\\_c \\,\\nu\\_c\\Big(\\frac{t'}{\\eta}, \\frac{\\vx'-\\ep\n \\vv\\_o't'}{\\ep/\\gamma} \\Big) \\right], \\label{eq:An9pp}\\end{aligned}$$ \nwith scaled standard deviations *σ̄**c*, *σ̄**v*, *σ̄**ρ* defined in.\nThe solution *ψ* of must have variations in *t*ʹ and $\\vx'$ on the same scale as the source term and the coefficients –, meaning that $\\partial\\_{t'} \\psi \\sim {{1}/{\\ep}}$, $|\\nabla\\_{\\vx'} \\psi| \\sim\n{1}/{\\ep}$. From equations – we obtain that in the scaled variables we have\n \n$$D\\_t = \\frac{1}{T\\_L c\\_o'} D\\_{t'}^\\ep, \\quad \\mbox{with}~D\\_{t'}^\\ep =\n\\partial\\_{t'} + \\ep \\vv'(t',\\vx') \\cdot \\nabla\\_{\\vx'}.\n\\label{eq:An8}$$ \nEquation gives\n \n$$\\begin{aligned}\nD\\_t \\ln{\\rho(t,\\vx)} &= \\frac{\\sqrt{\\ep\n \\gamma} \\bar{\\sigma}\\_\\rho}{ T\\_L } D\\_{t'}^\\ep \\nu\\_\\rho\n\\Big(\\frac{t'}{\\eta}, \\frac{\\vx'-\\ep \\vv\\_o't'}{\\ep/\\gamma} \\Big) =\n\\frac{O(\\sqrt{\\ep})}{T\\_L},\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n\\frac{D\\_t^2 \\sqrt{\\rho(t,\\vx)}}{\n \\sqrt{\\rho(t,\\vx)}} & = \\frac{\\sqrt{\\ep \\gamma} \\bar{\\sigma}\\_\\rho}{2\n T\\_L^2} (D\\_{t'}^\\ep)^2 \\nu\\_\\rho \\Big(\\frac{t'}{\\eta}, \\frac{\\vx'-\\ep\n \\vv\\_o't'}{\\ep/\\gamma} \\Big) + \\frac{\\ep \\gamma\n \\bar{\\sigma}\\_\\rho^2}{4 T\\_L^2} \\left[ D\\_{t'}^\\ep \\nu\\_\\rho\n \\Big(\\frac{t'}{\\eta}, \\frac{\\vx'-\\ep \\vv\\_o't'}{\\ep/\\gamma}\n \\Big)\\right]^2 \\nonumber \\\\&= \\frac{O(\\sqrt{\\ep})}{T\\_L^2}. \\end{aligned}$$ \nFrom equation we get\n \n$$D\\_t \\left[ \\frac{1}{c^2(t,\\vx)} \\right] = \\frac{\\sqrt{\\ep\n \\gamma} \\bar{\\sigma}\\_c}{(c\\_o c\\_o')^2 T\\_L} D\\_{t'}^\\ep \\nu\\_c \\Big(\\frac{t'}{\\eta},\n\\frac{\\vx'-\\ep \\vv\\_o't'}{\\ep/\\gamma} \\Big) = \\frac{O(\\sqrt{\\ep})}{c\\_o L},$$ \nand *q* defined in takes the form\n \n$$q(t,\\vx) = \\frac{ \\gamma^{5/2} \\bar{\\sigma}\\_\\rho}{2 \\ep^{3/2} L^2}\\left[\nQ^\\ep\\Big(\\frac{t'}{\\eta}, \\frac{\\vx'-\\ep \\vv\\_o't'}{\\ep/\\gamma}\n\\Big) + O(\\ep^2)\\right],\n\\label{eq:An15}$$ \nwith\n \n$$Q^\\ep(\\tau,\\vr) = \\Delta\\_{\\vr} \\nu\\_\\rho(\\tau,\\vr) + \\frac{\\sqrt{\\ep\n \\gamma} \\bar{\\sigma}\\_\\rho}{2} \\left|\\nabla\\_{\\vr} \\nu\\_\\rho(\\tau,\\vr)\n\\right|^2.\n\\label{eq:An16}$$ \nSubstituting in and multiplying both sides by $\\ep\nL^2$, we obtain that the potential denoted by $\\psi'(t',\\vx')$ in the scaled variables satisfies\n \n$$\\begin{aligned}\n&\\hspace{-0.15in}\\ep \\left\\{ \\frac{\\left[1 + \\sqrt{\\ep \\gamma}\\,\n \\bar{\\sigma}\\_c \\nu\\_c\\big(\\frac{t'}{\\eta}, \\frac{\\vx'-\\ep \\vv\\_o't'}{\\ep/\\gamma}\n \\big)\\right]}{(c\\_o')^2} \\partial\\_{t'}^2 + \\frac{2 \\ep}{(c\\_o')^2}\n \\vv\\_o' \\cdot \\nabla\\_{\\vx'} \\partial\\_{t'} - \\Delta\\_{\\vx'}\n \\right\\}\\psi'(t',\\vx') \\nonumber\n \\\\ & \\hspace{-0.1in}+\\frac{\\bar{\\sigma}\\_\\rho \\gamma^{5/2}}{2\n \\sqrt{\\ep}} Q^\\ep\\Big(\\frac{t'}{\\eta}, \\frac{\\vx'-\\ep\n \\vv\\_o't'}{\\ep/\\gamma} \\Big) \\psi'(t',\\vx') \\approx\n \\frac{1}{\\eta\\_s} \\Big(\\frac{\\gamma\\_s}{\\ep}\\Big)^d e^{-i\n \\frac{\\om\\_o'}{\\ep} t'} S\n \\Big(\\frac{t'}{\\eta\\_s},\\frac{\\bx'}{\\ep/\\gamma\\_s} \\Big) \\delta (z'),\n\\label{eq:An18}\\end{aligned}$$ \nwith initial condition obtained from and,\n \n$$\\psi'(t',\\vx') \\equiv 0, \\quad t' \\ll - \\eta\\_s.\n\\label{eq:An18p}$$ \nThe approximation in equation is because we neglect $O(\\sqrt{\\ep})$ terms that tend to zero in the limit $\\ep \\to 0.$ Note in particular that the random perturbations $\\vnu$ of the velocity of the flow appear in these terms and are negligible in our regime.\nAll variables are assumed scaled in the remainder of the section and we simplify notation by dropping the primes.\nMoving frame\n------------\nLet us introduce the notation $\\vv\\_o = (\\bv\\_o,v\\_{oz})$ for the scaled mean velocity of the ambient flow, and change the range coordinate *z* to\n \n$$\\zeta = z - \\ep v\\_{oz} t.\n\\label{eq:An19}$$ \nWe denote the potential in this moving frame by\n \n$$u^\\ep(t,\\bx,\\zeta) = \\psi(t,\\bx,\\zeta + \\ep v\\_{oz} t), \n\\label{eq:An20}$$ \nand obtain from that it satisfies the wave equation\n \n$$\\begin{aligned}\n&\\ep \\left\\{ \\frac{\\left[1 + \\sqrt{\\ep \\gamma}\\,\n \\bar{\\sigma}\\_c \\nu\\_c\\Big(\\frac{t}{\\eta}, \\frac{\\bx-\\ep \\bv\\_ot}{\\ep/\\gamma},\n \\frac{\\gamma \\zeta}{\\ep} \\Big)\\right]}{c\\_o^2} \\partial\\_{t}^2 +\n \\frac{2 \\ep}{c\\_o^2} \\bv\\_o \\cdot \\nabla\\_{\\bx} \\partial\\_t -\n \\Delta\\_{\\bx} - \\partial\\_\\zeta^2 \\right\\}u^\\ep(t,\\bx,\\zeta) \n \\nonumber \\\\ & +\\frac{\\bar{\\sigma}\\_\\rho \\gamma^{5/2}}{2\n \\sqrt{\\ep}}Q^\\ep\\Big(\\frac{t}{\\eta}, \\frac{\\bx-\\ep\n \\bv\\_ot}{\\ep/\\gamma},\\frac{\\gamma \\zeta}{\\ep} \\Big)\n u^\\ep(t,\\bx,\\zeta) \\approx \\Big(\\frac{\\gamma\\_s}{\\ep}\\Big)^d\n \\frac{e^{-i \\frac{\\om\\_o}{\\ep} t}}{\\eta\\_s} S\n \\Big(\\frac{t}{\\eta\\_s},\\frac{\\gamma\\_s \\bx}{\\ep} \\Big) \\delta (\\zeta +\n \\ep v\\_{oz} t),\n\\label{eq:An21}\\end{aligned}$$ \nwhere again we neglect the terms that become negligible in the limit $\\ep \\to 0$. The gradient $\\nabla\\_{\\bx}$ and Laplacian $\\Delta\\_{\\bx}$ are in the cross-range variable $\\bx \\in \\RR^d$.\nWave decomposition\n------------------\nThe interaction of the waves with the random medium depends on the frequency and direction of propagation, so we decompose $u^\\ep(t,\\bx,\\zeta)$ using the Fourier transform\n \n$$\\hat u^\\ep(\\om,\\bk, \\zeta) = \\int\\_{\\RR} {\\rm d}t \\int\\_{\\RR^d} {\\rm d} \\bx \\,\nu^\\ep(t,\\bx,\\zeta) e^{i \\big(\\frac{\\om\\_o}{\\ep} + \\om\\big)t - i\n \\frac{\\bk}{\\ep} \\cdot \\bx},\n\\label{eq:An22}$$ \nwith inverse\n \n$$u^\\ep(t,\\bx, \\zeta) = \\int\\_{\\RR} \\frac{{\\rm d} \\om}{2 \\pi}\n\\int\\_{\\RR^d} \\frac{{\\rm d} \\bk}{(2 \\pi \\ep)^d} \\, \\hat\nu^\\ep(\\om,\\bk,\\zeta) e^{-i \\big(\\frac{\\om\\_o}{\\ep} + \\om\\big)t + i\n \\frac{\\bk}{\\ep} \\cdot \\bx}.\n\\label{eq:An23}$$ \nThe Fourier transform of equation is\n \n$$\\begin{aligned}\n&\\hspace{-0.15in}\\left[ -\\frac{\\beta^2(\\bk)}{\\ep} - 2 k\\_o\n \\Big(\\frac{\\om}{c\\_o} - \\frac{\\bv\\_o \\cdot \\bk}{c\\_o} \\Big) - \n \\ep\n \\partial\\_\\zeta^2 \\right] \\hat u^\\ep(\\om,\\bk,\\zeta) \n- \\frac{\\eta\n \\gamma^{1/2 -d}}{\\sqrt{\\ep}} \\int\\_{\\RR} \\frac{{\\rm d} \\om'}{2\\pi}\n \\int\\_{\\RR^d} \\frac{{\\rm d} \\bk'}{(2\\pi)^d} \\nonumber\n \\\\ &\\hspace{0.95in}\\hat u^\\ep(\\om',\\bk',\\zeta) \\left[k\\_o^2\n \\bar{\\sigma}\\_c \\hat\n \\nu\\_c\\Big( \\eta\\big(\\om-\\om'-(\\bk-\\bk')\\cdot \\bv\\_o\\big),\n \\frac{\\bk-\\bk'}{\\gamma}, \\frac{\\gamma \\zeta}{\\ep} \\Big)\n \\right. \\nonumber \\\\ &\\left. \\hspace{1.2in} -\n \\frac{\\gamma^2\\bar{\\sigma}\\_\\rho}{2} \\hat Q^\\ep \\Big(\n \\eta\\big(\\om-\\om'-(\\bk-\\bk')\\cdot \\bv\\_o\\big),\n \\frac{\\bk-\\bk'}{\\gamma}, \\frac{\\gamma \\zeta}{\\ep} \\Big) \\right]\n \\nonumber \\\\ &\\hspace{0.95in}\\approx \\frac{e^{-i \\frac{\\om \\zeta}{\\ep\n v\\_{oz}}}}{\\ep \\eta\\_s v\\_{oz}} \\widecheck{S}\\Big(\n -\\frac{\\zeta}{\\ep \\eta\\_s v\\_{oz}}, \\frac{\\bk}{\\gamma\\_s}\n \\Big), \\label{eq:An25}\\end{aligned}$$ \nwhere\n \n$$\\beta(\\bk) = \\sqrt{k\\_o^2 - |\\bk|^2}, \\quad k\\_o = \\frac{\\om\\_o}{c\\_o},\n\\label{eq:An24}$$ \nand \n$$\\widecheck{S}(\\tau,\\boldsymbol{\\kappa}) = \\int\\_{\\RR^d} {\\rm d}\\br \\,\nS(\\tau,\\br) e^{-i \\boldsymbol{\\kappa} \\cdot \\br}.$$ \nNote that the right-hand side in is supported at $|\\bk| = O(\\gamma\\_s)$, so by keeping *γ**s* small, we ensure that $\\beta(\\bk)$ remains real valued in our regime. Physically, this means that $\\hat u^\\ep(\\om,\\bk,\\zeta)$ is a propagating wave, not evanescent.\nNote also that if the mean velocity $\\vv\\_o$ is orthogonal to the range direction, the source term satisfies \n$$\\lim\\_{v\\_{oz} \\to 0} \\frac{e^{-i \\frac{\\om \\zeta}{\\ep\n v\\_{oz}}}}{\\ep \\eta\\_s\n v\\_{oz}} \\widecheck{S}\\Big( -\\frac{\\zeta}{\\ep \\eta\\_s v\\_{oz}},\n \\frac{\\bk}{\\gamma\\_s} \\Big) \\to \\hat S\\Big(\\eta\\_s \\om,\n \\frac{\\bk}{\\gamma\\_s} \\Big) \\delta(\\zeta),$$ \n in the sense of distributions, where \n$$\\hat S(\\omega,\\boldsymbol{\\kappa}) = \\int\\_{\\RR} {\\rm d}\\tau \\int\\_{\\RR^d}\n {\\rm d}\\br \\, S(\\tau,\\br) e^{i \\om \\tau -i \\boldsymbol{\\kappa}\n \\cdot \\br}.$$ \nWe introduce\n \n$$\\begin{aligned}\n a^\\ep(\\om,\\bk,\\zeta) & = \\Big[ \\frac{\\sqrt{\\beta(\\bk)}}{2} \n\\hat u^\\ep(\\om, \\bk,\\zeta)\n+\\frac{\\ep}{2i \\sqrt{\\beta(\\bk)}} \\partial\\_\\zeta \\hat u^\\ep(\\om, \\bk,\\zeta)\n\\Big] e^{- i \\beta(\\bk) \\frac{\\zeta}{\\ep}}, \\\\\n a\\_-^\\ep(\\om,\\bk,\\zeta) \n &= \\Big[ \\frac{\\sqrt{\\beta(\\bk)}}{2} \n\\hat u^\\ep(\\om, \\bk,\\zeta) -\\frac{\\ep}{2i \\sqrt{\\beta(\\bk)}} \\partial\\_\\zeta \\hat u^\\ep(\\om, \\bk,\\zeta)\n\\Big] e^{ i \\beta(\\bk) \\frac{\\zeta}{\\ep}},\\end{aligned}$$ \nso that we have the decomposition\n \n$$\\begin{aligned}\n\\hat u^\\ep(\\om, \\bk,\\zeta) = \\frac{1}{\\sqrt{\\beta(\\bk)}} \\left[\n a^\\ep(\\om,\\bk,\\zeta) e^{i \\beta(\\bk) \\frac{\\zeta}{\\ep}} +\n a\\_-^\\ep(\\om,\\bk,\\zeta) e^{-i \\beta(\\bk) \\frac{\\zeta}{\\ep}}\\right],\n\\label{eq:An26}\\end{aligned}$$ \nand the complex amplitudes $a^\\ep$ and $a\\_-^\\ep$ satisfy the relation\n \n$$\\partial\\_\\zeta a^\\ep(\\om,\\bk,\\zeta) e^{i \\beta(\\bk) \\frac{\\zeta}{\\ep}} +\n\\partial\\_\\zeta a\\_-^\\ep(\\om,\\bk,\\zeta) e^{-i \\beta(\\bk) \\frac{\\zeta}{\\ep}} = 0.\n\\label{eq:An27}$$ \nThis gives that\n \n$$\\begin{aligned}\n\\partial\\_\\zeta \\hat u^\\ep(\\om, \\bk,\\zeta) = \\frac{i\n \\sqrt{\\beta(\\bk)}}{\\ep} \\left[ a^\\ep(\\om,\\bk,\\zeta) e^{i\n \\beta(\\bk) \\frac{\\zeta}{\\ep}} - a\\_-^\\ep(\\om,\\bk,\\zeta) e^{-i \\beta(\\bk)\n \\frac{\\zeta}{\\ep}}\\right],\n\\label{eq:An28}\\end{aligned}$$ \nand moreover, that\n \n$$\\begin{aligned}\n\\partial^2\\_\\zeta \\hat u^\\ep(\\om, \\bk,\\zeta) =\n-\\frac{\\beta^2(\\bk)}{\\ep^2} \\hat u^\\ep(\\om, \\bk,\\zeta) + \\frac{2 i\n \\sqrt{\\beta(\\bk)}}{\\ep} \\partial\\_\\zeta a^\\ep(\\om,\\bk,\\zeta) e^{i\n \\beta(\\bk) \\frac{\\zeta}{\\ep}}.\n\\label{eq:An29}\\end{aligned}$$ \nThe decomposition in and is a decomposition of $u^\\ep(t,\\bx,\\zeta)$ into a superposition of plane waves with wave vectors\n \n$$\\vec{\\bk}\\_{\\pm} = (\\bk,\\pm \\beta(\\bk)),$$ \nwhere the plus sign denotes the waves propagating in the positive range direction and the negative sign denotes the waves propagating in the negative range direction. The amplitudes $a^\\ep$ and $a\\_-^\\ep$ of these waves are random fields, which evolve in range according to and the equation\n \n$$\\begin{aligned}\n&\\partial\\_\\zeta a^\\ep(\\om,\\bk,\\zeta) \\approx \\frac{i k\\_o (\\om -\\bv\\_o\n \\cdot \\bk)}{c\\_o\\beta(\\bk)} \\left[ a^\\ep(\\om,\\bk,\\zeta) +\n a\\_-^\\ep(\\om,\\bk,\\zeta)e^{-2 i \\beta(\\bk) \\frac{\\zeta}{\\ep}}\n \\right] \\nonumber \\\\ &\\hspace{0.3in}+\\frac{i \\eta\n \\gamma^{1/2-d}}{2 \\sqrt{\\ep}} \\int\\_{\\RR} \\frac{{\\rm d} \\om'}{2\\pi}\n \\int \\frac{{\\rm d} \\bk'}{(2 \\pi)^d} \\left[ k\\_o^2 \\bar{\\sigma}\\_c \\hat \\nu\\_c\\Big(\n \\eta\\big(\\om-\\om'-(\\bk-\\bk')\\cdot \\bv\\_o\\big),\n \\frac{\\bk-\\bk'}{\\gamma}, \\frac{\\gamma \\zeta}{\\ep} \\Big)\n \\right. \\nonumber \\\\ &\\hspace{1.55in}\\left.\n -\\frac{\\gamma^2\\bar{\\sigma}\\_\\rho}{2} \\hat Q^\\ep \\Big(\n \\eta\\big(\\om-\\om'-(\\bk-\\bk')\\cdot \\bv\\_o\\big),\n \\frac{\\bk-\\bk'}{\\gamma}, \\frac{\\gamma \\zeta}{\\ep} \\Big)\\right]\n \\nonumber \\\\&\\hspace{0.3in}\\times \\frac{1}{\\sqrt{\\beta(\\bk)\n \\beta(\\bk')}}\\left[a^\\ep(\\om',\\bk',\\zeta)e^{i\n \\big[\\beta(\\bk')-\\beta(\\bk)\\big] \\frac{\\zeta}{\\ep}} +\n a\\_-^\\ep(\\om',\\bk',\\zeta)e^{-i \\big[\\beta(\\bk') +\n \\beta(\\bk)\\big]\\frac{\\zeta}{\\ep}} \\right] \\nonumber\n \\\\ &\\hspace{0.3in}+\\frac{i}{2 \\sqrt{\\beta(\\bk)} \\ep \\eta\\_s\n v\\_{oz}} \\widecheck{S}\\Big( -\\frac{\\zeta}{\\ep \\eta\\_s v\\_{oz}},\n \\frac{\\bk}{\\gamma\\_s} \\Big)e^{-i \\frac{\\om \\zeta}{\\ep v\\_{oz}}- i\n \\beta(\\bk) \\frac{\\zeta}{\\ep}},\n\\label{eq:An30}\\end{aligned}$$ \nderived by substituting – into.\nForward scattering approximation\n--------------------------------\nEquation shows that the amplitudes $a^\\ep$ are coupled to each other and to $a^\\ep\\_-$. In our scaling regime, where the cone of directions of propagation has small opening angle controlled by the parameter *γ**s*, and where the covariance of the fluctuations is smooth, the coupling between $a^\\ep$ and $a\\_{-}^\\ep$ becomes negligible in the limit $\\ep \\to\n0$. We refer to and for a more detailed explanation of this fact.\nUsing the assumption that the random fluctuations are supported at finite range (see section [sect:form]), we require that the wave be outgoing at ∣*ζ*∣ → ∞. This radiation condition and the negligible coupling between $a^\\ep$ and $a^\\ep\\_-$ in the limit $\\ep\n\\to 0$ imply that we can neglect the backward going waves, and we can write\n \n$$\\hat u^\\ep(\\om,\\bk,\\zeta) \\approx\n\\frac{a^\\ep(\\om,\\bk,\\zeta)}{\\sqrt{\\beta(\\bk)}} e^{i \\beta(\\bk)\n \\frac{\\zeta}{\\ep}}, \\quad \\zeta > O(\\ep).\n\\label{eq:An31f}$$ \nThe starting value of $a^\\ep(\\om,\\bk,\\zeta)$ is determined by the source term in, which contributes only for $\\zeta =\nv\\_{oz}O(\\ep)$. For such small *ζ*, we can change variables $\\zeta\n= \\ep \\xi$ in and obtain that\n \n$$\\begin{aligned}\n\\partial\\_\\xi a^\\ep(\\om,\\bk,\\ep \\xi) = \\frac{i}{2 \\sqrt{\\beta(\\bk)}\n \\eta\\_s v\\_{oz}} \\widecheck{S}\\Big( -\\frac{\\xi}{ \\eta\\_s v\\_{oz}},\n\\frac{\\bk}{\\gamma\\_s} \\Big)e^{-i \\frac{\\om \\xi}{ v\\_{oz}}- i \\beta(\\bk)\n \\xi} + O(\\sqrt{\\ep}).\\end{aligned}$$ \nIntegrating in *ξ* and using that $a^\\ep(\\om,\\bk,\\zeta)$ vanishes for $\\zeta \\ll - O(\\ep)$, we obtain that\n \n$$\\begin{aligned}\na^\\ep(\\om,\\bk,\\ep \\xi) &\\approx \\frac{i}{2 \\sqrt{\\beta(\\bk)} \\eta\\_s\n v\\_{oz}} \\int\\_{\\RR} {\\rm d} \\xi \\, \\widecheck{S}\\Big( -\\frac{\\xi}{\n \\eta\\_s v\\_{oz}}, \\frac{\\bk}{\\gamma\\_s} \\Big)e^{-i \\frac{\\om \\xi}{\n v\\_{oz}}- i \\beta(\\bk) \\xi} \\nonumber \\\\ &= \\frac{i}{2\n \\sqrt{\\beta(\\bk)}} \\hat S \\Big(\\eta\\_s(\\om +\n\\beta(\\bk)v\\_{oz}),\\frac{\\bk}{\\gamma\\_s}\\Big).\\end{aligned}$$ \nWe use this expression as the initial condition for the forward going amplitudes\n \n$$a^\\ep(\\om,\\bk,0+) \\approx \\frac{i}{2\\sqrt{\\beta(\\bk)}} \\hat S\n\\Big(\\eta\\_s(\\om + \\beta(\\bk)v\\_{oz}),\\frac{\\bk}{\\gamma\\_s}\\Big),\n\\label{eq:An35}$$ \nand drop the source term and the backward going amplitudes $a\\_{-}^\\ep$ in equation for range *ζ* > 0.\nThe acoustic pressure field in the Markovian limit\n--------------------------------------------------\nBy definitions,, and the scaling relations, the acoustic pressure is\n \n$$p(T\\_Lt,L\\bx,Lz) \\approx - \\frac{ {\\rho\\_o}}{T\\_L}\\partial\\_t\nu^\\ep(t,\\bx,z-\\ep v\\_{oz} t).$$ \nFurthermore, equation and the Fourier decomposition give that\n \n$$\\hspace{-0.15in}\\frac{p(T\\_Lt,L \\bx,Lz)}{2 \\pi c\\_o {\\rho\\_o}/\\la\\_o} \\approx\n\\int\\_{\\RR} \\frac{\\rm{d} \\om}{2 \\pi} \\int\\_{\\RR^d} \\frac{{\\rm d} \\bk}{(2\n \\pi \\ep)^d} \\frac{i a^\\ep(\\om,\\bk,z)}{\\sqrt{\\beta(\\bk)}} e^{ - i\n \\big(\\frac{\\om\\_o}{\\ep} + \\om + \\beta(\\bk) v\\_{oz}\\big)t+ i\n \\frac{(\\bk, \\beta(\\bk))}{\\ep} \\cdot (\\bx,z)},$$ \nwhere we have used equation to write $\na^\\ep(\\om,\\bk,z-\\ep v\\_{oz}t) = a^\\ep(\\om,\\bk,z) + O(\\sqrt{\\ep})$.\nThe shifted scaled frequency $\\om + \\beta(\\bk) v\\_{oz}$ appears in the initial condition, and the random processes $\\hat\n\\nu\\_c$ and $\\hat Q^\\ep$ in equation depend on $\\bk/\\gamma$. Thus, it is convenient to introduce the variables\n \n$$\\Omega = \\eta \\big[ \\om + \\beta(\\bk) v\\_{oz}\\big], \\quad \\bK = \\bk/\\gamma,\n\\label{eq:An38}$$ \nand rewrite the expression of the pressure as\n \n$$\\hspace{-0.11in}\\frac{p(T\\_Lt,L \\bx,Lz)}{\\om\\_o {\\rho\\_o}} \\approx\n\\int\\_{\\RR} \\frac{{\\rm d} \\Omega}{2 \\pi \\eta}\n\\int\\_{\\RR^d} \\hspace{-0.02in}\\frac{{\\rm d} \\bK}{(2 \\pi \\ep/\\gamma)^d}\n\\frac{A^\\ep(\\Omega,\\bK,z)}{\\sqrt{\\beta(\\gamma \\bK)}} e^{ - i\n \\big(\\frac{\\om\\_o}{\\ep} + \\frac{\\Omega}{\\eta} \\big)t+ i \\frac{(\\gamma \\bK,\n \\beta(\\gamma \\bK))}{\\ep} \\cdot (\\bx,z)},\n\\label{eq:An40}$$ \nwith redefined amplitude\n \n$$A^\\ep(\\Omega,\\bK,z) = i a^\\ep \\Big(\\frac{\\Omega}{\\eta} -\n\\beta(\\gamma \\bK)v\\_{oz},\\gamma \\bK,z\\Big) + o(1).\n\\label{eq:An39}$$ \nThe *o*(1) term, which tends to zero as $\\ep \\to 0$, is used in this definition so that we have an equal sign in the evolution equation for $A^\\ep$, derived from, after neglecting the backward going amplitudes,\n \n$$\\begin{aligned}\n& \\partial\\_z A^\\ep(\\Omega,\\bK,z) = \\frac{i\n k\\_o}{\\beta(\\gamma \\bK)} \\Big[\\frac{\\Omega}{\\eta\n c\\_o}-\\frac{\\vv\\_o}{c\\_o} \\cdot \\big(\\gamma \\bK, \\beta(\\gamma\n \\bK)\\big)\\Big]A^\\ep(\\Omega,\\bK,z) \\nonumber \\\\ \n &\n +\\frac{i}{2} \\sqrt{\\frac{\\gamma}{\\ep}} \\int\\_{\\RR} \\frac{{\\rm d}\n \\Omega'}{2\\pi} \\int\\_{\\RR^d} \\frac{{\\rm d} \\bK'}{(2 \\pi)^d}\n \\frac{A^\\ep(\\Omega',\\bK',z)}{\\sqrt{\\beta(\\gamma \\bK) \\beta(\\gamma\n \\bK')}}e^{i \\big[\\beta(\\gamma \\bK')-\\beta(\\gamma \\bK)\\big]\n \\frac{z}{\\ep}} \\nonumber \\\\\n & \n \\quad \\times \\Big[k\\_o^2 \\bar{\\sigma}\\_c\\hat\n \\nu\\_c\\Big(\\Omega - \\Omega' - \\eta \\big(\\gamma \\bK-\\gamma\n \\bK',\\beta(\\gamma \\bK)-\\beta(\\gamma \\bK'))\\cdot \\vv\\_o, \\bK-\\bK',\n \\frac{\\gamma z}{\\ep} \\Big) \\nonumber\n \\\\ &\\quad-\\frac{\\gamma^2\\bar{\\sigma}\\_\\rho}{2} \\hat\n Q^{\\ep}\\Big(\\Omega - \\Omega' - \\eta \\big(\\gamma \\bK-\\gamma\n \\bK',\\beta(\\gamma \\bK)-\\beta(\\gamma \\bK'))\\cdot \\vv\\_o, \\bK-\\bK',\n \\frac{\\gamma z}{\\ep} \\Big) \\Big],\n\\label{eq:An41}\\end{aligned}$$ \nfor *z* > 0. The initial condition becomes\n \n$$A^\\ep(\\Omega,\\bK,0+) = A\\_o(\\Omega,\\bK) = -\\frac{1}{2\n \\sqrt{\\beta(\\gamma \\bK)}} \\hat S \\Big(\\frac{\\eta\\_s}{\\eta} \\Omega\n,\\frac{\\gamma}{\\gamma\\_s} \\bK\\Big).\n\\label{eq:An42}$$ \nThe Markovian limit\n-------------------\nLet *L*2(O, C) be the space of complex-valued, square-integrable functions defined on the set\n \n$$\\label{def:calO}\n\\mathscr{O} =\\{\\Omega \\in\n\\RR\\} \\times \\{\\bK\\in \\RR^d, \\gamma |\\bK|< k\\_o\\}$$ \nand denote ${\\itbf\n A}^\\ep (z) = (A^\\eps(\\Omega,\\bK,z))\\_{(\\Omega,\\bK)\\in\n \\mathscr{O}}$ for *z* ≥ 0. From equation we obtain the conservation of energy relation\n \n$$\\partial\\_{z} \n\\int\\_{\\mathscr{O}} {\\rm d} \\Omega {\\rm d} \\bK \\,\n|\\tae(\\Omega,\\bK,{z})|^2 = 0,$$ \nso the Markov process ${\\itbf A}^\\ep (z) \\in\nL^2(\\mathscr{O},\\mathbb{C})$ lives on the surface of the ball with center at the origin and $\\ep$ independent radius $R\\_{{\\itbf A}}$ defined by\n \n$$R\\_{{\\itbf A}}^2 =\n\\int\\_{\\mathscr{O}} {\\rm d} \\Omega {\\rm d} \\bK \\,\n |\\tae(\\Omega,\\bK,{z})|^2 =\n \\int\\_{\\mathscr{O}} {\\rm d} \\Omega {\\rm d} \\bK \\,\n |A\\_o(\\Omega,\\bK)|^2.\n \\label{eq:radius}$$ \nWe describe the Markovian limit $\\ep \\to 0$ in Appendix [ap:DL]. The result is that the process of ${\\itbf A}^\\ep(z)$ converges weakly in ${\\mathcal C}([0,\\infty),L^2)$ to a Markov process whose infinitesimal generator can be identified. The first and second moments of the limit process are described below.\n### The mean amplitude\nThe expectation of $\\tae(\\Omega,\\bK,z)$ in the limit $\\ep \\to 0$ is given by\n \n$$\\lim\\_{\\ep \\to 0} \\EE \\big[\\tae(\\Omega,\\bK,z)\\big] = A\\_o(\\Omega,\\bK)\ne^{i \\theta(\\Omega,\\bK) z + D(\\bK) z },\n\\label{eq:mean}$$ \nwhere\n \n$$\\begin{aligned}\n\\theta(\\Omega,\\bK) =\\frac{k\\_o}{\\beta(\\gamma \\bK)}\n\\Big[\\frac{\\Omega}{\\eta c\\_o} - \\frac{\\vv\\_o}{c\\_o} \\cdot (\\gamma\n \\bK,\\beta(\\gamma \\bK)) \\Big] + \n \\frac{\\gamma^3 \\bar{\\sigma}\\_\\rho^2}{8 \\beta(\\gamma \\bK)}\n\\Delta\\_{\\vr} \\cR\\_{\\rho \\rho}(0,\\vr)\\big|\\_{\\vr = {\\bf 0}},\n\\label{eq:mean1}\\end{aligned}$$ \nis a real phase and\n \n$$\\begin{aligned}\nD(\\bK) =& - \\int\\_{|\\bK'| < k\\_o/\\gamma}\n \\frac{{\\rm d}\\bK'}{(2\\pi)^d}\\, \\frac{1}{4\n \\beta(\\gamma\\bK)\\beta(\\gamma\\bK')} \\int\\_{\\RR^d} {\\rm d} \\br\n \\int\\_0^\\infty {\\rm d}r\\_z \\, e^{- i \\big(\\bK- \\bK',\\frac{\\beta(\\gamma\n \\bK)-\\beta(\\gamma \\bK')}{\\gamma} \\big)\\cdot {\\vr}}\\nonumber\n \\\\ &\\times \\left\\{k\\_o^4 \\bar{\\sigma}\\_c^2 {\\cR}\\_{cc}(0,\\vr) +\n \\frac{\\gamma^4 \\bar{\\sigma}\\_\\rho^2}{4}\n \\Delta\\_{\\vr}^2\\cR\\_{\\rho\\rho}(0,\\vr)- {k\\_o^2 \\gamma^2\n \\bar{\\sigma}\\_c \\bar{\\sigma}\\_\\rho}\\Delta\\_{\\vr} \\cR\\_{c\\rho}(0,\\vr) \n\\right\\},\\label{eq:mean2}\\end{aligned}$$ \nwith $\\vr = (\\br,r\\_z)$. Moreover, $ \\mbox{Re}\n\\big[ D(\\bK)\\big] < 0$, because \n$$\\int\\_{\\RR^{d+1}} {\\rm d} \\br \\, e^{- i \\vec{\\bK} \\cdot\n {\\vr}} \\left\\{k\\_o^4 \\bar{\\sigma}\\_c^2 {\\cR}\\_{cc}(0,\\vr) +\n \\frac{\\gamma^4 \\bar{\\sigma}\\_\\rho^2}{4}\n \\Delta\\_{\\vr}^2\\cR\\_{\\rho\\rho}(0,\\vr)- {k\\_o^2 \\gamma^2\n \\bar{\\sigma}\\_c\\bar{\\sigma}\\_\\rho}\\Delta\\_{\\vr} \\cR\\_{c\\rho}(0,\\vr) \n\\right\\} \\ge 0,$$ \n is the power spectral density of the process [eq:P] X(t,) = k02 |c c(t,) - (t,), in the variable $\\vr$ (for fixed *t*), which is non-negative by Bochner’s theorem. Thus, the mean amplitude decays on the range scale\n \n$$\\mathscr{S}(\\bK) = - \\frac{1}{\\mbox{Re} \\big[ D(\\bK)\\big]},\n\\label{eq:mean3}$$ \ncalled the scattering mean free path. In the relatively high frequency regime the damping is mainly due to the fluctuations of the wave speed, while in the relatively low frequency regime the damping is mainly due to the fluctuations of the density. This damping is the mathematical manifestation of the randomization of the wave due to cumulative scattering.\nRecall that we have assumed *σ̄**ρ* = *O*(1). Therefore, in the regime *γ* ≪ 1, the $ \\cR\\_{cc}$ term dominates in.\n### The mean intensity\nThe expectation of the intensity\n \n$$\\label{eq:TR1}\nI(\\Omega,\\bK,{z}) = \\lim\\_{\\ep \\to 0} \\EE \\big[|\\tae(\\Omega,\\bK,{z})|^2\\big]$$ \nsatisfies\n \n$$\\begin{aligned}\n\\label{eq:TR2}\n \\partial\\_{z} I(\\Omega,\\bK,{z}) = \n \\int\\_{\\mathscr{O}} \\frac{{\\rm d} \\Omega'}{2\\pi} \\frac{ {\\rm d} \\bK'}{(2\\pi)^d} \\,\n Q(\\Omega,\\Omega',\\bK,\\bK') \\left[\n I(\\Omega',\\bK',{z})-I(\\Omega,\\bK,{z})\\right],\\end{aligned}$$ \nfor *z* > 0, with initial condition obtained from equation :\n \n$$I(\\Omega,\\bK,0) = |A\\_o(\\Omega,\\bK)|^2.\n\\label{eq:TR3}$$ \nDenoting the power spectrum of $X(t, \\vr)$ in ([eq:P]) by $P(\\Omega, \\vec{\\bK})$, and letting \n$$\\left( {\\widetilde}{\\Omega}, \\vec{\\bK} \\right) \n = \\left(\\Omega - \\eta\n\\big(\\gamma \\bK, \\beta(\\gamma \\bK))\\cdot\n\\vv\\_o\n,\\bK,\\frac{\\beta(\\gamma \\bK)}{\\gamma}\\right),$$ \n the kernel in isgiven by \n$$Q(\\Omega,\\Omega',\\bK,\\bK')\n = \\frac{ P( {\\widetilde}{\\Omega} -{\\widetilde}{\\Omega}', \\vec{\\bK} - \\vec{\\bK}' ) }\n {4 \\beta(\\gamma\\bK)\\beta(\\gamma \\bK')},$$ \n that is explicitly\n \n$$\\begin{aligned}\nQ&(\\Omega,\\Omega',\\bK,\\bK') = \\left\\{k\\_o^4 \\bar{\\sigma}\\_c^2 \\widetilde\n\\cR\\_{cc} + \\frac{\\gamma^4 \\bar\\sigma\\_{\\rho}^2}{4}\\left[ |\\bK-\\bK'|^2 +\n \\Big(\\frac{\\beta(\\gamma \\bK)-\\beta(\\gamma \\bK')}{\\gamma} \\Big)^2\n \\right]^2 \\widetilde \\cR\\_{\\rho \\rho}\\right. \\nonumber\n\\\\&\\left. +k\\_o^2 \\gamma^2 \\bar{\\sigma}\\_c\\bar{\\sigma}\\_\\rho \\left[ |\\bK-\\bK'|^2 +\n \\Big(\\frac{\\beta(\\gamma \\bK)-\\beta(\\gamma \\bK')}{\\gamma} \\Big)^2\n \\right] \\widetilde \\cR\\_{\\rho c} \\right\\}\\frac{1}\n { 4 \\beta(\\gamma\\bK)\\beta(\\gamma \\bK')},\n\\label{eq:IntKer}\\end{aligned}$$ \nwhere $\\widetilde \\cR\\_{cc}$ is the power spectral density, evaluated as \n$$\\widetilde \\cR\\_{cc} = \\widetilde \\cR\\_{cc} \\Big(\\Omega - \\Omega' - \\eta\n\\big(\\gamma \\bK-\\gamma \\bK',\\beta(\\gamma \\bK)-\\beta(\\gamma \\bK'))\\cdot\n\\vv\\_o, \\bK-\\bK', \\frac{\\beta(\\gamma \\bK)-\\beta(\\gamma \\bK')}{\\gamma}\n\\Big),$$ \n and similar for $\\widetilde \\cR\\_{ \\rho c}$ and $\\widetilde \\cR\\_{\\rho\n \\rho}$.\nNote that the kernel satisfies\n \n$$\\int\\_{\\mathscr{O}} \\frac{{\\rm d} \\Omega'}{2 \\pi} \\frac{{\\rm d}\\bK'}{(2 \\pi)^d}\\,\n Q(\\Omega,\\Omega',\\bK,\\bK') = - 2\n\\mbox{Re}[\\mathscr{Q}(\\bK)] = \\frac{2}{\\mathscr{S}(\\bK)},\n\\label{eq:TR5}$$ \nwhere $\\mathscr{S}(\\bK)$ is the scattering mean free path defined in.\n### The Wigner transform\nThe wave amplitudes decorrelate at distinct frequencies Ω ≠ Ωʹ and wave vectors $\\bK \\ne \\bK'$, meaning that\n \n$$\\lim\\_{\\ep \\to 0} \\EE \\big[\\tae(\\Omega,\\bK,{z})\n \\overline{\\tae}(\\Omega',\\bK',z)\\big] = \\lim\\_{\\ep \\to 0} \\EE\n\\big[\\tae(\\Omega,\\bK,{z})] \\lim\\_{\\ep \\to 0}\n\\EE[\\overline{\\tae}(\\Omega',\\bK',z)\\big].\n\\label{eq:Adecorrel}$$ \nThe right-hand side is the product of the means of the mode amplitudes, which decay on the range scale defined by the scattering mean free path.\nHowever, the amplitudes are correlated for $|\\Omega - \\Omega'| =\nO(\\ep)$ and $|\\bK-\\bK'| = O(\\ep)$. We are interested in the second moment \n$$\\EE \\Big[\\tae\\Big(\\Omega,\\bK + \\frac{\\ep \\bq}{2},{z}\\Big)\n \\overline{\\tae\\Big(\\Omega,\\bK- \\frac{\\ep \\bq}{2},z\\Big)}\\Big],$$ \n whose Fourier transform in $\\bq$ gives the energy density resolved over frequencies and directions of propagation. This is the Wigner transform defined by\n \n$$W^\\ep(\\Omega,\\bK,\\bx,{z}) = \\int\\_{\\mathbb{R}^d} \\frac{{\\rm d} \\bq}{(2\n \\pi)^d} e^{i \\bq \\cdot ( \\nabla \\beta(\\gamma\\bK) {z} + \\bx)} \\EE\n\\hspace{-0.03in}\\left[ \\tae\\Big(\\Omega,\\bK + \\frac{\\ep\n \\bq}{2},{z}\\Big) \\overline{\\tae\\Big(\\Omega,\\bK - \\frac{\\ep\n \\bq}{2},{z}\\Big)}\\right].\n\\label{eq:defWigner}$$ \nWe show in Appendix [sec:appW] that the Wigner transform converges in the limit $\\ep \\to 0$ to $W(\\Omega,\\bK,\\bx,{z})$, the solution of the transport equation\n \n$$\\begin{aligned}\n\\left[\\partial\\_{z} - \\nabla\\beta(\\gamma \\bK) \\cdot \\nabla\\_{\\bx}\n \\right] W(\\Omega,\\bK,\\bx,{z}) =&\n\\int\\_{\\mathscr{O}} \\frac{{\\rm d} \\Omega'}{2 \\pi} \\frac{{\\rm d}\\bK'}{(2 \\pi)^d}\\,\nQ(\\Omega,\\Omega',\\bK,\\bK') \\nonumber \\\\ &\\times \\left[\n W(\\Omega',\\bK',\\bx,{z})-W(\\Omega,\\bK,\\bx,{z})\\right],\n\\label{eq:TR6}\\end{aligned}$$ \nfor *z* > 0, with initial condition\n \n$$W(\\Omega,\\bK,\\bx,0) = |A\\_o(\\Omega,\\bK)|^2 \\delta(\\bx)\n\\label{eq:TR7}$$ \nThe transport equation in the physical scales is obtained from as explained in Appendix [ap:unscaled]. In the next section we will show how this equation simplifies in the paraxial regime, when *γ* ≪ 1. This is the result used for the imaging applications discussed in section [sect:Inverse].\nThe paraxial limit\n------------------\nEquation shows that the energy is transported on the characteristic\n \n$$\\bx = - \\gamma \\frac{\\bK}{\\beta(\\gamma \\bK)} {z},\n\\label{eq:TR7p}$$ \nparametrized by *z*, and depending on the wave-vector $\\bK$. Here $|\\bx|/{z} = O(\\gamma)$ quantifies the opening angle of the cone (beam) of propagation with axis *z*. We write this explicitly as\n \n$$\\bX = \\bx/\\gamma, \\quad \\mbox{where} ~ |\\bX| = O(1).\n\\label{eq:ParX}$$ \nThe paraxial regime corresponds to a narrow beam, modeled by *γ* → 0 and\n \nΓ = *γ*/*γ**s* = *O*(1). \nAt the range *z* = 0 of the source we have from and that\n \n$$W(\\Omega,\\bK,\\bx,0) = \\frac{\\big|\\hat S\\big( \\frac{\\eta\\_s}{\\eta}\n \\Omega,\\Gamma\\bK\\big)\\big|^2}{4 \\gamma^d k\\_o } \\delta(\\bX),\n\\label{eq:TR10pp}$$ \nand to obtain a finite limit as *γ* → 0 we rescale the Wigner transform as\n \n$${\\cal W}(\\Omega,\\bK,\\bX,{z}) = \\gamma^d W(\\Omega,\\bK,\\gamma \\bX,{z}).\n\\label{eq:Wparax}$$ \nWe also change variables in,\n \n$$\\begin{aligned}\n \\Omega - \\Omega' - \\eta \\big(\\gamma(\\bK-\\bK'),\\beta(\\gamma\n \\bK)-\\beta(\\gamma\\bK')\\big)\\cdot \\vv\\_o \\leadsto \\Omega', \\qquad\n \\bK-\\bK' \\leadsto \\bK', \\end{aligned}$$ \nand obtain the transport equation\n \n$$\\begin{aligned}\n&\\hspace{-0.1in}\\left[\\partial\\_{z} +\\frac{\\bK}{\\beta(\\gamma \\bK)}\n \\cdot \\nabla\\_{\\bX} \\right] {\\cal W} (\\Omega,\\bK,\\bX,{z}) =\n \\frac{k\\_o^4}{4 }\n \\int\\_{\\mathscr{O}} \\frac{{\\rm d} \\Omega'}{2 \\pi} \\frac{{\\rm d}\\bK'}{(2 \\pi)^d}\\, \n \\frac{1}{\n \\beta(\\gamma \\bK) \\beta(\\gamma \\bK')} \\nonumber\\\\ & \\hspace{0.1in}\n \\times \\Big[ \\bar{\\sigma}\\_c^2 \\widetilde \\cR\\_{cc} \\Big(\\Omega',\\bK',\n \\frac{\\beta(\\gamma \\bK) - \\beta(\\gamma \\bK - \\gamma \\bK')}{\\gamma}\n \\Big) + O(\\gamma^2)\\Big]\\nonumber \\\\ &\\hspace{0.1in}\\times \\Big[\n {\\cal W}\\left( \\Omega - \\Omega' - \\eta \\gamma \\bv\\_o \\cdot \\bK' -\n \\eta v\\_{oz} (\\beta(\\gamma \\bK) - \\beta(\\gamma \\bK - \\gamma \\bK')),\n \\bK-\\bK',\\bX,{z}\\right) \\nonumber \\\\ &\\hspace{3.4in}\n - {\\cal W}(\\Omega,\\bK,\\bX,{z})\\Big],\n\\label{eq:TR11}\\end{aligned}$$ \nfor *z* > 0 and a finite *γ* ≪ 1, where *O*(*γ*2) denotes the $\\widetilde \\cR\\_{\\rho c}$ and $\\widetilde \\cR\\_{\\rho \\rho}$ terms in the kernel.\nRecall that *γ* ≪ 1 so, in order to observe a significant effect of the ambient motion, we rescale the transversal speed as\n \n$$\\bv\\_o = \\frac{{\\itbf V}\\_o}{\\eta \\gamma}, \\quad \\mbox{with} ~ |{\\itbf\n V}\\_o| = O(1).\n\\label{eq:TR12}$$ \nWith a similar scaling of the range velocity\n \n$$v\\_{oz} = \\frac{V\\_{oz}}{\\eta \\gamma}, \\quad \\mbox{with} ~ |V\\_{oz}| =\nO(1),\n\\label{eq:TR13}$$ \nwe obtain that the range motion plays no role in equation as *γ* → 0, because \n$$\\beta(\\gamma \\bK) = k\\_o + O(\\gamma^2), \\qquad \\beta(\\gamma \\bK) -\n\\beta(\\gamma \\bK - \\gamma \\bK') = O(\\gamma^2).$$ \nThe transport equation satisfied by the Wigner transform ${\\cal\n W}(\\Omega,\\bK,\\bX,{z})$ in the paraxial limit *γ* → 0 is\n \n$$\\begin{aligned}\n\\nonumber \\Big[\\partial\\_{z} + \\frac{\\bK}{k\\_o} \\cdot \\nabla\\_{\\bX} \\Big]\n {\\cal W}(\\Omega,\\bK,\\bX,{z}) = \\frac{k\\_o^2}{4 }\n \\int\\_{\\mathbb{R}^d} \\frac{{\\rm d} \\bK'}{(2 \\pi)^d}\n \\int\\_{\\mathbb{R}}\\frac{{\\rm d} \\Omega'}{2 \\pi} \\, \\bar{\\sigma}\\_c^2\\widetilde\n \\cR\\_{cc} (\\Omega',\\bK', 0) \\nonumber \\\\ \\times \\big[\n \\mathcal{W}\\left( \\Omega - \\Omega' - \\bK' \\cdot {\\itbf\n V}\\_o, \\bK-\\bK',\\bX,{z}\\right)\n -\\mathcal{W}(\\Omega,\\bK,\\bX,{z})\\big],\n\\label{eq:TR15}\\end{aligned}$$ \nfor *z* > 0, with initial condition\n \n$$\\mathcal{W}(\\Omega,\\bK,\\bX,0) = \\frac{\\big|\\hat S\\big(\n \\frac{\\eta\\_s}{\\eta} \\Omega,\\Gamma\\bK\\big)\\big|^2}{4k\\_o}\n\\delta(\\bX).\n\\label{eq:TR16}$$ \nThe transport equation in the physical scales is obtained from using the scaling relations explained in Appendix [ap:unscaled].\nSummary\n=======\nWe introduced an analysis of sound wave propagation in a time dependent random medium which moves due to an ambient flow at speed $\\vv(t,\\vx)$, and is modeled by the wave speed $c(t,\\vx)$ and mass density $\\rho(t,\\vx)$. The random fields $\\vv(t,\\vx)$, $c(t,\\vx)$ and $\\rho(t,\\vx)$ have small, statistically correlated fluctuations about the constant values $\\vv\\_o$, *c**o* and *ρ**o*, on the length scale ℓ and time scale *T*. The analysis starts from Pierce’s equation, which is obtained from the linearization of the fluid dynamics equations about an ambient flow, and applies to waves with central wavelength $\\la\\_o \\ll \\ell$. The excitation is from a stationary source with radius ℓ*s*, which emits a narrowband signal of duration *T**s*.\nThe analysis is in a forward wave propagation regime to a large distance (range) *L* ≫ ℓ, within a cone with small opening angle. Using the diffusion approximation theory, we showed that the coherent part (the expectation) of the wave decays exponentially in $L/{\\mathscr S}$, and quantified the frequency- and wavevector-dependent scattering mean free path ${\\mathscr S}$. We also derived transport equations for the energy density (Wigner transform) of the wave, which show explicitly the effect of the ambient flow and net scattering in the time dependent random medium.\nWe used the wave propagation theory to study the inverse problem of localizing (imaging) the source from measurements at a stationary array of receivers located at range *L*. This study is in the regime of paraxial wave propagation, where the Wigner transform can be computed explicitly, and assumes a large range $L \\gg {\\mathscr S}$, so that the wave is incoherent due to strong scattering in the random medium. The temporal variation of the medium is at time scale *T* ≪ *T**s*, and it has two beneficial effects for imaging: First, it causes broadening of the bandwidth of the recorded waves, which leads to improved travel time estimation and consequently, better range resolution. Second, it allows a robust (statistically stable) estimation of the Wigner transform from the array measurements. We presented an explicit analysis of imaging based on this Wigner transform and showed how one can estimate the source location, the mean velocity $\\vv\\_o$ and the statistics of the random medium.\nAcknowledgments\n===============\nLiliana Borcea’s research is supported in part by the Air Force Office of Scientific Research under award number FA9550-18-1-0131 and in part by the U.S. Office of Naval Research under award number N00014-17-1-2057. Knut Sølna’s research is supported in part by the Air Force Office of Scientific Research under award number FA9550-18-1-0217 and NSF grant 1616954.\nThe Markovian limit theorem\n===========================\nIn this appendix we obtain the $\\ep \\to 0$ limit of the Markov process ${\\itbf A}^\\ep (z) = (A^\\eps(\\Omega,\\bK,z))\\_{(\\Omega,\\bK)\\in\n \\mathscr{O}}$, which lies on the surface of the sphere with radius $R\\_{\\itbf A}$ given in equation. The set O is defined by ([def:calO]). The process ${\\itbf A}^\\ep (z)$ starts from\n \n$${\\itbf A}^\\ep (0) = \\left(-\\frac{1}{2 \\sqrt{\\beta(\\gamma\\bK})} \\hat S\n\\Big(\\frac{\\eta\\_s}{\\eta} \\Omega,\\frac{\\gamma}{\\gamma\\_s} \\bK\\Big)\n\\right)\\_{(\\Omega,\\bK)\\in \\mathscr{O}},\n\\label{eq:startA}$$ \nwhich is independent of $\\ep$, and evolves at *z* > 0 according to the stochastic equation\n \n$$\\frac{{\\rm d} {\\itbf A}^\\ep}{{\\rm d}z} = {\\mathcal\n G}\\Big(\\frac{z}{\\ep},\\frac{z}{\\ep}\\Big) {\\itbf A}^\\ep +\n\\frac{1}{\\sqrt{\\ep}} {\\mathcal F}\\Big(\\frac{z}{\\ep},\\frac{z}{\\ep}\\Big)\n {\\itbf A}^\\ep.\n\\label{eq:StochA}$$ \nHere ${\\mathcal G}$ and ${\\mathcal F}$ are integral operators\n \n$$\\begin{aligned}\n[{\\mathcal G}(z,\\zeta) {\\itbf A}](\\Omega,\\bK)&= \\int\\_{\\mathscr{O}}\n{\\rm d}\\Omega' {\\rm d}\\bK'\\, G(z,\\zeta,\\Omega,\\Omega',\\bK,\\bK')\nA(\\Omega',\\bK'), \\\\ [{ \\mathcal F}(z,\\zeta) {\\itbf A}](\\Omega,\\bK)&=\n\\int\\_{\\mathscr{O}}{\\rm d}\\Omega' {\\rm d}\\bK'\\,\nF(z,\\zeta,\\Omega,\\Omega',\\bK,\\bK') A(\\Omega',\\bK'),\\end{aligned}$$ \nwith kernels depending on the random processes $\\nu\\_c(\\tau,\\vr)$ and $Q^\\ep(\\tau,\\vr)$. Recall the definition of $Q^\\ep(\\tau,\\vr)$. We rewrite it here as\n \n$$Q^\\ep(\\tau,\\vr) = Q^{(0)}(\\tau,\\vr) + \\frac{\\sqrt{\\ep \\gamma}\n \\bar{\\sigma}\\_\\rho}{2}Q^{(1)}(\\tau,\\vr), \n\\label{eq:QepDef}$$ \nwith\n \n$$Q^{(0)}(\\tau,\\vr) = \\Delta\\_{\\vr} \\nu\\_\\rho(\\tau,\\vr) ~~ \\mbox{and} ~~\nQ^{(1)}(\\tau,\\vr) = \\left|\\nabla\\_{\\vr} \\nu\\_\\rho(\\tau,\\vr) \\right|^2.\n\\label{eq:Q01Def}$$ \nThe kernel *G* has a deterministic part supported at Ωʹ = Ω and $\\bK' = \\bK$, and a random part determined by *Q*(1),\n \n$$\\begin{aligned}\nG(z,\\zeta,&\\Omega,\\Omega',\\bK,\\bK') = \\frac{i k\\_o}{\\beta(\\gamma \\bK)}\n\\Big[\\frac{\\Omega}{\\eta c\\_o}-\\frac{\\vv\\_o}{c\\_o}"},"data_id":{"kind":"string","value":"arxiv_0183866"}}},{"rowIdx":45996,"cells":{"content":{"kind":"string","value":"*P**A* and gives the updating scheme\n \n$$\\begin{aligned}\n h\\_{ij}^{t+1} \\leftarrow h\\_{ij}^t \\frac{(W^TXQ+2\\lambda H(PA))\\_{ij}}{(W^TWHQ + 2\\lambda H(PD))\\_{ij}}. \\end{aligned}$$ \n*k*-NN induced Persistent Laplacian\n-----------------------------------\nOne major issue with top-GNMF and top-rGNMF is that the parameters {*ζ**t*}*t* = 1*T* have to be chosen. For the parameters, we let *ζ**t* ∈ {0, 1, 1/2, ⋯, 1/*T*} for a total of *T* + 1 parameters. Therefore, the number of parameters that needs to be chosen increases exponentially as the number of filtration *T* increases. Therefore, we propose an approximation to the original formulation using *k*-NN based persistent Laplacian.\nLet N*t*(**x***j*) be the *t*-nearest neighbors of sample **x***j*. Then, define the *t*-persistent directed adjacency matrix *Ã**t* as\n \n$$\\begin{aligned}\n \\tilde{A}^t = \\{\\tilde{a}\\_{ij}^t\\}, \\quad \\tilde{a}\\_{ij}^t = \\begin{cases}\n 1 & \\mathbf{x\\_j} \\in \\mathcal{N}\\_t(\\mathbf{x}\\_i) \\\\\n 0 & \\text{otherwise}.\n \\end{cases}\\end{aligned}$$ \nThen, the *k*-NN based directed adjacency Laplacian is the weighted sum of {*A**t*}\n \n$$\\begin{aligned}\n \\tilde{A} := \\sum\\_{t=1}^T \\zeta\\_t \\tilde{A}^t.\\end{aligned}$$ \nThen, the undirected persistent adjacency matrix can be obtained via symmetrization\n \n$$\\begin{aligned}\n PA = \\tilde{A} + \\tilde{A}^T - \\tilde{A}\\cdot \\tilde{A}^T,\\end{aligned}$$ \nwhere ⋅ denote Hadamard product. Then, the persistent Laplacian can be constructed using the persistent degree matrix\n \n$$\\begin{aligned}\n PL = PD - PA, \\quad PD\\_{ii} = \\sum\\_{j\\ne i}PA\\_{ij}.\\end{aligned}$$ \nOne advantage of utilizing the *k*-NN induced persistent Laplacian is that the parameter space is much smaller. We can set *ζ**t* ∈ {0, 1}, where *ζ**t* = 0 would ’turn-off’ the particular neighbor’s connectivity. In essence, the number of parameters will be reduced to 2*T*, a significant decrease from *T*(*T* + 1) of the original formulation.\nEvaluation metrics\n------------------\nLet *Y* = {*Y*1, ..., *Y**L*} and *C* = {*C*1, ..., *C**L*} be 2 partitions of the data. Here, we let *Y* be the true label partition and *C* be the cluster label partition. Let {*y**i*}*i* = 1*N* and {*c**i*}*i* = 1*N* be the true and predicted labels of sample *i*.\n#### Adjusted Rand Index\nAdjusted random index (ARI) measures the similarity between two clustering by observing all pairs of samples that belong to the same cluster, and seeing if the other clustering result also have the same pair of samples in the same cluster. Let *n**i**j* = ∣*T**i* ∩ *S**j*∣ be the number of samples that belong to true label *i* and cluster label *j*, and define *a**i* = ∑*j**n**i**j* and *b**j* = ∑*i**n**i**j*. Then, the ARI is defined as\n \n$$\\begin{aligned}\n \\text{ARI} = \\frac{\\sum\\_{ij} \\begin{pmatrix}\n n\\_{ij} \\\\ 2\n \\end{pmatrix} - \\left[ \\sum\\_i \\begin{pmatrix}\n a\\_i \\\\ 2\\end{pmatrix} \\sum\\_j \\begin{pmatrix}\n b\\_j \\\\ 2\\end{pmatrix}\\right] / \\begin{pmatrix}\n N \\\\ 2\\end{pmatrix} }{\\frac{1}{2} \\left[\\begin{pmatrix}\n a\\_i \\\\ 2\\end{pmatrix} + \\begin{pmatrix}\n b\\_j \\\\ 2\\end{pmatrix}\\right] - \\left[\\begin{pmatrix}\n a\\_i \\\\ 2\\end{pmatrix} \\sum\\_j \\begin{pmatrix}\n b\\_j \\\\ 2\\end{pmatrix}\\right] / \\begin{pmatrix}\n N \\\\ 2\\end{pmatrix}}.\\end{aligned}$$ \nThe ARI takes on a value between -1 and 1, where 1 is a perfect match between two clustering methods, and 0 is a completely random assignment of labels, and -1 indicates that the two clusterings are completely different.\n#### Normalized Mutual Information\nThe normalized mutual information (NMI) measures the mutual information between two clustering results and normalized according to cluster size. We fix the true labels *Y* as one of the clustering result, and use the predicted labels as the other to calculate NMI. The NMI is calculated as the following\n \n$$\\begin{aligned}\n \\text{NMI} = \\frac{2I(Y; C)}{H(Y)H(C)},\\end{aligned}$$ \nwhere *H*( ⋅ ) is the entropy and *I*(*Y*; *C*) is the mutual information between true labels *Y* and predicted labels *C*. NMI has a range of 0 and 1, where 1 is a perfect mutual correlation between the two sets of labels and 0 means no mutual information.\n#### Accuracy\nAccuracy (ACC) calculates the percentage of correctly predicted class labels. The accuracy is given by\n \n$$\\begin{aligned}\n \\text{ACC} = \\frac{1}{N} \\sum\\_{i=1}^N \\delta(y^i, f(c^i)),\\end{aligned}$$ \nwhere *δ*(*a*, *b*) is the indicator function, where if *a* = *b*, *δ*(*a*, *b*) = 1, and 0 otherwise. *f* : *C* → *Y* maps the cluster labels to the true labels, where the mapping is the optimal permutation of the cluster labels and true labels obtained from the Hungarian algorithm.\n#### Purity\nFor purity calculation, each predicted label *C**i* is assigned to a true label *Y**j* such that the ∣*C**i* ∩ *Y**j*∣ is maximized. Taking the average over all the predicted label, we obtain the following\n \n$$\\begin{aligned}\n \\text{Purity} = \\frac{1}{N}\\max\\_j |C\\_i \\cap Y\\_j|.\\end{aligned}$$ \nNote that unlike accuracy, purity does not map the predicted labels to the true labels.\nResults\n=======\nBenchmark Data\n--------------\nWe have performed benchmark on 12 publicly available datasets. The GEO accession number, reference, organism, number of cell types, and number of samples are recorded in. For each data, cell types with less than 15 cells were removed. Log-normalization was applied, and scaled the data to have unit length. For GNMF and rGNMF, *k* = 8 neighbors were used. For TNMF and rTNMF, 8 filtration values were used to construct PL, and for each scale, binary selection *ζ**p* = {0, 1} was used. for *k*-TNMF and *k*-rTNMF, *k* = 8 was used with *ζ**p* = {0, 1}. For each test, double nonnegative singular value decomposition with zeros filled with the average of *X* (NNDSVDA) was used for the initialization. The *k*-means clustering was applied to obtain the clustering results.\n[H]\n| Geo Accession | Reference | Organism | Cell type | Number of Samples | Number of Genes |\n| --- | --- | --- | --- | --- | --- |\n| GSE67835 | Dramanis | Human | 8 | 420 | 22084 |\n| GSE75748 time | Chu | Human | 6 | 758 | 19189 |\n| GSE82187 | Gokce | Mouse | 8 | 705 | 18840 |\n| GSE84133human1 | Baron | Human | 9 | 1895 | 20125 |\n| GSE84133human2 | Baron | Human | 9 | 1702 | 20125 |\n| GSE84133human3 | Baron | Human | 9 | 3579 | 20125 |\n| GSE84133human4 | Baron | Human | 6 | 1275 | 20125 |\n| GSE84133mouse1 | Baron | Mouse | 6 | 782 | 14878 |\n| GSE84133mouse2 | Baron | Mouse | 8 | 1036 | 14878 |\n| GSE57249 | Biase | Human | 3 | 49 | 25737 |\n| GSE64016 | Leng | Human | 4 | 460 | 19084 |\n| GSE94820 | Villani | Human | 5 | 1140 | 26593 |\n[tab: data]\nBenchmarking PL regularized NMF\n-------------------------------\nIn order to benchmark persistent Laplacain regularized NMF, we compared our methods to other commonly used NMF methods, namely the GNMF, rGNMF, rNMF and NMF. For a fair comparison, We omitted supervised or semi-supervised methods. For *k*-rTNMF, rTNMF, *k*-TNMF, TNMF, GNMF and rGNMF, we set *α* = 1 for all tests.\nshows the ARI values of the NMF methods for the 12 data we have tested. The bold number indicate the highest performance. depicts the average ARI value over the 12 datasets for each method.\n[H]\n| data | *k*-rTNMF | rTNMF | *k*-TNMF | TNMF | rGNMF | GNMF | rNMF | NMF |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| GSE67835 | **0.9454** | 0.9236 | 0.9306 | 0.8533 | 0.9391 | 0.9109 | 0.7295 | 0.7314 |\n| GSE64016 | **0.2569** | 0.1544 | 0.2237 | 0.1491 | 0.1456 | 0.1605 | 0.1455 | 0.1466 |\n| GSE75748time | 0.6421 | **0.6581** | 0.5963 | 0.6099 | 0.6104 | 0.5790 | 0.5969 | 0.5996 |\n| GSE82187 | **0.9877** | 0.9815 | 0.9676 | 0.9809 | 0.7558 | 0.7577 | 0.8221 | 0.8208 |\n| GSE84133human1 | 0.8310 | **0.8969** | 0.8301 | 0.8855 | 0.8220 | 0.7907 | 0.7080 | 0.6120 |\n| GSE84133human2 | **0.9469** | 0.9072 | 0.9433 | 0.9255 | 0.9350 | 0.9255 | 0.8930 | 0.8929 |\n| GSE84133human3 | 0.8504 | 0.9179 | 0.8625 | **0.9181** | 0.8447 | 0.8361 | 0.7909 | 0.8089 |\n| GSE84133human4 | 0.8712 | **0.9692** | 0.8712 | 0.9692 | 0.8699 | 0.8681 | 0.8311 | 0.8311 |\n| GSE84133mouse1 | **0.8003** | 0.7894 | 0.8003 | 0.7913 | 0.7945 | 0.7918 | 0.6428 | 0.6348 |\n| GSE84133mouse2 | 0.6953 | 0.8689 | 0.7005 | **0.9331** | 0.6808 | 0.6957 | 0.5436 | 0.5470 |\n| GSE57249 | **1.0000** | 0.9638 | 1.0000 | 0.9483 | 1.0000 | 1.0000 | 0.9483 | 0.9483 |\n| GSE94820 | **0.6101** | 0.5480 | 0.4916 | 0.5574 | 0.5139 | 0.5189 | 0.5440 | 0.5556 |\n[tab: ARI]\n![Average ARI of k-rTNMF, rTNMF, k-TNMF,TNMF, rGNMF, GNMF, rNMF and NMF for the 12 datasets](figures/mean_ARI.png \"fig:\") [fig: ARI]\nOverall, PL regularized rNMF and NMF have the highest ARI value across all the datasets. *k*-rTNMF outperforms other NMF methods by at least 0.09 for GSE64016. All PL regularized NMF methods outperform other NMF methods by at least 0.14 for GSE82187. For GSE84133 human 3, both rTNMF and TNMF outperform other methods by 0.07. TNMF improves other methods by more than 0.2 for GSE84133 mouse 2. Lastly, *k*-rTNMF has the highest ARI value for GSE94820. Moreover, rTNMF improves rGNMF by 0.05, and TNMF improves GNMF by about 0.06. *k*-TNMF and *k*-rTNMF also improve GNMF and rGNMF by about 0.03.\nshows the NMI values of of the NMF methods for the 12 datasets we have tested. The bold number indicate the highest performance. shows the average NMI value over the 12 datasets.\n[H]\n| data | *k*-rTNMF | rTNMF | *k*-TNMF | TNMF | rGNMF | GNMF | rNMF | NMF |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| GSE67835 | **0.9235** | 0.8999 | 0.9107 | 0.8607 | 0.9104 | 0.8858 | 0.7975 | 0.8017 |\n| GSE64016 | 0.**3057** | 0.2059 | 0.3136 | 0.1869 | 0.2593 | 0.2562 | 0.1896 | 0.1849 |\n| GSE75748time | 0.7522 | **0.7750** | 0.7159 | 0.7343 | 0.7235 | 0.6971 | 0.7227 | 0.7244 |\n| GSE82187 | **0.9759** | 0.9691 | 0.9298 | 0.9668 | 0.8802 | 0.8754 | 0.9124 | 0.9117 |\n| GSE84133human1 | **0.8802** | 0.8716 | 0.8785 | 0.8780 | 0.8713 | 0.8310 | 0.8226 | 0.7949 |\n| GSE84133human2 | **0.9363** | 0.8937 | 0.9313 | 0.9070 | 0.9237 | 0.9145 | 0.8835 | 0.8829 |\n| GSE84133human3 | 0.8500 | **0.8718** | 0.8577 | 0.8677 | 0.8439 | 0.8357 | 0.8215 | 0.8260 |\n| GSE84133human4 | 0.8795 | **0.9542** | 0.8795 | 0.9542 | 0.8775 | 0.8753 | 0.8694 | 0.8694 |\n| GSE84133mouse1 | **0.8664** | 0.8498 | 0.8664 | 0.8495 | 0.8596 | 0.8565 | 0.7634 | 0.7593 |\n| GSE84133mouse2 | 0.8218 | **0.8355** | 0.8299 | 0.8713 | 0.8005 | 0.8129 | 0.7258 | 0.7272 |\n| GSE57249 | **1.0000** | 0.9505 | 1.0000 | 0.9293 | 1.0000 | 1.0000 | 0.9293 | 0.9293 |\n| GSE94820 | **0.7085** | 0.6657 | 0.6157 | 0.6716 | 0.6195 | 0.6258 | 0.6624 | 0.6693 |\n[tab: NMI]\n![Average NMI values of k-rTNMF, rTNMF, k-TNMF,TNMF, rGNMF, GNMF, rNMF and NMF for the 12 datasets](figures/mean_NMI.png \"fig:\") [fig: NMI]\nInterestingly, *k*-rTNMF and *k*-TNMF on average have higher NMI values than rTNMF and TNMF, respectively. However, all PL regularized methods outperform rGNMF, GNMF, rNMF and NMF. Overall, PL regularized methods outperform other methods. Most noticeably, *k*-rTNMF, rTNMF and TNMF outperform standard NMF methods by 0.06 for GSE82187. Both rTNMF and TNMf outperform rGNMF and GNMF by 0.08 for GSE84133 human 4.\nshows the purity values of the NMF methods for the 12 datasets we have tested. The bold number indicate the highest performance. shows the average purity over the 12 datasets.\n[H]\n| data | *k*-rTNMF | rTNMF | *k*-TNMF | TNMF | rGNMF | GNMF | rNMF | NMF |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| GSE67835 | **0.9643** | 0.9267 | 0.9595 | 0.9024 | 0.9595 | 0.9476 | 0.8726 | 0.8719 |\n| GSE64016 | **0.6048** | 0.4913 | 0.5846 | 0.5013 | 0.5339 | 0.5398 | 0.5080 | 0.5050 |\n| GSE75748time | **0.7736** | 0.7512 | 0.7533 | 0.7454 | 0.7553 | 0.7387 | 0.7467 | 0.7455 |\n| GSE82187 | **0.9927** | 0.9895 | 0.9620 | 0.9888 | 0.9620 | 0.9594 | 0.9693 | 0.9692 |\n| GSE84133human1 | **0.9543** | 0.9357 | 0.9536 | 0.9382 | 0.9490 | 0.9187 | 0.9189 | 0.9099 |\n| GSE84133human2 | **0.9818** | 0.9614 | 0.9806 | 0.9661 | 0.9777 | 0.9736 | 0.9602 | 0.9600 |\n| GSE84133human3 | 0.9472 | **0.9485** | 0.9531 | 0.9460 | 0.9452 | 0.9420 | 0.9464 | 0.9466 |\n| GSE84133human4 | 0.9427 | **0.9882** | 0.9427 | 0.9882 | 0.9427 | 0.9420 | 0.9412 | 0.9412 |\n| GSE84133mouse1 | **0.9565** | 0.9540 | 0.9565 | 0.9540 | 0.9552 | 0.9540 | 0.9309 | 0.9299 |\n| GSE84133mouse2 | 0.9585 | 0.9410 | **0.9604** | 0.9373 | 0.9466 | 0.9507 | 0.9185 | 0.9199 |\n| GSE57249 | **1.0000** | 0.9857 | 1.0000 | 0.9796 | 1.0000 | 1.0000 | 0.9796 | 0.9796 |\n| GSE94820 | **0.7893** | 0.7462 | 0.6658 | 0.7550 | 0.6421 | 0.6421 | 0.7429 | 0.7531 |\n[tab: Purity]\n![Average purity values of k-rTNMF, rTNMF, k-TNMF,TNMF, rGNMF, GNMF, rNMF and NMF for the 12 datasets](figures/mean_Purity.png \"fig:\") [fig: Purity]\nIn general, PL-regularized methods achieve higher purity values compared to other NMF methods. Purity measures the maximum intersection between true and predicted classes, which is why we do not observe a significant difference, as seen in ARI and NMI. Furthermore, since purity does not account for the size of a class, and given the imbalanced class sizes in scNRA-seq data, it is not surprising that the purity values are similar.\nshows the ACC of the NMF methods for the 12 datasets we have tested. The bold number indicate the highest performance. shows the average ACC over the 12 datasets.\n[H]\n| data | *k*-rTNMF | rTNMF | *k*-TNMF | TNMF | rGNMF | GNMF | rNMF | NMF |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| GSE67835 | **0.9643** | 0.9243 | 0.9595 | 0.9000 | 0.9595 | 0.9383 | 0.8357 | 0.8364 |\n| GSE64016 | **0.5700** | 0.4870 | 0.5502 | 0.4746 | 0.4891 | 0.4537 | 0.4691 | 0.4759 |\n| GSE75748time | **0.7565** | 0.7438 | 0.7414 | 0.6917 | 0.7355 | 0.7241 | 0.6873 | 0.6875 |\n| GSE82187 | **0.9927** | 0.9895 | 0.9599 | 0.9888 | 0.8512 | 0.8514 | 0.8896 | 0.8889 |\n| GSE84133human1 | 0.8973 | **0.9194** | 0.8974 | 0.9088 | 0.8889 | 0.8364 | 0.7988 | 0.7370 |\n| GSE84133human2 | **0.9260** | 0.9069 | 0.9242 | 0.9447 | 0.9224 | 0.9177 | 0.8998 | 0.8994 |\n| GSE84133human3 | 0.8539 | **0.9456** | 0.8597 | 0.9419 | 0.8498 | 0.8228 | 0.8032 | 0.8178 |\n| GSE84133human4 | 0.8831 | **0.9882** | 0.8831 | 0.9882 | 0.8824 | 0.8816 | 0.8847 | 0.8847 |\n| GSE84133mouse1 | **0.8581** | 0.8542 | 0.8581 | 0.8542 | 0.8555 | 0.8542 | 0.7361 | 0.7311 |\n| GSE84133mouse2 | 0.8232 | **0.9101** | 0.8263 | 0.9305 | 0.7903 | 0.8155 | 0.7239 | 0.7294 |\n| GSE57249 | **1.0000** | 0.9857 | 1.0000 | 0.9796 | 1.0000 | 1.0000 | 0.9796 | 0.9796 |\n| GSE94820 | **0.7533** | 0.7119 | 0.6482 | 0.7201 | 0.6088 | 0.6107 | 0.7091 | 0.7189 |\n[tab: ACC]\n![Average ACC of k-rTNMF, rTNMF, k-TNMF,TNMF, rGNMF, GNMF, rNMF and NMF for the 12 datasets](figures/mean_ACC.png \"fig:\") [fig: ACC]\nOnce again, we see that PL regularized methods have higher ACC than other NMF methods. RTNMF and TNMF improves rGNMF and GNMF by 0.05, and *k*-rTNMF and *k*-TNMF improves rGNMF and GNMF by 0.04. We see an improvement in ACC for both *k*-rTNMF and *k*-TNMF for GSE64016. All 4 PL regularized methods improve ACC of GSE82187 by 0.1. RTNMF and TNMF improve GSE84133 mouse 2 by at least 0.1 as well.\nOverall performance\n-------------------\nshows the average ARI, NMI, purity and ACC of *k*-rTNMF, rTNMF, *k*-TNMF, TNMF, rGNMF, GNMF, rNMF, NMF across 10 datasets. All PL regularized NMF methods outperform the traditional rGNMF, GNMF, rNMF and NMF. Both rTNMF and TNMF have higher average ARI and purity than the *k*-NN based PL counterparts. However, *k*-rTNMF and *k*-TNMF have higher average NMI than rTNMF and TNMF, respectively. *k*-rTNMF has a significantly higher purity than other methods.\n![Average ARI, NMI, purity and ACC of k-rTNMF, rTNMF, k-TNMF, TNMF, rGNMF, GNMF, rNMF, NMF across 10 datasets](figures/results.png \"fig:\") [fig: overall]\nDiscussion\n==========\nVisualization of meta-genes based UMAP and t-SNE\n------------------------------------------------\nBoth UMAP and t-SNE are well-known for their effectiveness in visualization. However, these methods may not perform as competitively in clustering or classification tasks. Therefore, it is beneficial to employ NMF-based methods to enhance the visualization capabilities of UMAP and t-SNE.\nIn this process, we generate meta-genes and subsequently utilize UMAP or t-SNE to further reduce the data to 2 dimensions for visualization. For a dataset with *M* cells, the number of meta-genes will be the integer value of $\\sqrt{M}$. To compare the standard UMAP and t-SNE plots with the top-NMF-assisted and top-rNMF-assisted UMAP and t-SNE visualizations, we used the default settings of the Python implementation of UMAP and the Scikit-learn implementation of t-SNE. For unassisted UMAP and t-SNE, we first removed low-abundance genes and performed log-transformation before applying UMAP and t-SNE.\nshows the visualization of PL regularized NMF methods through UMAP. Each row corresponds to GSE67835, GSE75748 time, GSE94820 and GSE84133 mouse 2 data. The columns from left to right are the *k*-rTNMF assisted UMAP, rTNMF assisted UMAP, *k*-TNMF assisted UMAP, TNMF assisted UMAP and UMAP visualization. Samples were colored according to their true cell types.\n![Visualization of top-NMF and top-rNMF meta-genes through UMAP. Each row corresponds to GSE67835, GSE75748 time, GSE94820 and GSE84133 mouse 2 data. The columns from left to right are the k-rTNMF assisted UMAP, rTNMF assisted UMAP, k-TNMF assisted UMAP, TNMF assisted UMAP and UMAP visualization. Samples were colored according to their true cell types](./figures/topology_umap.png \"fig:\") [fig: top umap]\nshows the visualization of PL regularized NMF through t-SNE. Each row corresponds to GSE67835, GSE75748 time, GSE94820 and GSE84133 mouse 2 data. The columns from left to right are the *k*-rTNMF assisted t-SNE, rTNMF assisted t-SNE, *k*-TNMF assisted t-SNE, TNMF assisted t-SNE and t-SNE visualization. Samples were colored according to their true cell types.\n![Visualization of top-NMF and top-rNMF meta-genes through t-SNE. Each row corresponds to GSE67835, GSE75748 time, GSE94820 and GSE84133 mouse 2 data. The columns from left to right are the k-rTNMF assisted t-SNE, rTNMF assisted t-SNE, k-TNMF asssited t-SNE, TNMF assisted t-SNE and t-SNE visualization. Samples were colored according to their true cell types](./figures/topology_tsne.png \"fig:\") [fig: top tsne]\nWe see a considerable improvement in both top-NMF assisted and top-rNMF assisted UMAP and t-SNE visualization.\n#### GSE67835\nIn the assisted UMAP and t-SNE visualizations of GSE67835, we observe a more distinct cluster, which includes a supercluster of fetal quiescent (Fetal-Q) and fetal replicating (Fetal-R) cells. Darmanis et al. conducted a study that involved obtaining differential gene expression data for human adult brain cells and sequencing fetal brain cells for comparison. It is not surprising that the undeveloped Fetal-Q and Fetal-R cells do not exhibit significant differences and cluster together.\n#### GSE75748 time\nIn GSE75748 time data, Chu et al. sequenced human embryonic stem cells at times 0hr, 12hr, 24hr, 36hr, 72hr, and 96hr under hypoxic conditions to observe differentiation. In unassisted UMAP and t-SNE, although some clustering is visible, there is no clear separation between the clusters. Additionally, two subclusters of 12hr cells are observed.\nNotably, in the PL-regularized assisted UMAP and t-SNE visualizations, there is a distinct supercluster comprising the 72hr and 96hr cells, while cells from different time points form their own separate clusters. This finding aligns with Chu’s observation that there was no significant difference between the 72hr and 96hr cells, suggesting that differentiation may have already occurred by the 72hr mark.\n#### GSE94820\nNotice that in both t-SNE and UMAP, although there is a boundary, the cells do not form distinct clusters. This lack of distinct clustering can pose challenges in many clustering and classification methods. On the other hand, all PL-regularized NMF methods result in distinct clusters.\nAmong the PL-regularized NMF approaches, cutoff-based PL, rTNMF, and TNMF form a single CD1C+ (CD1C1) cluster, whereas the *k*-NN induced PL, *k*-rTNMF, and *k*-TNMF exhibit two subclusters. Villani et al. previously noted the similarity in the expression profile of CD1C1−CD141− (DoubleNeg) cells and monocytes. PL-regularized NMF successfully differentiates between these two types.\n#### GSE84133 mouse 2\nPL-regularized NMF yields significantly more distinct clusters compared to unassisted UMAP and t-SNE. Notably, the beta and gamma cells form distinct clusters in PL-regularized NMF. Additionally, when PL-regularized NMF is applied to assist UMAP, potential outliers within the beta cell population become visible. Baron et al. previously highlighted heterogeneity within the beta cell population, and we observe potential outliers in all visualizations.\nRS analysis\n-----------\nAlthough UMAP and t-SNE are excellent tools for visualizing clusters, they may struggle to capture heterogeneity within clusters. Moreover, these methods can be less effective when dealing with a large number of classes. Therefore, it is essential to explore alternative visualization techniques.\n![RS plots of GSE67835 data. The columns from left to right correspond to k-rTNMF, rTNMF, k-TNMF, and TNMF. Each row corresponds to a cell type. For each section, the x-axis and y-axis correspond to the S-score and R-score, respectively. K-means was used to obtain a cluster label, and the Hungarian algorithm was used to map the cluster labels to the true labels. Each sample was colored according to their true labels.](figures/RSplot.png \"fig:\") [fig: RS plot]\nIn our approach, we visualize each cluster using RS plots. RS plots depict the relationship between the residue score (R score) and similarity score (S score) and have proven useful in various applications for visualizing data with multiple class types.\nLet {(**x***m*, *y**m*)∣**x***m* ∈ R*N*, *y**m* ∈ Z*L*, 1 ≤ *m* ≤ *M*} be the data, where **x***m* is the *m*th sample, *y**m* is the cell type or cluster label. *L* is the number of class. That is, *C**l* = {**x***m* ∈ X∣*y**m* = *l*} and ⊎ 0*L* − 1C*l* = X.\nThe residue (R) score is defined as the inter-class sum of distance. For a given data **x***m* with assignment *y**m* = *l*, the R-score is defined as\n \n$$\\begin{aligned}\n & R\\_m = R(\\mathbf{x}\\_m) = \\frac{1}{R\\_{\\max}}\\sum\\_{\\mathbf{x}\\_j \\notin \\mathcal{C}\\_l} \\|\\mathbf{x}\\_m - \\mathbf{x}\\_j\\|,\\end{aligned}$$ \nwhere *R*max = max**x***m*, **x***m* ∈ X*R**m*. The similarity (S) score is defined as the intra-class average of distance, defined as\n \n$$\\begin{aligned}\n S\\_m = S(\\mathbf{x}\\_m) = \\frac{1}{|\\mathcal{C}\\_l|}\\sum\\_{\\mathbf{x}\\_j \\in \\mathcal{C}\\_l} \\left(1 - \\frac{\\|\\mathbf{x}\\_m - \\mathbf{x}\\_j\\|}{d\\_{\\max}}\\right),\\end{aligned}$$ \nwhere *d*max = max**x***i*, **x***j* ∈ X∥**x***i* − **x***j*∥ and ∣C*l*∣ is the number of data in class C*l*. Both *R**m* and *S**m* are bounded by 0 and 1, and the larger the better for a given dataset.\nThe class residue index (CRI) and the class similarity index (CSI) can then be defined as the average of the R-score and S-score of each of the classes. That is $ \\text{CRI}\\_l = \\frac{1}{|\\mathcal{C}\\_l|}\\sum\\_{m} R\\_m$ and $ \\text{CSI}\\_l = \\frac{1}{|\\mathcal{C}\\_l|}\\sum\\_{ m } S\\_m$. Then, the residue index (RI) and the similarity index (SI) can be defined $ \\text{RI} = \\frac{1}{L}\\text{CRI}\\_l$ and $ \\text{SI} = \\frac{1}{L}\\text{CSI}\\_l $, respectively.\nUsing the RI and SI, the residue similarity disparity can be computed by taking RSD = RI − SI, and the residue-similarity index (RSI) can be computed as RSI = 1 − ∣RI − SI∣.\nshows the RS plots of PL-regularized NMF methods for GSE67835 data. The columns from left to right correspond to *k*-rTNMF, rTNMF, *k*-TNMF, and TNMF, while the rows correspond to the cell types. The x-axis and y-axis represent the S-score and R-score for each sample, respectively. The samples are colored according to their predicted cell types. Predictions were obtained using k-means clustering, and the Hungarian algorithm was employed to find the optimal mapping from the cluster labels to the true cell types.\nWe can see that TNMF fails to identify OP cells, whereas *k*-rTNMF, rTNMF, and *k*-TNMF are able to identify OPC cells. Notably, the S-score is quite low, indicating that the OPC did not form a cluster for TNMF. For fetal quiescent and replicating cells, *k*-rTNMF correctly identifies these two types, and the few misclassified samples are located on the boundaries. RTNMF is able to correctly identify fetal replicating cells but could not distinguish fetal quiescent cells from fetal replicating cells. The S-score is low for neurons in both rTNMF and TNMF, which shows a direct correlation with the number of misclassified cells.\nConclusion\n==========\nPersistent Laplacian-regularized NMF is a dimensionality reduction technique that incorporates multiscale topological interactions between the cells. Traditional graph Laplacian-based regularization only represents a single scale and cannot capture the multiscale features of the data. We have also shown that the *k*-NN induced persistent Laplacian outperforms other NMF methods and is comparable to the cutoff-based persistent Laplacian-regularized NMF methods. However, PL methods do come with their downside. In particular, the weights for each filtration must be determined prior to the reduction. If there are *T* filtrations, then the hyperparameter space is (*T* + 1)*T*. However, *k*-NN induced PL reduces the number of parameters to 2*T*. In addition, we have shown that we can achieve a significant improvement even if we limit the hyperparameter space to 2*T*. We would like to further explore possible parameter-free versions of topological NMF. Additionally, NMF methods are not globally convex, but we have shown that with NNDSVDA initialization, our methods perform the best. One possible extension to the proposed methods is to incorporate higher-order persistent Laplacians in the regularization framework, which will reveal higher-order interactions. In addition, we would like to expand the ideas to tensor decomposition, such as Canonical Polyadic Decomposition (CPD) and Tucker decomposition, multimodal omics data, and spatial transcriptomics data.\nData availability and code\n==========================\nThe data and model used to produce these results can be obtained at <https://github.com/hozumiyu/TopologicalNMF-scRNAseq>.\nAcknowledgment\n==============\nThis work was supported in part by NIH grants R01GM126189, R01AI164266, and R35GM148196, National Science Foundation grants DMS2052983, DMS-1761320, and IIS-1900473, NASA grant 80NSSC21M0023, Michigan State University Research Foundation, and Bristol-Myers Squibb 65109.\n10\nAaron TL Lun, Davis J McCarthy, and John C Marioni. A step-by-step workflow for low-level analysis of single-cell rna-seq data with bioconductor., 2016.\nByungjin Hwang, Ji Hyun Lee, and Duhee Bang. 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Nonnegative matrix factorization (NMF) offers a unique approach due to its meta-gene interpretation of resulting low-dimensional components. However, NMF approaches suffer from the lack of multiscale analysis. This work introduces two persistent Laplacian regularized NMF methods, namely, topological NMF (TNMF) and robust topological NMF (rTNMF). By employing a total of 12 datasets, we demonstrate that the proposed TNMF and rTNMF significantly outperform all other NMF-based methods. We have also utilized TNMF and rTNMF for the visualization of popular Uniform Manifold Approximation and Projection (UMAP) and t-distributed stochastic neighbor embedding (t-SNE).\nkeywords: Algebraic topology, Persistent Laplacian, scRNA-seq, dimensionality reduction, machine learning\nIntroduction\n============\nSingle-cell RNA sequencing (scRNA-seq) is a relatively new technology that has unveiled the heterogeneity within cell populations, providing valuable insights into complex biological interactions and pathways, such as cell-cell interactions, differential gene expression, signal transduction pathways, and more.\nUnlike traditional microarray analysis, often referred to as bulk sequencing, scRNA-seq offers the transcriptomic profile of individual cells. With current technology, it’s possible to sequence more than 20,000 genes and 10,000 samples simultaneously. Standard experimental procedures involve cell isolation, RNA extraction, sequencing, library preparation, and data analysis.\nOver the years, numerous data analysis pipelines have been proposed, typically encompassing data preprocessing, batch correction, normalization, dimensionality reduction, feature selection, cell type identification, and downstream analyses to uncover relevant biological functions and pathways.\nHowever, scRNA-seq data, in addition to their high dimensionality, are characterized by nonuniform noise, sparsity due to drop-out events and low reading depth, as well as unlabeled data. Consequently, dimensionality reduction and feature selection are essential for successful downstream analysis.\nPrincipal components analysis (PCA), uniform manifold approximation and projection (UMAP), and t-distributed stochastic neighbor embedding (t-SNE) are among the most commonly used dimensionality reduction tools for scRNA-seq data. PCA is often employed as an initial step in analysis pipelines, such as trajectory analysis and data integration. In PCA, the first few components are referred to as the principal components, where the variance of the projected data is maximized. In PCA, each *i*th component is orthogonal to all the *i* − 1 components, maximizing the residual data projected onto the *i*th component. Numerous successful extensions to the original formulation have been proposed. However, due to the orthogonality constraint of PCA, the reduced data may contain negative values, making it challenging to interpret.\nUMAP and t-SNE are nonlinear dimensionality reduction methods often used for visualization. UMAP constructs a *k*-dimensional weighted graph based on *k*-nearest neighbors and computes the edge-wise cross-entropy between the embedded low-dimensional weighted graph representation, utilizing the fuzzy set cross-entropy loss function. t-SNE computes the pairwise similarity between cells by constructing a conditional probability distribution over pairs of cells. Then, a student t-distribution is used to obtain the probability distribution in the embedded space, and the Kullback-Leibler (KL) divergence between the two probability distributions is minimized to obtain the reduced data. However, due to the stochastic nature of these methods and their instability at dimensions greater than 3, they may not be suitable for downstream analysis.\nNonnegative matrix factorization (NMF) is another dimensionality reduction method in which the objective is to decompose the original count matrix into two nonnegative factor matrices. The resulting basis matrices are often referred to as meta-genes and represent nonnegative linear combinations of the original genes. Consequently, NMF results are highly interpretable. However, the original formulation employs a least-squares optimization scheme, making the method susceptible to outlier errors.\nTo address this issue, Kong et al. introduced robust NMF (rNMF), or *l*2, 1-NMF, which utilizes the *l*2, 1-norm and can better handle outliers while maintaining comparable computational efficiency to standard NMF. Manifold regularization has also been employed to incorporate geometric structures into dimensionality reduction, utilizing a graph Laplacian, leading to Graph Regularized NMF (GNMF). Semi-supervised methods, such as those incorporating marker genes, similarity and dissimilarity constraints, have been proposed to enhance NMF’s robustness. Additionally, various other NMF derivatives have been introduced.\nDespite these advancements in NMF, manifold regularization remains an essential component to ensure that the lower-dimensional representation of the data can form meaningful clusters. However, using graph Laplacians can only capture a single scale of the data, specifically the scaling factor in the heat kernel. Therefore, single-scale graph Laplacians lack multiscale information.\nEckmann et al. introduced simplicial complexes to the graph Laplacian defined on point cloud data, leading to the combinatorial Laplacian. This can be viewed as a discrete counterpart of the de Rham-Hodge Laplacian on manifolds. Both the Hodge Laplacian and the combinatorial Laplacian are topological Laplacians that give rise to topological invariants in their kernel space, specifically the harmonic spectra. However, the nonharmonic spectra contain algebraic connectivity that cannot be revealed by the topological invariants.\nA significant development in topological Laplacians occurred in 2019 with the introduction of persistent topological Laplacians. Specifically, evolutionary de Rham theory was introduced to obtain persistent Hodge Laplacians on manifolds. Meanwhile, persistent combinatorial Laplacian, also known as the persistent spectral graph or persistent Laplacian (PL), was introduced for point cloud data. These methods have spurred numerous theoretical developments and code construction, as well as remarkable applications in various fields, including protein engineering, forecasting emerging SARS-CoV-2 variants BA.4/BA.5, and predicting protein-ligand binding affinity. Recently, PL has been shown to improve PCA performance.\nThis growing interest arises from the fact that persistent topological Laplacians represent a new generation of topological data analysis (TDA) methods that address certain limitations of the popular persistent homology. In persistent homology, the goal is to represent data as a topological space, often as simplicial complexes. Then, ideas from algebraic topology, such as connected components, holes, and voids, are used to extract topological invariants during a multiscale filtration. Persistent homology has facilitated topological deep learning (TDL), an emerging field. However, persistent homology is unable to capture the homotopic shape evolution of data. PLs overcome this limitation by tracking changes in non-harmonic spectra, revealing the homotopic shape evolution. Additionally, the persistence of PL’s harmonic spectra recovers all topological invariants from persistent homology.\nIn this work, we introduce PL-regularized NMF, namely the topological NMF (TNMF) and robust topological NMF (rTNMF). Both TNMF and rTNMF can better capture multiscale geometric information than the standard GNMF and rGNMF. To achieve improved performance, PL is constructed by observing cell-cell interactions at multiple scales through filtration, creating a sequence of simplicial complexes. We can then view the spectra at each complex associated with a filtration to capture both topological and geometric information. Additionally, we introduce *k*-NN based PL to TNMF and rTNMF, referred to as *k*-TNMF and *k*-rTNMF, respectively. The *k*-NN based PL reduces the number of hyperparameters compared to the standard PL algorithm.\nThe outline of this work is as follows. First, we provide a brief overview of NMF, rNMF, GNMF, and rGNMF. Next, we present a concise theoretical formulation of PL and derive the multiplicative updating scheme for TNMF and rTNMF. Additionally, we introduce an alternative construction of PL, termed *k*-NN PL. Following that, we present a benchmark using 12 publicly available datasets. We have observed that PL can improve NMF performance by up to 0.16 in ARI, 0.08 in NMI, 0.04 in purity, and 0.1 in accuracy.\nMethods\n=======\nIn this section, we provide a brief overview of NMF methods, namely NMF, rNMF, GNMF, and rGNMF. We then give persistent Laplacian and its construction. Finally, we formulate various PL regularized NMF methods.\nPrior Work\n----------\n#### NMF\nThe original formulation of NMF utilizes the Frobenius norm, which assumes that the noise of the data is sample from Gaussian distribution.\n \n$$\\begin{aligned}\n \\min\\_{W, H}\\|X - WH\\|\\_F^2, \\quad \\text{s.t. } W, H \\ge 0\\end{aligned}$$ \nwhere ∥*A*∥*F*2 = ∑*i*, *j**a**i**j*2. Lee et al. proposed a multiplicative updating scheme, which preserves the nonnegativity. For the *t* + 1th iteration,\n \n$$\\begin{aligned}\n & w^{t+1} = w\\_{ij}^t \\frac{(XH^T)\\_{ij}}{(WHH^T)\\_{ij}} \\\\\n & h^{t+1} = h\\_{ij}^t \\frac{(W^TX)\\_{ij}}{(W^TWH)\\_{ij}}\\end{aligned}$$ \nAlthough the updating scheme is simple and effective in many biological data applications, scRNA-seq data is sparse and contains large amount of noise. Therefore, a model that is more robust to noise is necessary for feature selection and dimensionality reduction\n#### rNMF\nThe robust NMF (rNMF) utilizes the *l*2, 1 norm, which assumes that the noise of the data is sampled from a Laplace distribution, which may be more suitable for a count-based data matrix, like scRNA-seq. The minimization function is given as the following\n \n$$\\begin{aligned}\n \\min\\_{W,H}\\| X - WH\\|\\_{2,1}, \\quad \\text{s.t. } W, H \\ge 0,\\end{aligned}$$ \nwhere ∥*A*∥2, 1 = ∑*j*∥**a***j*∥2. Because *l*2, 1-norm utilizes summation over the *l*2 distance of the original cell feature and the reduced feature, the effect of the outlier will not dominate the loss function as much as the Frobenius norm formulation. RNMF has the following updating scheme\n \n$$\\begin{aligned}\n & w^{t+1}\\_{ij} = w\\_{ij}^{t} \\frac{(XQH^T)\\_{ij}}{(WHQH^T)\\_{ij}} \\\\\n & h\\_{ij}^{t+1} = h\\_{ij}^t \\frac{(W^TXQ)\\_{ij}}{(W^TWHQ)\\_{ij}},\\end{aligned}$$ \nwhere *Q**j**j* = 1/∥*X* − *W***h***j*∥2.\n#### GNMF amd rGNM\nManifold regularization has been widely utilized in scRNA-seq. Let *G*(*V*, *E*, *W*) be a graph, where *V* = {**x***j*}*j* = 1*N* is the set of vertices, *E* = {(**x***i*, **x***j*)∣**x***i* ∈ N*k*(**x***j*) ∪ **x***j* ∈ N*k*(**x***i*)} is the set of edges, and *W* is the weight associated with the edges. Here, N*k*(**x***j*) denotes the *k*-th nearest neighbors of vertex *j*. The heat kernel is often used to construct the weight, and we can construct the adjacency matrix *A* as the following.\n \n$$\\begin{aligned}\n A\\_{ij} = \\begin{cases}\n \\exp\\left(-\\frac{\\|\\mathbf{x}\\_i - \\mathbf{x}\\_j\\|^2}{\\sigma}\\right) & \\mathbf{x}\\_j \\in \\mathcal{N}\\_k (\\mathbf{x}\\_i)\\\\\n 0, & \\text{otherwise}.\n \\end{cases}\\end{aligned}$$ \nSince heat kernel satisfies the conditions *W**i**j* → 0 as ∥**x***i* − **x***j*∥ → ∞ and *W**i**j* → 1 as ∥**x***i* − **x***j*∥ → 0, we can construct the graph regularization term, *R**G*, by looking at the distance ∥**h***i* − **h***j*∥2.\n \n$$\\begin{aligned}\n R\\_G & = \\frac{1}{2}\\sum\\_{i,j} A\\_{ij} \\|\\mathbf{h}\\_i - \\mathbf{h}\\_j\\|^2 \\\\\n & = \\sum\\_{i}D\\_{ii}\\mathbf{h}\\_i^T\\mathbf{h}\\_i -\\sum\\_{ij} A\\_{ij} \\mathbf{h}\\_i^T\\mathbf{h}\\_j \\\\\n & = \\text{Tr}(HDH^T) - \\text{Tr}(HAH^T) \\\\\n & = \\text{Tr}(HLH^T).\\end{aligned}$$ \nHere, *L* and *D* are the Laplacian and the degree matrix, given by *L* = *D* − *A* and *D**i**i* = ∑*j**A**i**j*, respectively. Tr( ⋅ ) denotes the trace of the matrix. Utilizing the regularization parameters, *λ* ≥ 0, we get the objective function of GNMF\n \n$$\\begin{aligned}\n \\min\\_{W,H} \\|X - WH\\|\\_F^2 + \\lambda \\text{Tr}(HLH^T).\\end{aligned}$$ \nand the objective function for rGNMF\n \n$$\\begin{aligned}\n \\min\\_{W,H} \\|X - WH\\|\\_{2,1} + \\lambda \\text{Tr}(HLH^T).\\end{aligned}$$ \nTopological NMF\n---------------\nWhile graph regularization improves the traditional NMF and rNMF, the choice of *σ* and vastly change the result. Furthermore, graph regularization only captures a single scale, and may not be able to capture the mutliscale geometric information in data. Here, we give a brief introduction to persistent homology and persistent Laplacian and derive the updating scheme for the topological NMF.\n### Persistent Laplacians\nPersistent homology and persistent spectral graphs have been successfully used in biomolecular data. Similar to persistent homology, persistent spectral graphs track birth and death of topological features, i.e., holes, over different scales. However, unlike persistent homology, persistent spectral graphs can further capture the homotopic shape evolution of data during the filtration. Through the filtration process, these methods offer the multiscale analysis of data.\nWe begin by the definition of simplex. Let *σ**q* = [*v*0, ⋯, *v**q*] denote *q*-simplex, where *v**i* is a vertex. *σ*0 is a node, *σ*1 is an edge, *σ*2 is a triangle *σ*3 is a tetrahedron, and so on. A simplicial complex *K* is a union of simplicies such that\n1. If *σ**q* ∈ *K* and *σ**p* is a face of *σ**q*, then *σ**p* ∈ *K*\n2. The nonempty intersection of any 2 simplicies in *K* is a face of both simplicies.\nWe can think of *K* as gluing lower dimensional simplicies that satisfies the above 2 properties.\nA *q*-chain is a formal sum of *q*-simplicies in *K* with the coefficients Z2 = {0, 1}. The set of all *q*-chains has contains the basis for the set of *q*-simplicies in *K*. Such set forms a finitely generated free Abelian group *C**q*(*K*). We can relate the chain groups via a boundary operator, which is a group homomorphism ∂*q* : *C**q*(*K*) → *C**q* − 1(*K*). The boundary operator is defined as the following.\n \n$$\\begin{aligned}\n \\partial\\_q\\sigma\\_q := \\sum\\_{i=0}^q(-1)^i\\sigma\\_{q-1}^i\\end{aligned}$$ \nwhere *σ**q* − 1*i* = [*v*0, , , , *v**i*\\*, ..., *v**q*], where *σ**q* − 1*i* is a (*q* − 1)-simplex with vertex *v**i* removed. The sequence of chain group connected by the boundary operator defines the chain complex.\n \n$$\\begin{aligned}\n...\\xrightarrow{\\partial\\_{q+2}} C\\_{q+1} \\xrightarrow{\\partial\\_{q+1}} C\\_q(K) \\xrightarrow{\\partial\\_{q}}... \\end{aligned}$$ \nThe chain complex associated with a simplicial complex *K* defines the *q*-th homology group *H**q* = Ker∂*q*/Im∂*q*, and the dimension of *H**q* is the *q*-dimensional holes, or the *q*th Betti number denoted as *β**q*. For example, *β*0 is the number of connected components, *β*1 is the number of loops and *β*2 is the number of cavities.\nWe can now define the dual chain complex through the adjoint operator of ∂*q*. The dual space is defined as *C**q*(*K*) ≅ *C**q*\\*(*K*), and the coboundary operator ∂*q*\\* is defined as ∂*q*\\* : *C**q* − 1(*K*) → *C**q*(*K*). For *ω**q* − 1 ∈ *C**q* − 1(*K*) and *c**q* ∈ *C**q*(*K*), the coboundary operator is defined as\n \n$$\\begin{aligned}\n \\partial^\\*\\omega^{q-1}(c\\_q) \\equiv \\omega^{q-1}(\\partial c\\_q).\\end{aligned}$$ \nHere *ω**q* − 1 is a (*q* − 1) cochain, or a homomorphic mapping from a chain to the coefficient group. The homology of the dual chain complex is called the cohomology.\nWe then define the *q*-combinatorial Laplacian operator △*q* : *C**q*(*K*) → *C**q*(*K*)\n \n$$\\begin{aligned}\n \\triangle\\_q:= \\partial\\_{q+1}\\partial\\_{q+1}^\\* + \\partial\\_q^\\*\\partial\\_q.\\end{aligned}$$ \nLet B*q* be the standard basis for the matrix representation of *q*-boundary operator from *C**q*(*K*) and *C**q* − 1(*K*), and B*q**T* be th *q*-coboundary operator. The matrix representation of the *q*-th order Laplacian operator L*q* is defined as\n \n$$\\begin{aligned}\n \\mathcal{L}\\_q = \\mathcal{B}\\_{q+1}\\mathcal{B}\\_{q+1}^T + \\mathcal{B}\\_q^T\\mathcal{B}\\_q.\\end{aligned}$$ \nThe multiplicity of zero eigenvalue of L*q* is the *q*-th Betti number of the simplicail complex. The nonzero eigenvalues (non-harmonic spectrum) contains other topological and geometrical features.\nAs stated before, simplicial complex does not provide sufficient information to understand the geometry of the data. To this end, we utilize simplicial complex induced by filtration\n \n$$\\begin{aligned}\n \\{\\emptyset\\} = K\\_0 \\subseteq K\\_1 \\subseteq \\cdots \\subseteq K\\_p = K,\\end{aligned}$$ \nwhere *p* is the number of filtration.\nFor each *K**t* 0 ≤ *t* ≤ *p*, denote *C**q*(*K**t*) as chain group induced by *K**t*, and the corresponding boundary operator ∂*q**t* : *C**q*(*K**t*) → *C**q* − 1(*K**t*), resulting in\n \n$$\\begin{aligned}\n \\partial\\_q^t \\sigma\\_q = \\sum\\_{i=1}^q (-1)^i\\sigma\\_{q-1}^{i-1},\\end{aligned}$$ \nfor *σ**q* ∈ *K**t*. The adjoint operator of ∂*q**t* is similarity defined as ∂*q**t* \\* : *C**q* − 1(*K**t*) → *C**q*(*K**t*), which we regard as the mapping *C**q* − 1(*K**t*) → *C**q*(*K**t*) via the isomorphism between cochain and chain groups. Through these 2 operators, we can define the chain complexes induced by *K**t*.\nUtilizing filtration with simplicial complex, we can define persistence Laplacian spectra. Let *C**q**t* + *p* whose boundary is in *C**q* − 1*t* be **C***q**t* + *p*, assuming an inclusion mapping *C**q* − 1*t* → *C**q* − 1*t* + *p*. On this set, we can define the *p*-persistent *q*-boundary operator denoted $\\hat{\\partial}\\_q^{t,p}:\\mathbb{C}\\_q^{t,p} \\to C\\_{q-1}^t$ and the corresponding adjoint operator $(\\hat{\\partial}^{t,p})^\\*:C\\_{q-1}^t \\to \\mathbb{C}\\_q^{t,p}$. Then, the *q*-order *p*-persistent Laplacian operator is computed as\n \n$$\\begin{aligned}\n \\triangle\\_q^{t,p} = \\hat{\\partial}\\_{q+1}^{t,p}(\\hat{\\partial}\\_{q+1}^{t,p})^\\* + (\\hat{\\partial}\\_q^{t})^\\*\\hat{\\partial}\\_q^{t},\\end{aligned}$$ \nand its matrix representation as\n \n$$\\begin{aligned}\n \\mathcal{L}\\_q^{t,p} = \\mathcal{B}\\_{q+1}^{t,p}(\\mathcal{B}\\_{q+1}^{t,p})^T + (\\mathcal{B}\\_q^t)^T\\mathcal{B}\\_q^t.\\end{aligned}$$ \nLikewise as before, the multiplicity of the zero-eigenvalue is the *q*-th order *p*-persistent Betti number *β**q**t*, *p*, which is the *q*-dimensional hole in *K**t* that persists in *K**t* + *p*. Moreover, the *q*-th order Laplacian is just a particular case of L*q**t*, *p*, where *p* = 0, which is a snapshot of the topology at the filtration step *t*.\nWe can utilize the 0-persistent Laplacian to capture the interactions between the data at different filtration values. In particular, we can perform filtration by computing a family of subgraphs induced by a threshold distance *r*, which is called the Vietoris Rips complex. Alternatively, we can compute a Gaussian Kernel induced distance to construct the subgraphs.\n### TNMF and rTNMF\nFor scRNA-seq data, we calculate the 0-persistent Laplacian using the Vietoris-Rips (VR) complexes by increasing the filtration distance. We can then take a weighted sum over the 0-persistent Laplacian induced by the changes in the filtration distance. For persistent Laplacian enhanced NMF, we will provide a computationally efficient algorithm to construct the persistent Laplacian matrix.\nLet *L* be a Laplacian matrix induced by some weighted graph, and note the following\n \n$$\\begin{aligned}\n L = \\begin{cases}\n l\\_{ij}, & i \\ne j \\\\\n -\\sum\\_{j=1}^N l\\_{ij} & i = j.\n \\end{cases}\\end{aligned}$$ \nThen, let *l*max = max*i* ≠ *j**l**i**j*, *l*min = min*i* ≠ *j**l**i**j* and *d* = *l*max − *l*min. The *t*-th Persistent Laplacian *L**t*, *t* = 1, ..., *T* is defined as *L**t* = {*l**i**j**t*}, where\n \n$$\\begin{aligned}\n & l\\_{ij}^t = \\begin{cases}\n 0 \\quad l\\_{ij} \\le (t/T)d + l\\_{\\min} \\\\\n 1 & \\text{otherwise}\n \\end{cases} \\\\\n & l\\_{ii}^t = -\\sum\\_{i\\ne j}l\\_{ij}^t.\\end{aligned}$$ \nThen, we can take the weighted sum over the all the persistent Laplacians\n \n$$\\begin{aligned}\n PL := \\sum\\_{t=1}^T \\zeta\\_t L^t.\\end{aligned}$$ \nUnlike the standard Laplacian matrix *L*, PL captures the topological features that persists over different filtration, thus providing a multiscale view of the data that standard Laplacian lacks. Here, *ζ**t* is the hyper-parameter and must be chosen as a hyperparameter. Then, the topological NMF (TNMF) is defined as\n \n$$\\begin{aligned}\n \\|X - WH\\|\\_F^2 + \\text{Tr}(H^T(PL)H) \\end{aligned}$$ \nand the topological rNMF (rTNMF) is defined as\n \n$$\\begin{aligned}\n \\|X-WH\\|\\_{2,1} + \\text{Tr}(H^T(PL)H).\\end{aligned}$$ \n### Multiplicative Updating scheme\nThe updating scheme follows the same principle as the standard GNMF and rGNMF.\n#### TNMF\nFor top-NMF, the Lagrangian function is defined as\n \n$$\\begin{aligned}\n \\mathcal{L} & = \\|X - WH\\|\\_F^2 + \\lambda \\text{Tr}(H^T(PL)H) + \\text{Tr}(\\Phi W) + \\text{Tr}(\\Psi H) \\\\\n & = \\text{Tr}(X^TX) - 2\\text{Tr}(XH^TW^T) + \\text{Tr}(WHH^TW^T)+ \\lambda \\text{Tr}(H^T(PL)H) + \\text{Tr}(\\Phi W) + \\text{Tr}(\\Psi H).\\end{aligned}$$ \nTaking the partial with respect to *W*, we get\n \n$$\\begin{aligned}\n \\frac{\\partial \\mathcal{L}}{\\partial W} = -2H^TXH + 2WHH^T + \\Phi.\\end{aligned}$$ \nUsing the KKT condition Φ*i**j**w**i**j* = 0, we get the following\n \n$$\\begin{aligned}\n (-2XH^T)\\_{ij}w\\_{ij} +(2WHH^T)\\_{ij} w\\_{ij} = 0.\\end{aligned}$$ \nTherefore, the updating scheme is\n \n$$\\begin{aligned}\n w\\_{ij}^{t+1} \\leftarrow w\\_{ij}^t \\frac{(XH^T)\\_{ij}}{(WHH^T)\\_{ij}}.\\end{aligned}$$ \nFor updating *H*, we take the derivative of the Lagrangian function with respect to *H*\n \n$$\\begin{aligned}\n \\frac{\\partial \\mathcal{L}}{\\partial H} = -2 W^TX + 2W^TWH + 2\\lambda H (PL)+ \\Psi.\\end{aligned}$$ \nUsing the Karush–Kuhn–Tucker (KKT) condition, we have Ψ*i**j**h**i**j* = 0 and obtain\n \n$$\\begin{aligned}\n -2(W^TX + \\lambda H(PA))\\_{ij}h\\_{ij} + 2(W^TWH + \\lambda H(PD))\\_{ij}h\\_{ij} = 0,\\end{aligned}$$ \nwhere *P**L* = *P**D* − *P**A* and *P**D**i**i* = ∑*i* ≠ *j**P**A**i**j*. The updating scheme is then given by\n \n$$\\begin{aligned}\n h\\_{ij}^{t+1} \\leftarrow h\\_{ij}^t\\frac{(W^TWH + \\lambda H(PD))\\_{ij}}{(W^TX + \\lambda H(PA))\\_{ij}}.\\end{aligned}$$ \n#### rTNMF\nFor the updating scheme for top-rNMF, we utilize the fact that ∥*A*∥2, 1 = Tr(*A**Q**A**T*), where $Q\\_{ii} = \\frac{1}{2\\|A\\_i\\|\\_2}$. The Lagrangian is given by\n \n$$\\begin{aligned}\n \\mathcal{L} & = \\|X - WH\\|\\_{2,1} + \\lambda \\text{Tr}(H^T(PL)H) + \\text{Tr}(\\Phi W) + \\text{Tr}(\\Psi H) \\\\\n & = \\text{Tr}( (X-WH)Q(X-WH)^T ) + \\lambda \\text{Tr}(H^T(PL)H) + \\text{Tr}(\\Phi W) + \\text{Tr}(\\Psi H) \\\\\n & = \\text{Tr}( XQX^T ) -2\\text{Tr}(WHQ) + \\lambda \\text{Tr}(H^T(PL)H) + \\text{Tr}(\\Phi W) + \\text{Tr}(\\Psi H),\\end{aligned}$$ \nwhere $Q\\_{ii} = \\frac{1}{\\|\\mathbf{x}\\_j - W\\mathbf{h}\\_j\\|}$. Taking the partial with respect to *W*, we get\n \n$$\\begin{aligned}\n \\frac{\\partial L}{\\partial W} = -(XQH^T) + WHQH^T - \\Phi.\\end{aligned}$$ \nUsing the KKT conditions Φ*i**j**w**i**j* = 0, we get\n \n$$\\begin{aligned}\n -(XQH^T)\\_{ij}w\\_{ij} + (WHQH^T)\\_{ij}w\\_{ij} = 0,\\end{aligned}$$ \nwhich gives the updating scheme\n \n$$\\begin{aligned}\n w\\_{ij}^{t+1} \\leftarrow w\\_{ij}^t \\frac{(XQH^T)\\_{ij}}{ (WHQH^T)\\_{ij}}.\\end{aligned}$$ \nFor *H*, we take the partial with respect to *H*.\n \n$$\\begin{aligned}\n \\frac{\\partial L}{\\partial H} = -W^TXQ + W^TWHQ + 2\\lambda H(PL) + \\Psi.\\end{aligned}$$ \nThen, using the KKT conditions Ψ*i**j**h**i**j* = 0, we get\n \n$$\\begin{aligned}\n (-W^TXQ - 2\\lambda H(PA))\\_{ij}h\\_{ij} + (W^TWHQ + 2\\lambda H(PD))\\_{ij}h\\_{ij} = 0,\\end{aligned}$$ \nwhere *P**L* = *P**D* − *P**A* and gives the updating scheme\n \n$$\\begin{aligned}\n h\\_{ij}^{t+1} \\leftarrow h\\_{ij}^t \\frac{(W^TXQ+2\\lambda H(PA))\\_{ij}}{(W^TWHQ + 2\\lambda H(PD))\\_{ij}}. \\end{aligned}$$ \n*k*-NN induced Persistent Laplacian\n-----------------------------------\nOne major issue with top-GNMF and top-rGNMF is that the parameters {*ζ**t*}*t* = 1*T* have to be chosen. For the parameters, we let *ζ**t* ∈ {0, 1, 1/2, ⋯, 1/*T*} for a total of *T* + 1 parameters. Therefore, the number of parameters that needs to be chosen increases exponentially as the number of filtration *T* increases. Therefore, we propose an approximation to the original formulation using *k*-NN based persistent Laplacian.\nLet N*t*(**x***j*) be the *t*-nearest neighbors of sample **x***j*. Then, define the *t*-persistent directed adjacency matrix *Ã**t* as\n \n$$\\begin{aligned}\n \\tilde{A}^t = \\{\\tilde{a}\\_{ij}^t\\}, \\quad \\tilde{a}\\_{ij}^t = \\begin{cases}\n 1 & \\mathbf{x\\_j} \\in \\mathcal{N}\\_t(\\mathbf{x}\\_i) \\\\\n 0 & \\text{otherwise}.\n \\end{cases}\\end{aligned}$$ \nThen, the *k*-NN based directed adjacency Laplacian is the weighted sum of {*A**t*}\n \n$$\\begin{aligned}\n \\tilde{A} := \\sum\\_{t=1}^T \\zeta\\_t \\tilde{A}^t.\\end{aligned}$$ \nThen, the undirected persistent adjacency matrix can be obtained via symmetrization\n \n$$\\begin{aligned}\n PA = \\tilde{A} + \\tilde{A}^T - \\tilde{A}\\cdot \\tilde{A}^T,\\end{aligned}$$ \nwhere ⋅ denote Hadamard product. Then, the persistent Laplacian can be constructed using the persistent degree matrix\n \n$$\\begin{aligned}\n PL = PD - PA, \\quad PD\\_{ii} = \\sum\\_{j\\ne i}PA\\_{ij}.\\end{aligned}$$ \nOne advantage of utilizing the *k*-NN induced persistent Laplacian is that the parameter space is much smaller. We can set *ζ**t* ∈ {0, 1}, where *ζ**t* = 0 would ’turn-off’ the particular neighbor’s connectivity. In essence, the number of parameters will be reduced to 2*T*, a significant decrease from *T*(*T* + 1) of the original formulation.\nEvaluation metrics\n------------------\nLet *Y* = {*Y*1, ..., *Y**L*} and *C* = {*C*1, ..., *C**L*} be 2 partitions of the data. Here, we let *Y* be the true label partition and *C* be the cluster label partition. Let {*y**i*}*i* = 1*N* and {*c**i*}*i* = 1*N* be the true and predicted labels of sample *i*.\n#### Adjusted Rand Index\nAdjusted random index (ARI) measures the similarity between two clustering by observing all pairs of samples that belong to the same cluster, and seeing if the other clustering result also have the same pair of samples in the same cluster. Let *n**i**j* = ∣*T**i* ∩ *S**j*∣ be the number of samples that belong to true label *i* and cluster label *j*, and define *a**i* = ∑*j**n**i**j* and *b**j* = ∑*i**n**i**j*. Then, the ARI is defined as\n \n$$\\begin{aligned}\n \\text{ARI} = \\frac{\\sum\\_{ij} \\begin{pmatrix}\n n\\_{ij} \\\\ 2\n \\end{pmatrix} - \\left[ \\sum\\_i \\begin{pmatrix}\n a\\_i \\\\ 2\\end{pmatrix} \\sum\\_j \\begin{pmatrix}\n b\\_j \\\\ 2\\end{pmatrix}\\right] / \\begin{pmatrix}\n N \\\\ 2\\end{pmatrix} }{\\frac{1}{2} \\left[\\begin{pmatrix}\n a\\_i \\\\ 2\\end{pmatrix} + \\begin{pmatrix}\n b\\_j \\\\ 2\\end{pmatrix}\\right] - \\left[\\begin{pmatrix}\n a\\_i \\\\ 2\\end{pmatrix} \\sum\\_j \\begin{pmatrix}\n b\\_j \\\\ 2\\end{pmatrix}\\right] / \\begin{pmatrix}\n N \\\\ 2\\end{pmatrix}}.\\end{aligned}$$ \nThe ARI takes on a value between -1 and 1, where 1 is a perfect match between two clustering methods, and 0 is a completely random assignment of labels, and -1 indicates that the two clusterings are completely different.\n#### Normalized Mutual Information\nThe normalized mutual information (NMI) measures the mutual information between two clustering results and normalized according to cluster size. We fix the true labels *Y* as one of the clustering result, and use the predicted labels as the other to calculate NMI. The NMI is calculated as the following\n \n$$\\begin{aligned}\n \\text{NMI} = \\frac{2I(Y; C)}{H(Y)H(C)},\\end{aligned}$$ \nwhere *H*( ⋅ ) is the entropy and *I*(*Y*; *C*) is the mutual information between true labels *Y* and predicted labels *C*. NMI has a range of 0 and 1, where 1 is a perfect mutual correlation between the two sets of labels and 0 means no mutual information.\n#### Accuracy\nAccuracy (ACC) calculates the percentage of correctly predicted class labels. The accuracy is given by\n \n$$\\begin{aligned}\n \\text{ACC} = \\frac{1}{N} \\sum\\_{i=1}^N \\delta(y^i, f(c^i)),\\end{aligned}$$ \nwhere *δ*(*a*, *b*) is the indicator function, where if *a* = *b*, *δ*(*a*, *b*) = 1, and 0 otherwise. *f* : *C* → *Y* maps the cluster labels to the true labels, where the mapping is the optimal permutation of the cluster labels and true labels obtained from the Hungarian algorithm.\n#### Purity\nFor purity calculation, each predicted label *C**i* is assigned to a true label *Y**j* such that the ∣*C**i* ∩ *Y**j*∣ is maximized. Taking the average over all the predicted label, we obtain the following\n \n$$\\begin{aligned}\n \\text{Purity} = \\frac{1}{N}\\max\\_j |C\\_i \\cap Y\\_j|.\\end{aligned}$$ \nNote that unlike accuracy, purity does not map the predicted labels to the true labels.\nResults\n=======\nBenchmark Data\n--------------\nWe have performed benchmark on 12 publicly available datasets. The GEO accession number, reference, organism, number of cell types, and number of samples are recorded in. For each data, cell types with less than 15 cells were removed. Log-normalization was applied, and scaled the data to have unit length. For GNMF and rGNMF, *k* = 8 neighbors were used. For TNMF and rTNMF, 8 filtration values were used to construct PL, and for each scale, binary selection *ζ**p* = {0, 1} was used. for *k*-TNMF and *k*-rTNMF, *k* = 8 was used with *ζ**p* = {0, 1}. For each test, double nonnegative singular value decomposition with zeros filled with the average of *X* (NNDSVDA) was used for the initialization. The *k*-means clustering was applied to obtain the clustering results.\n[H]\n| Geo Accession | Reference | Organism | Cell type | Number of Samples | Number of Genes |\n| --- | --- | --- | --- | --- | --- |\n| GSE67835 | Dramanis | Human | 8 | 420 | 22084 |\n| GSE75748 time | Chu | Human | 6 | 758 | 19189 |\n| GSE82187 | Gokce | Mouse | 8 | 705 | 18840 |\n| GSE84133human1 | Baron | Human | 9 | 1895 | 20125 |\n| GSE84133human2 | Baron | Human | 9 | 1702 | 20125 |\n| GSE84133human3 | Baron | Human | 9 | 3579 | 20125 |\n| GSE84133human4 | Baron | Human | 6 | 1275 | 20125 |\n| GSE84133mouse1 | Baron | Mouse | 6 | 782 | 14878 |\n| GSE84133mouse2 | Baron | Mouse | 8 | 1036 | 14878 |\n| GSE57249 | Biase | Human | 3 | 49 | 25737 |\n| GSE64016 | Leng | Human | 4 | 460 | 19084 |\n| GSE94820 | Villani | Human | 5 | 1140 | 26593 |\n[tab: data]\nBenchmarking PL regularized NMF\n-------------------------------\nIn order to benchmark persistent Laplacain regularized NMF, we compared our methods to other commonly used NMF methods, namely the GNMF, rGNMF, rNMF and NMF. For a fair comparison, We omitted supervised or semi-supervised methods. For *k*-rTNMF, rTNMF, *k*-TNMF, TNMF, GNMF and rGNMF, we set *α* = 1 for all tests.\nshows the ARI values of the NMF methods for the 12 data we have tested. The bold number indicate the highest performance. depicts the average ARI value over the 12 datasets for each method.\n[H]\n| data | *k*-rTNMF | rTNMF | *k*-TNMF | TNMF | rGNMF | GNMF | rNMF | NMF |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| GSE67835 | **0.9454** | 0.9236 | 0.9306 | 0.8533 | 0.9391 | 0.9109 | 0.7295 | 0.7314 |\n| GSE64016 | **0.2569** | 0.1544 | 0.2237 | 0.1491 | 0.1456 | 0.1605 | 0.1455 | 0.1466 |\n| GSE75748time | 0.6421 | **0.6581** | 0.5963 | 0.6099 | 0.6104 | 0.5790 | 0.5969 | 0.5996 |\n| GSE82187 | **0.9877** | 0.9815 | 0.9676 | 0.9809 | 0.7558 | 0.7577 | 0.8221 | 0.8208 |\n| GSE84133human1 | 0.8310 | **0.8969** | 0.8301 | 0.8855 | 0.8220 | 0.7907 | 0.7080 | 0.6120 |\n| GSE84133human2 | **0.9469** | 0.9072 | 0.9433 | 0.9255 | 0.9350 | 0.9255 | 0.8930 | 0.8929 |\n| GSE84133human3 | 0.8504 | 0.9179 | 0.8625 | **0.9181** | 0.8447 | 0.8361 | 0.7909 | 0.8089 |\n| GSE84133human4 | 0.8712 | **0.9692** | 0.8712 | 0.9692 | 0.8699 | 0.8681 | 0.8311 | 0.8311 |\n| GSE84133mouse1 | **0.8003** | 0.7894 | 0.8003 | 0.7913 | 0.7945 | 0.7918 | 0.6428 | 0.6348 |\n| GSE84133mouse2 | 0.6953 | 0.8689 | 0.7005 | **0.9331** | 0.6808 | 0.6957 | 0.5436 | 0.5470 |\n| GSE57249 | **1.0000** | 0.9638 | 1.0000 | 0.9483 | 1.0000 | 1.0000 | 0.9483 | 0.9483 |\n| GSE94820 | **0.6101** | 0.5480 | 0.4916 | 0.5574 | 0.5139 | 0.5189 | 0.5440 | 0.5556 |\n[tab: ARI]\n![Average ARI of k-rTNMF, rTNMF, k-TNMF,TNMF, rGNMF, GNMF, rNMF and NMF for the 12 datasets](figures/mean_ARI.png \"fig:\") [fig: ARI]\nOverall, PL regularized rNMF and NMF have the highest ARI value across all the datasets. *k*-rTNMF outperforms other NMF methods by at least 0.09 for GSE64016. All PL regularized NMF methods outperform other NMF methods by at least 0.14 for GSE82187. For GSE84133 human 3, both rTNMF and TNMF outperform other methods by 0.07. TNMF improves other methods by more than 0.2 for GSE84133 mouse 2. Lastly, *k*-rTNMF has the highest ARI value for GSE94820. Moreover, rTNMF improves rGNMF by 0.05, and TNMF improves GNMF by about 0.06. *k*-TNMF and *k*-rTNMF also improve GNMF and rGNMF by about 0.03.\nshows the NMI values of of the NMF methods for the 12 datasets we have tested. The bold number indicate the highest performance. shows the average NMI value over the 12 datasets.\n[H]\n| data | *k*-rTNMF | rTNMF | *k*-TNMF | TNMF | rGNMF | GNMF | rNMF | NMF |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| GSE67835 | **0.9235** | 0.8999 | 0.9107 | 0.8607 | 0.9104 | 0.8858 | 0.7975 | 0.8017 |\n| GSE64016 | 0.**3057** | 0.2059 | 0.3136 | 0.1869 | 0.2593 | 0.2562 | 0.1896 | 0.1849 |\n| GSE75748time | 0.7522 | **0.7750** | 0.7159 | 0.7343 | 0.7235 | 0.6971 | 0.7227 | 0.7244 |\n| GSE82187 | **0.9759** | 0.9691 | 0.9298 | 0.9668 | 0.8802 | 0.8754 | 0.9124 | 0.9117 |\n| GSE84133human1 | **0.8802** | 0.8716 | 0.8785 | 0.8780 | 0.8713 | 0.8310 | 0.8226 | 0.7949 |\n| GSE84133human2 | **0.9363** | 0.8937 | 0.9313 | 0.9070 | 0.9237 | 0.9145 | 0.8835 | 0.8829 |\n| GSE84133human3 | 0.8500 | **0.8718** | 0.8577 | 0.8677 | 0.8439 | 0.8357 | 0.8215 | 0.8260 |\n| GSE84133human4 | 0.8795 | **0.9542** | 0.8795 | 0.9542 | 0.8775 | 0.8753 | 0.8694 | 0.8694 |\n| GSE84133mouse1 | **0.8664** | 0.8498 | 0.8664 | 0.8495 | 0.8596 | 0.8565 | 0.7634 | 0.7593 |\n| GSE84133mouse2 | 0.8218 | **0.8355** | 0.8299 | 0.8713 | 0.8005 | 0.8129 | 0.7258 | 0.7272 |\n| GSE57249 | **1.0000** | 0.9505 | 1.0000 | 0.9293 | 1.0000 | 1.0000 | 0.9293 | 0.9293 |\n| GSE94820 | **0.7085** | 0.6657 | 0.6157 | 0.6716 | 0.6195 | 0.6258 | 0.6624 | 0.6693 |\n[tab: NMI]\n![Average NMI values of k-rTNMF, rTNMF, k-TNMF,TNMF, rGNMF, GNMF, rNMF and NMF for the 12 datasets](figures/mean_NMI.png \"fig:\") [fig: NMI]\nInterestingly, *k*-rTNMF and *k*-TNMF on average have higher NMI values than rTNMF and TNMF, respectively. However, all PL regularized methods outperform rGNMF, GNMF, rNMF and NMF. Overall, PL regularized methods outperform other methods. Most noticeably, *k*-rTNMF, rTNMF and TNMF outperform standard NMF methods by 0.06 for GSE82187. Both rTNMF and TNMf outperform rGNMF and GNMF by 0.08 for GSE84133 human 4.\nshows the purity values of the NMF methods for the 12 datasets we have tested. The bold number indicate the highest performance. shows the average purity over the 12 datasets.\n[H]\n| data | *k*-rTNMF | rTNMF | *k*-TNMF | TNMF | rGNMF | GNMF | rNMF | NMF |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| GSE67835 | **0.9643** | 0.9267 | 0.9595 | 0.9024 | 0.9595 | 0.9476 | 0.8726 | 0.8719 |\n| GSE64016 | **0.6048** | 0.4913 | 0.5846 | 0.5013 | 0.5339 | 0.5398 | 0.5080 | 0.5050 |\n| GSE75748time | **0.7736** | 0.7512 | 0.7533 | 0.7454 | 0.7553 | 0.7387 | 0.7467 | 0.7455 |\n| GSE82187 | **0.9927** | 0.9895 | 0.9620 | 0.9888 | 0.9620 | 0.9594 | 0.9693 | 0.9692 |\n| GSE84133human1 | **0.9543** | 0.9357 | 0.9536 | 0.9382 | 0.9490 | 0.9187 | 0.9189 | 0.9099 |\n| GSE84133human2 | **0.9818** | 0.9614 | 0.9806 | 0.9661 | 0.9777 | 0.9736 | 0.9602 | 0.9600 |\n| GSE84133human3 | 0.9472 | **0.9485** | 0.9531 | 0.9460 | 0.9452 | 0.9420 | 0.9464 | 0.9466 |\n| GSE84133human4 | 0.9427 | **0.9882** | 0.9427 | 0.9882 | 0.9427 | 0.9420 | 0.9412 | 0.9412 |\n| GSE84133mouse1 | **0.9565** | 0.9540 | 0.9565 | 0.9540 | 0.9552 | 0.9540 | 0.9309 | 0.9299 |\n| GSE84133mouse2 | 0.9585 | 0.9410 | **0.9604** | 0.9373 | 0.9466 | 0.9507 | 0.9185 | 0.9199 |\n| GSE57249 | **1.0000** | 0.9857 | 1.0000 | 0.9796 | 1.0000 | 1.0000 | 0.9796 | 0.9796 |\n| GSE94820 | **0.7893** | 0.7462 | 0.6658 | 0.7550 | 0.6421 | 0.6421 | 0.7429 | 0.7531 |\n[tab: Purity]\n![Average purity values of k-rTNMF, rTNMF, k-TNMF,TNMF, rGNMF, GNMF, rNMF and NMF for the 12 datasets](figures/mean_Purity.png \"fig:\") [fig: Purity]\nIn general, PL-regularized methods achieve higher purity values compared to other NMF methods. Purity measures the maximum intersection between true and predicted classes, which is why we do not observe a significant difference, as seen in ARI and NMI. Furthermore, since purity does not account for the size of a class, and given the imbalanced class sizes in scNRA-seq data, it is not surprising that the purity values are similar.\nshows the ACC of the NMF methods for the 12 datasets we have tested. The bold number indicate the highest performance. shows the average ACC over the 12 datasets.\n[H]\n| data | *k*-rTNMF | rTNMF | *k*-TNMF | TNMF | rGNMF | GNMF | rNMF | NMF |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| GSE67835 | **0.9643** | 0.9243 | 0.9595 | 0.9000 | 0.9595 | 0.9383 | 0.8357 | 0.8364 |\n| GSE64016 | **0.5700** | 0.4870 | 0.5502 | 0.4746 | 0.4891 | 0.4537 | 0.4691 | 0.4759 |\n| GSE75748time | **0.7565** | 0.7438 | 0.7414 | 0.6917 | 0.7355 | 0.7241 | 0.6873 | 0.6875 |\n| GSE82187 | **0.9927** | 0.9895 | 0.9599 | 0.9888 | 0.8512 | 0.8514 | 0.8896 | 0.8889 |\n| GSE84133human1 | 0.8973 | **0.9194** | 0.8974 | 0.9088 | 0.8889 | 0.8364 | 0.7988 | 0.7370 |\n| GSE84133human2 | **0.9260** | 0.9069 | 0.9242 | 0.9447 | 0.9224 | 0.9177 | 0.8998 | 0.8994 |\n| GSE84133human3 | 0.8539 | **0.9456** | 0.8597 | 0.9419 | 0.8498 | 0.8228 | 0.8032 | 0.8178 |\n| GSE84133human4 | 0.8831 | **0.9882** | 0.8831 | 0.9882 | 0.8824 | 0.8816 | 0.8847 | 0.8847 |\n| GSE84133mouse1 | **0.8581** | 0.8542 | 0.8581 | 0.8542 | 0.8555 | 0.8542 | 0.7361 | 0.7311 |\n| GSE84133mouse2 | 0.8232 | **0.9101** | 0.8263 | 0.9305 | 0.7903 | 0.8155 | 0.7239 | 0.7294 |\n| GSE57249 | **1.0000** | 0.9857 | 1.0000 | 0.9796 | 1.0000 | 1.0000 | 0.9796 | 0.9796 |\n| GSE94820 | **0.7533** | 0.7119 | 0.6482 | 0.7201 | 0.6088 | 0.6107 | 0.7091 | 0.7189 |\n[tab: ACC]\n![Average ACC of k-rTNMF, rTNMF, k-TNMF,TNMF, rGNMF, GNMF, rNMF and NMF for the 12 datasets](figures/mean_ACC.png \"fig:\") [fig: ACC]\nOnce again, we see that PL regularized methods have higher ACC than other NMF methods. RTNMF and TNMF improves rGNMF and GNMF by 0.05, and *k*-rTNMF and *k*-TNMF improves rGNMF and GNMF by 0.04. We see an improvement in ACC for both *k*-rTNMF and *k*-TNMF for GSE64016. All 4 PL regularized methods improve ACC of GSE82187 by 0.1. RTNMF and TNMF improve GSE84133 mouse 2 by at least 0.1 as well.\nOverall performance\n-------------------\nshows the average ARI, NMI, purity and ACC of *k*-rTNMF, rTNMF, *k*-TNMF, TNMF, rGNMF, GNMF, rNMF, NMF across 10 datasets. All PL regularized NMF methods outperform the traditional rGNMF, GNMF, rNMF and NMF. Both rTNMF and TNMF have higher average ARI and purity than the *k*-NN based PL counterparts. However, *k*-rTNMF and *k*-TNMF have higher average NMI than rTNMF and TNMF, respectively. *k*-rTNMF has a significantly higher purity than other methods.\n![Average ARI, NMI, purity and ACC of k-rTNMF, rTNMF, k-TNMF, TNMF, rGNMF, GNMF, rNMF, NMF across 10 datasets](figures/results.png \"fig:\") [fig: overall]\nDiscussion\n==========\nVisualization of meta-genes based UMAP and t-SNE\n------------------------------------------------\nBoth UMAP and t-SNE are well-known for their effectiveness in visualization. However, these methods may not perform as competitively in clustering or classification tasks. Therefore, it is beneficial to employ NMF-based methods to enhance the visualization capabilities of UMAP and t-SNE.\nIn this process, we generate meta-genes and subsequently utilize UMAP or t-SNE to further reduce the data to 2 dimensions for visualization. For a dataset with *M* cells, the number of meta-genes will be the integer value of $\\sqrt{M}$. To compare the standard UMAP and t-SNE plots with the top-NMF-assisted and top-rNMF-assisted UMAP and t-SNE visualizations, we used the default settings of the Python implementation of UMAP and the Scikit-learn implementation of t-SNE. For unassisted UMAP and t-SNE, we first removed low-abundance genes and performed log-transformation before applying UMAP and t-SNE.\nshows the visualization of PL regularized NMF methods through UMAP. Each row corresponds to GSE67835, GSE75748 time, GSE94820 and GSE84133 mouse 2 data. The columns from left to right are the *k*-rTNMF assisted UMAP, rTNMF assisted UMAP, *k*-TNMF assisted UMAP, TNMF assisted UMAP and UMAP visualization. Samples were colored according to their true cell types.\n![Visualization of top-NMF and top-rNMF meta-genes through UMAP. Each row corresponds to GSE67835, GSE75748 time, GSE94820 and GSE84133 mouse 2 data. The columns from left to right are the k-rTNMF assisted UMAP, rTNMF assisted UMAP, k-TNMF assisted UMAP, TNMF assisted UMAP and UMAP visualization. Samples were colored according to their true cell types](./figures/topology_umap.png \"fig:\") [fig: top umap]\nshows the visualization of PL regularized NMF through t-SNE. Each row corresponds to GSE67835, GSE75748 time, GSE94820 and GSE84133 mouse 2 data. The columns from left to right are the *k*-rTNMF assisted t-SNE, rTNMF assisted t-SNE, *k*-TNMF assisted t-SNE, TNMF assisted t-SNE and t-SNE visualization. Samples were colored according to their true cell types.\n![Visualization of top-NMF and top-rNMF meta-genes through t-SNE. Each row corresponds to GSE67835, GSE75748 time, GSE94820 and GSE84133 mouse 2 data. The columns from left to right are the k-rTNMF assisted t-SNE, rTNMF assisted t-SNE, k-TNMF asssited t-SNE, TNMF assisted t-SNE and t-SNE visualization. Samples were colored according to their true cell types](./figures/topology_tsne.png \"fig:\") [fig: top tsne]\nWe see a considerable improvement in both top-NMF assisted and top-rNMF assisted UMAP and t-SNE visualization.\n#### GSE67835\nIn the assisted UMAP and t-SNE visualizations of GSE67835, we observe a more distinct cluster, which includes a supercluster of fetal quiescent (Fetal-Q) and fetal replicating (Fetal-R) cells. Darmanis et al. conducted a study that involved obtaining differential gene expression data for human adult brain cells and sequencing fetal brain cells for comparison. It is not surprising that the undeveloped Fetal-Q and Fetal-R cells do not exhibit significant differences and cluster together.\n#### GSE75748 time\nIn GSE75748 time data, Chu et al. sequenced human embryonic stem cells at times 0hr, 12hr, 24hr, 36hr, 72hr, and 96hr under hypoxic conditions to observe differentiation. In unassisted UMAP and t-SNE, although some clustering is visible, there is no clear separation between the clusters. Additionally, two subclusters of 12hr cells are observed.\nNotably, in the PL-regularized assisted UMAP and t-SNE visualizations, there is a distinct supercluster comprising the 72hr and 96hr cells, while cells from different time points form their own separate clusters. This finding aligns with Chu’s observation that there was no significant difference between the 72hr and 96hr cells, suggesting that differentiation may have already occurred by the 72hr mark.\n#### GSE94820\nNotice that in both t-SNE and UMAP, although there is a boundary, the cells do not form distinct clusters. This lack of distinct clustering can pose challenges in many clustering and classification methods. On the other hand, all PL-regularized NMF methods result in distinct clusters.\nAmong the PL-regularized NMF approaches, cutoff-based PL, rTNMF, and TNMF form a single CD1C+ (CD1C1) cluster, whereas the *k*-NN induced PL, *k*-rTNMF, and *k*-TNMF exhibit two subclusters. Villani et al. previously noted the similarity in the expression profile of CD1C1−CD141− (DoubleNeg) cells and monocytes. PL-regularized NMF successfully differentiates between these two types.\n#### GSE84133 mouse 2\nPL-regularized NMF yields significantly more distinct clusters compared to unassisted UMAP and t-SNE. Notably, the beta and gamma cells form distinct clusters in PL-regularized NMF. Additionally, when PL-regularized NMF is applied to assist UMAP, potential outliers within the beta cell population become visible. Baron et al. previously highlighted heterogeneity within the beta cell population, and we observe potential outliers in all visualizations.\nRS analysis\n-----------\nAlthough UMAP and t-SNE are excellent tools for visualizing clusters, they may struggle to capture heterogeneity within clusters. Moreover, these methods can be less effective when dealing with a large number of classes. Therefore, it is essential to explore alternative visualization techniques.\n![RS plots of GSE67835 data. The columns from left to right correspond to k-rTNMF, rTNMF, k-TNMF, and TNMF. Each row corresponds to a cell type. For each section, the x-axis and y-axis correspond to the S-score and R-score, respectively. K-means was used to obtain a cluster label, and the Hungarian algorithm was used to map the cluster labels to the true labels. Each sample was colored according to their true labels.](figures/RSplot.png \"fig:\") [fig: RS plot]\nIn our approach, we visualize each cluster using RS plots. RS plots depict the relationship between the residue score (R score) and similarity score (S score) and have proven useful in various applications for visualizing data with multiple class types.\nLet {(**x***m*, *y**m*)∣**x***m* ∈ R*N*, *y**m* ∈ Z*L*, 1 ≤ *m* ≤ *M*} be the data, where **x***m* is the *m*th sample, *y**m* is the cell type or cluster label. *L* is the number of class. That is, *C**l* = {**x***m* ∈ X∣*y**m* = *l*} and ⊎ 0*L* − 1C*l* = X.\nThe residue (R) score is defined as the inter-class sum of distance. For a given data **x***m* with assignment *y**m* = *l*, the R-score is defined as\n \n$$\\begin{aligned}\n & R\\_m = R(\\mathbf{x}\\_m) = \\frac{1}{R\\_{\\max}}\\sum\\_{\\mathbf{x}\\_j \\notin \\mathcal{C}\\_l} \\|\\mathbf{x}\\_m - \\mathbf{x}\\_j\\|,\\end{aligned}$$ \nwhere *R*max = max**x***m*, **x***m* ∈ X*R**m*. The similarity (S) score is defined as the intra-class average of distance, defined as\n \n$$\\begin{aligned}\n S\\_m = S(\\mathbf{x}\\_m) = \\frac{1}{|\\mathcal{C}\\_l|}\\sum\\_{\\mathbf{x}\\_j \\in \\mathcal{C}\\_l} \\left(1 - \\frac{\\|\\mathbf{x}\\_m - \\mathbf{x}\\_j\\|}{d\\_{\\max}}\\right),\\end{aligned}$$ \nwhere *d*max = max**x***i*, **x***j* ∈ X∥**x***i* − **x***j*∥ and ∣C*l*∣ is the number of data in class C*l*. Both *R**m* and *S**m* are bounded by 0 and 1, and the larger the better for a given dataset.\nThe class residue index (CRI) and the class similarity index (CSI) can then be defined as the average of the R-score and S-score of each of the classes. That is $ \\text{CRI}\\_l = \\frac{1}{|\\mathcal{C}\\_l|}\\sum\\_{m} R\\_m$ and $ \\text{CSI}\\_l = \\frac{1}{|\\mathcal{C}\\_l|}\\sum\\_{ m } S\\_m$. Then, the residue index (RI) and the similarity index (SI) can be defined $ \\text{RI} = \\frac{1}{L}\\text{CRI}\\_l$ and $ \\text{SI} = \\frac{1}{L}\\text{CSI}\\_l $, respectively.\nUsing the RI and SI, the residue similarity disparity can be computed by taking RSD = RI − SI, and the residue-similarity index (RSI) can be computed as RSI = 1 − ∣RI − SI∣.\nshows the RS plots of PL-regularized NMF methods for GSE67835 data. The columns from left to right correspond to *k*-rTNMF, rTNMF, *k*-TNMF, and TNMF, while the rows correspond to the cell types. The x-axis and y-axis represent the S-score and R-score for each sample, respectively. The samples are colored according to their predicted cell types. Predictions were obtained using k-means clustering, and the Hungarian algorithm was employed to find the optimal mapping from the cluster labels to the true cell types.\nWe can see that TNMF fails to identify OP cells, whereas *k*-rTNMF, rTNMF, and *k*-TNMF are able to identify OPC cells. Notably, the S-score is quite low, indicating that the OPC did not form a cluster for TNMF. For fetal quiescent and replicating cells, *k*-rTNMF correctly identifies these two types, and the few misclassified samples are located on the boundaries. RTNMF is able to correctly identify fetal replicating cells but could not distinguish fetal quiescent cells from fetal replicating cells. The S-score is low for neurons in both rTNMF and TNMF, which shows a direct correlation with the number of misclassified cells.\nConclusion\n==========\nPersistent Laplacian-regularized NMF is a dimensionality reduction technique that incorporates multiscale topological interactions between the cells. Traditional graph Laplacian-based regularization only represents a single scale and cannot capture the multiscale features of the data. We have also shown that the *k*-NN induced persistent Laplacian outperforms other NMF methods and is comparable to the cutoff-based persistent Laplacian-regularized NMF methods. However, PL methods do come with their downside. In particular, the weights for each filtration must be determined prior to the reduction. If there are *T* filtrations, then the hyperparameter space is (*T* + 1)*T*. However, *k*-NN induced PL reduces the number of parameters to 2*T*. In addition, we have shown that we can achieve a significant improvement even if we limit the hyperparameter space to 2*T*. We would like to further explore possible parameter-free versions of topological NMF. Additionally, NMF methods are not globally convex, but we have shown that with NNDSVDA initialization, our methods perform the best. One possible extension to the proposed methods is to incorporate higher-order persistent Laplacians in the regularization framework, which will reveal higher-order interactions. In addition, we would like to expand the ideas to tensor decomposition, such as Canonical Polyadic Decomposition (CPD) and Tucker decomposition, multimodal omics data, and spatial transcriptomics data.\nData availability and code\n==========================\nThe data and model used to produce these results can be obtained at <https://github.com/hozumiyu/TopologicalNMF-scRNAseq>.\nAcknowledgment\n==============\nThis work was supported in part by NIH grants R01GM126189, R01AI164266, and R35GM148196, National Science Foundation grants DMS2052983, DMS-1761320, and IIS-1900473, NASA grant 80NSSC21M0023, Michigan State University Research Foundation, and Bristol-Myers Squibb 65109.\n10\nAaron TL Lun, Davis J McCarthy, and John C Marioni. A step-by-step workflow for low-level analysis of single-cell rna-seq data with bioconductor., 2016.\nByungjin Hwang, Ji Hyun Lee, and Duhee Bang. 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Email: weig@msu.edu[↩](#fnref1)\n"},"data_id":{"kind":"string","value":"arxiv_0183869"}}},{"rowIdx":45997,"cells":{"content":{"kind":"string","value":"Early-type Eclipsing Binaries with Intermediate Orbital Periods\n===============================================================\nWe analyze 221 eclipsing binaries (EBs) in the Large Magellanic Cloud with B-type main-sequence (MS) primaries (*M*1 ≈ 4-14 ) and orbital periods *P* = 20-50 days that were photometrically monitored by the Optical Gravitational Lensing Experiment. We utilize our three-stage automated pipeline to (1) classify all 221 EBs, (2) fit physical models to the light curves of 130 detached well-defined EBs from which unique parameters can be determined, and (3) recover the intrinsic binary statistics by correcting for selection effects. We uncover two statistically significant trends with age. First, younger EBs tend to reside in dustier environments with larger photometric extinctions, an empirical relation that can be implemented when modeling stellar populations. Second, younger EBs generally have large eccentricities. This demonstrates that massive binaries at moderate orbital periods are born with a Maxwellian “thermal” orbital velocity distribution, which indicates they formed via dynamical interactions. In addition, the age-eccentricity anticorrelation provides a direct constraint for tidal evolution in highly eccentric binaries containing hot MS stars with radiative envelopes. The intrinsic fraction of B-type MS stars with stellar companions *q* = *M*2/*M*1 > 0.2 and orbital periods *P*=20-50 days is (7 ± 2)%. We find early-type binaries at *P* = 20-50 days are weighted significantly toward small mass ratios *q* ≈ 0.2-0.3, which is different than the results from previous observations of closer binaries with *P* < 20 days. This indicates that early-type binaries at slightly wider orbital separations have experienced substantially less competitive accretion and coevolution during their formation in the circumbinary disk.\nIntroduction\n============\nIt has long been understood that the main-sequence (MS) binary star fraction increases with primary mass. Indeed, most massive stars with *M*1 > 10 will interact with a stellar companion before they explode as core-collapse supernovae. Throughout the decades, there have been significant advances in the detection of close and wide companions to massive stars. However, the intrinsic properties of binary companions to early-type primaries, e.g. their eccentricity and mass-ratio distributions, remain elusive at intermediate orbital periods. The major goal of this work is to help fill this particular portion of the parameter space.\nEclipsing binaries (EBs) offer a key to the accurate measurement of the binary properties of early-type stars. Large photometric surveys, such as the third phase of the Optical Gravitational Lensing Experiment (OGLE-III), have discovered tens of thousands of EBs. These populations of EBs are orders of magnitude larger than previous binary samples. Despite the geometrical selection effects, we can still achieve large sample statistics to reliably infer the intrinsic binary fraction and properties at intermediate orbital periods. We emphasize that EBs can probe a unique portion of the binary parameter space unavailable to other observational techniques.\nIn, we incorporated OGLE catalogs of EBs in the Large and Small Magellanic Clouds (LMC and SMC, respectively) as well as Hipparcos observations of EBs in the Milky Way. We compared the close binary properties (*P* < 20 days) of early-B MS primaries in the three different galaxies. The Milky Way and SMC EB samples are too small to warrant an analysis of period-dependent binary properties. The OGLE-III LMC EB catalog, on the other hand, contains ≈ 5-40 times more systems, is relatively complete toward shallow eclipse depths, and includes the full I-band and V-band light curves.\nIn, we developed a three-stage automated pipeline to analyze EBs with short orbital periods in the OGLE-III LMC database. This pipeline (1) classifies EBs according to their light curve characteristics, (2) measures the intrinsic physical properties of detached EBs, e.g. ages and component masses, based on the observed radii and temperatures, and (3) recovers the intrinsic binary statistics by correcting for selection effects.\nIn the present study, we utilize EBs in the OGLE-III LMC database to measure the binary fraction, mass-ratio distribution, and eccentricity distribution of B-type MS stars with intermediate orbital periods *P* = 20-50 days. We organize the rest of this paper as follows. In §2, we define our selection criteria for identifying EBs with B-type MS primaries, intermediate orbital periods, and well-defined eclipse parameters. We next describe an automated procedure we developed to fit detailed physical models to the observed EB light curves, and we present our results for the physical properties of the individual EBs (§3). In §4, we explain the observed trends in the measured EB parameters, paying special attention to the empirical age-extinction and age-eccentricity anticorrelations. We then perform Monte Carlo simulations to quantify selection effects, and present our results for the corrected binary statistics (§5). We summarize our main results and conclusions in §6.\nEB Selection and Classification (Stage I)\n=========================================\nIn Paper II, we developed a three-stage automated pipeline to fully analyze short-period EBs in the OGLE-III LMC database. In the present study, we adapt our routine to identify intermediate-period EBs with well-defined light curves (Stage I - this section), measure their physical properties (Stage II - §3), and correct for selection effects (Stage III - §5). EBs with intermediate orbital periods exhibit two major differences that must be considered. First, the eclipse widths Θ1 and Θ2, which are expressed as a fraction of the orbital period *P*, become narrower with increasing orbital separation. Given the average number $\\langle {\\cal N}\\_{I} \\rangle$ ≈ 470 of I-band measurements in the OGLE-III LMC survey, the light curves are not sufficiently sampled if either of the eclipse widths Θ < $\\langle {\\cal N}\\_{I} \\rangle^{-1}$ ≈ 0.0021 are too narrow. EBs with small MS components and long orbital periods *P* $\\gtrsim$ 50 days have narrow eclipses Θ $\\lesssim$ 0.002, and are therefore not Nyquist sampled. This subsampling leads to detection incompleteness, issues with aliasing, and the inability to fully characterize their intrinsic physical properties. Hence, it is the finite cadence of the OGLE-III observations, *not* geometrical selection effects, that limits our present study of EBs to *P* = 20-50 days (see also ).\nSecond, the majority of early-type EBs at *P* > 20 days are in eccentric orbits. We must therefore adapt our physical models to simultaneously fit the eccentricity *e* and argument of periastron *ω* (§3). In addition, it is possible for an eccentric binary to have a certain combination of eccentricity, periastron angle, and inclination that is sufficiently offset from edge-on (e.g., *i* $\\lesssim$ 86$^{\\rm o}$) so that there is only one eclipse per orbit. Indeed, there are many EBs with single eclipses in the OGLE-III LMC database (see below). Unfortunately, we cannot measure the physical properties of these systems. We therefore remove single-eclipse EBs from our well-defined sample, and we account for their removal when we correct for selection effects (see §5). In the following, we review our methods from Paper II, where we pay special attention to the nuances of EBs with intermediate orbital periods.\nIn this study, we select the ${\\cal N}\\_{\\rm B}$ ≈ 96,000 systems in the OGLE-III LMC catalog with mean magnitudes 16.0 < ⟨*I*⟩ < 17.6 and observed colors − 0.25 < ⟨*V* − *I*⟩ < 0.20. Given the distance modulus *μ* = 18.5 to the LMC and typical dust reddenings *E*(*V* − *I*) ≈ 0.1-0.3 mag toward hot young stars in the LMC, these stars have luminosities and surface temperatures that correspond to B-type MS primaries. From this sample, we analyze the 221 systems that were identified as EBs with orbital periods *P* = 20-50 days. In Table 1, we list the OGLE-III LMC EB identification numbers, observed colors ⟨*V* − *I*⟩, and numbers of I-band measurements ${\\cal N}\\_I$ for each of these 221 EBs.\nAs in Paper II, we measure the intrinsic rms scatter in the *I*-band light curve outside of eclipses for each EB. We then calculate the correction factor *f**σ*, *I* ≥ 1.0, i.e. the ratio between the actual rms scatter and photometric uncertainties reported in the catalog. For each *I*-band measurement in an EB light curve, we multiply the listed photometric uncertainties by the correction factor *f**σ*, *I* to determine the corrected uncertainties.\nWe classify EBs based on an analytic light curve model of two Gaussians with eight total free parameters. The orbital phase 0 ≤ *ϕ* < 1 is determined by the time of observation and two model parameters: the orbital period *P* (in days) and epoch of primary eclipse minimum $t\\_{\\rm o}$ (Julian date − 2450000). The six remaining analytic model parameters are the average I-band magnitude outside of eclipses ⟨*I*⟩, primary and secondary eclipse depths Δ*I*1 and Δ*I*2, primary and secondary eclipse widths Θ1 and Θ2, and the phase of secondary eclipse Φ2. The analytic model of Gaussians is:\n \n$$\\begin{aligned}\nI\\_{\\rm G}(\\phi) = \\langle I \\rangle &+\\Delta I\\_1 \\Big[{\\rm exp}\\Big(\\frac{-\\phi^2}{2\\Theta\\_1^2}\\Big)+\n \\rm{exp}\\Big(\\frac{-(\\phi-1)^2}{2\\Theta\\_1^2}\\Big)\\Big]\\nonumber \\\\\n &+ \\Delta I\\_2\\, \\rm{exp} \\Big(\\frac{-(\\phi-\\Phi\\_2)^2}{2\\Theta\\_2^2}\\Big)\\end{aligned}$$ \nWe fit this analytic model to each EB I-band light curve. Specifically, we utilize an automated Levenberg-Marquardt algorithm to minimize the $\\chi^2\\_{\\rm G}$ statistic. The MPFIT routine provides robust best-fit solutions and measurement uncertainties for the eight analytic model parameters. Some of the photometric measurements are clear outliers, so we clip up to ${\\cal N}\\_{\\rm c}$ ≤ 2 data points per light curve that exceed 4*σ* from the model. This results in *ν* = ${\\cal N}\\_I$ − ${\\cal N}\\_{\\rm c}$ − 8 degrees of freedom. For each EB, we report in Table 1 the eight fitted analytic model parameters and the fit statistics. Excluding the few EBs that exhibit variability or are evolved Roche-lobe filling systems (see below), the goodness-of-fit statistics $\\chi^2\\_{\\rm G}$/*ν* = 0.87-1.16 indicate the analytic models can adequately describe the EB light curves.\nWe can measure the physical properties of EBs based solely on the observed photometric light curves (see §3) only if: (1) the binary components are detached from their Roche lobes, (2) the light curves have two well-defined eclipses, and (3) there is no superimposed variability. To be considered well-defined, we require that the 1*σ* uncertainties in the measured eclipse depths Δ*I*1 and Δ*I*2 and eclipse widths Θ1 and Θ2 are < 20% their respective values. These criteria are not satisfied for 91 of the 221 EBs due to a variety of reasons, which we discuss below:\n[t!]\n*(A) No Secondary Eclipse*. For 16 of our EBs, there is no evidence for a secondary eclipse. These EBs may have secondary eclipses that are too shallow and below the sensitivity of the OGLE-III LMC survey, or have eclipse widths that are too narrow and therefore not detected given the cadence of the observations. Most likely, the EBs have a certain combination of *e*, *ω*, and *i* as discussed above so that there is only one eclipse per orbit. We list these 16 systems in Category 1 of Table 1, and we show an example in panel A of Fig. 1.\n*(B) Uncertain*. For 32 EBs, both eclipses are observed but one or more of their measured properties are uncertain by more than 20%. This is because one of the eclipses is too shallow and/or too narrow. We group these 32 systems in Category 2 of Table 1. In panel B of Fig. 1, we display an example of a long-period *P* ≈ 45 day EB with a secondary eclipse at Φ2 ≈ 0.55 that is too narrow to be accurately measured.\n*(C) Roche-lobe filling*. Three EBs have wide eclipses such that one or both components of the binary must be filling their Roche lobes. We list these three systems in Category 3 of Table 1, and we show an example in panel C of Fig. 1.\n*(D) Ambiguous Periods*. The orbital periods of 23 of our EBs are ambiguous. These 23 EBs can either have twin components *q* ≈ 1.0 in nearly circular orbits *e* ≈ 0.0 or have half the listed orbital periods and exhibit only one eclipse per very eccentric orbit. Using the orbital periods listed in the OGLE-III LMC catalog, these EBs have primary and secondary eclipses that are nearly identical and separated by almost precisely 50% in orbital phase. Quantitatively, we identify these systems to have values of and uncertainties in eclipse depths, widths, and phases that satisfy:\n \n$$\\begin{aligned}\n |\\Delta I\\_1 - \\Delta I\\_2| & \\le 3\\big[(\\sigma\\_{\\Delta I\\_1})^2 + (\\sigma\\_{\\Delta I\\_2})^2\\big]^{\\nicefrac{1}{2}} \\\\\n |\\Theta\\_1 - \\Theta\\_2| & \\le 3\\big[(\\sigma\\_{\\Theta\\_1})^2 + (\\sigma\\_{\\Theta\\_2})^2\\big]^{\\nicefrac{1}{2}} \\\\\n |\\Phi\\_2 - \\nicefrac{1}{2}| & \\le 3\\sigma\\_{\\Phi\\_2}\\end{aligned}$$ \nGiven the sensitivity of the data, the observed properties imply the 23 systems have large mass ratios *q* $\\gtrsim$ 0.9 with extremely small eccentricities *e* $\\lesssim$ 0.05 (see §3). However, none of the EBs in our sample have eclipse depths that satisfy Eqn. 2a (*q* $\\gtrsim$ 0.9) with secondary eclipse phases 3*σ*Φ2 < $|\\Phi\\_2 - \\nicefrac{1}{2}|$ ≤ 10*σ*Φ2 (*e* ≈ 0.05-0.10). Similarly, there is only one EB that satisfies Eqn. 2c (*e* $\\lesssim$ 0.05) with primary and secondary eclipse depths that are discrepant at the (3-10)*σ* level (*q* ≈ 0.8-0.9). Hence, there are no twin systems in slightly eccentric orbits, and there is only one moderate-mass companion in a nearly circular orbit. The prevalence of 23 twin systems in nearly circular orbits at these moderate orbital periods is therefore highly unlikely. If there is indeed an excess of twins in circular orbits relative to twins in eccentric orbits, our study does not include them. We expect only a few of the 23 EBs that appear to be twins in circular orbits to have the listed orbital periods. The majority of these EBs more likely have orbital periods that are half their listed values, and would therefore exhibit only one eclipse per orbit similar to the systems discussed in (A) above. In panel D of Fig. 1, we show one example where we fold the photometric data with the listed orbital period (in black) and the more plausible scenario that the binary has half the catalog orbital period (in red). We list these 23 EBs in Category 4 of Table 1. We further motivate the removal of these 23 systems in §4 when we show the intrinsic frequency of *q* > 0.6 companions with *e* < 0.2 is relatively sparse.\n*(E) Superimposed Variability*. Fifteen of the EBs exhibit superimposed variability. Three of these systems are intrinsic variables, two of which (ID-7651 and ID-22929) were already listed as such in the OGLE-III LMC EB catalog. The intrinsic variability is readily apparent in the unfolded light curves. Moreover, the measured intrinsic scatter outside of eclipses is substantially higher than the photometric errors, e.g. *f**σ*, *I* ≈ 2.8 for ID-3414. We note that a few additional systems with *f**σ*, *I* ≈ 1.5-1.9 may exhibit low-amplitude variations *δ**I* < 0.01 mag, but these variations are sufficiently small so as to not to interfere with the light curve modeling. We list the three systems that exhibit definitive intrinsic variability in Category 5 of Table 1. The other 12 EBs exhibit variability in the eclipses themselves, only one of which (ID-17017) was identified as such in the OGLE-III LMC catalog. For these systems, it is possible that more than two bad data points occur near the eclipse. More likely, these 12 EBs display changes in the eclipse depths and/or eclipse phases during the seven years of observations. Apsidal motion due to tidal and relativistic effects are negligible on timescales *d**t* ≈ 7 yrs at these wide orbital separations. Such evolution in the eclipse parameters are most likely caused by orbital motion with a tertiary component. We group these 12 EBs in Category 6 of Table 1, and we display an example in panel E of Fig. 1.\n*(F) Peculiar*. Finally, two EBs have peculiar light curves. ID-343 exhibits a pronounced peak in the folded light curve at *ϕ* = 0.8 between eclipses. This peak may be caused by ellipsoidal modulation in an extremely eccentric orbit. ID-4458, which is shown in panel F of Fig. 1, displays a sinusoidal variation between two eclipses of comparable depth. ID-4458 may contain a hot spot and/or disk, and is similar to the green systems in the top left corner of Fig. 3 in Paper II. We list these two systems in Category 7 of Table 1.\nAfter removing these 91 systems, our well-defined sample contains 130 EBs. We list these 130 systems in Category 8 of Table 1. When necessary, we switch the primary and secondary eclipses to ensure Δ*I*1 > Δ*I*2 in our well-defined sample. If the epoch of primary eclipse minimum $t\\_{\\rm o}$ substantially changed from the catalog value in order to satisfy this criterion, we place an asterisk next to our value of $t\\_{\\rm o}$ in Table 1.\nThe 130 EBs in our well-defined sample have uncertainties in eclipse depths Δ*I*1 and Δ*I*2 and eclipse widths Θ1 and Θ2 that are $\\lesssim$20% their respective values. The uncertainties in the Gaussian analytic fit parameters have been used only to determine which EBs have detectable and measurable eclipse properties. These uncertainties propagate into our Monte Carlo simulations when we calculate the fraction of binaries that produce detectable eclipses (see §5). The uncertainties in the Gaussian analytic fit parameters are *not* utilized to calculate the uncertainties in the physical properties of the EBs. Instead, we implement detailed light curve models to measure the values of and uncertainties in the physical model properties, which we now discuss.\nPhysical Models (Stage II)\n==========================\nAlgorithm\n---------\nThe physical properties of EBs are routinely measured by fitting detailed models to the observational data. Normally, spectroscopic radial velocity observations are required to dynamically measure the component masses *M*1 and *M*2. In the modern era of wide-field photometric surveys, the discovery of EBs is quickly outpacing our ability to obtain follow-up spectra. For large EB samples, the physical properties must be inferred based solely on the photometric light curves. MS constraints and isochrone fitting have helped ascertain EB properties from the photometric data. In general, however, these methods lead to large systematic uncertainties and/or solutions that are highly degenerate.\nIn Paper II, we developed a technique that uniquely and accurately characterizes the intrinsic physical properties of *detached EBs with known distances* using only the photometric data. The distances to EBs in the LMC are known. In fact, we have already utilized the observed magnitudes ⟨*I*⟩ and colors ⟨*V* − *I*⟩ to select EBs with B-type MS primaries. For detached EBs with MS primaries, both components are effectively evolving along their respective single-star evolutionary sequences. The photospheric properties of the stellar components, e.g. effective temperatures *T*1 and *T*2, radii *R*1 and *R*2, and luminosities *L*1 and *L*2, therefore depend entirely on the age *τ* and component masses *M*1 and *M*2. The systematic uncertainties in the evolutionary tracks are relatively small, e.g. ≈ 15% uncertainties in the masses and ≈ 30% uncertainties in the ages (see Paper II and §3.3 for further justification and a full assessment of the uncertainties). We can therefore measure the component masses *M*1 and *M*2 and ages *τ* of detached EBs with known distances based solely on the observed light curve features (Fig. 2).\nIn our physical models, detached EBs with B-type MS primaries can be uniquely described by nine independent properties. These nine physical model parameters are the orbital period *P*, epoch of primary eclipse minimum $t\\_{\\rm o}$, primary mass *M*1, secondary mass *M*2, age *τ*, inclination *i*, eccentricity *e*, argument of periastron *ω*, and I-band dust extinction *A**I*. Given the age *τ* and component masses *M*1 and *M*2 of the binary, we interpolate the radii *R*1 and *R*2, surface gravities *g*1 and *g*2, effective temperatures *T*1 and *T*2, and luminosities *L*1 and *L*2 from pre-MS and MS stellar evolutionary tracks with metallicity *Z* = 0.008. We then use the LMC distance modulus *μ* = 18.5, dust reddening law *E*(*V* − *I*) = 0.7*A**I*, and temperature-dependent color indices and bolometric corrections to transform the intrinsic properties of the binary into observed magnitudes and colors. Our physical model parameter space (*M*1, *M*2, *τ*, etc.) of EBs with detached configurations, pre-MS/MS evolutionary constraints, and known distances is quite different than the typical EB parameter space (*T*2/*T*1, (*R*1+*R*2)/*a*, etc.) where the distances and evolutionary status of the components are unknown.\nUsing the physical properties of a binary, e.g., *P*, *M*1, *M*2, *R*1, *R*2, *T*1, *T*2, *e*, *ω*, etc., we synthesize photometric light curves with the EB modeling package Nightfall[1](#fn1). We use the same Nightfall model options adopted in Paper II, e.g. a square-root limb darkening law, default gravity darkening coefficients, model atmospheres, etc., except for three notable distinctions. First, we do not assume circular orbits for our EBs at longer orbital periods, but instead solve for both the eccentricity *e* and periastron angle *ω*. Second, we set the albedo of the secondary to *A*2 = 0.7 and implement one iteration of reflection effects. Considering reflection effects are minuscule for our wider EBs in this study, different treatments of reflection have negligible effects on the synthesized light curves. Finally, we simulate an EB light curve at 1,000 uniformly-spaced discrete orbital phases to ensure narrow eclipses are sufficiently sampled.\nMost of our EBs with intermediate orbital periods have eccentric orbits and narrow eclipses. Nightfall and all other EB software packages that account for tidal effects are computationally expensive for eccentric binaries. This is because the three-dimensional photospheric surfaces of the stars need to be recalculated at each of the 1,000 discretely sampled orbital phases. We therefore adapt our algorithm from Paper II to guarantee fast, automated convergence. Namely, we choose initial values for our nine physical model properties that are sufficiently close to the true values to ensure *χ*2 minimization converges quickly to the global solution. The major goal of our algorithm is to synthesize light curves with Nightfall as few times as possible. Our routine can easily be adapted for any population of detached EBs with known distances, and can be used in combination with any EB light curve modeling software.\nWe decompose our algorithm into three steps.[2](#fn2) In Step 1, we select initial values for our nine physical model properties based on the observed light curve features quantified in §2. In Step 2, we make small adjustments in the physical model properties until the analytic model parameters of the synthesized light curve matches those of the observed light curve. In Step 3, we utilize a Levenberg-Marquardt technique, as done in Paper II, to minimize the *χ*2 statistic between the observed and simulated light curves. We elaborate on these three steps below. To help illustrate this procedure, we display in Fig. 3 the light curve of an example EB, ID-2142, and the solutions at the end of each of these three steps.\n**Step 1.** We use the eight analytic model parameters (*P*, $t\\_{\\rm o}$, ⟨*I*⟩, Δ*I*1, Θ1, Φ2, Δ*I*2, Θ2) and observed color ⟨*V* − *I*⟩ from Table 1 to estimate initial solutions for the nine physical model properties. In Fig. 2, we show how the nine observed light curve features can be used to approximate the nine physical properties of the binary. We select the physical parameters *P* and $t\\_{\\rm o}$ to match the analytic model values. We then estimate *e* and *ω* according to the observed phase of the secondary eclipse and the difference in eclipse widths :\n \n$$\\begin{aligned}\n e\\,{\\rm cos}\\,\\omega & \\approx \\frac{\\pi}{2}(\\Phi\\_2 - \\nicefrac{1}{2}) \\\\\n e\\,{\\rm sin}\\,\\omega & \\approx \\frac{\\Theta\\_2 - \\Theta\\_1}{\\Theta\\_2+ \\Theta\\_1} \\end{aligned}$$ \nIn this study, *ω* = 90$^{\\rm o}$ if periastron coincides with the observed primary eclipse. For our example ID-2142, Θ1 ≈ Θ2 and Φ2 ≈ 0.16, indicating *ω* ≈ 180$^{\\rm o}$ and *e* ≈ 0.5-0.6.\nThe intrinsic colors of B-type MS stars span a narrow interval − 0.3 $\\lesssim$ $\\langle V - I \\rangle\\_{\\rm o}$ $\\lesssim$ − 0.1. We therefore initially assume the intrinsic color of an EB to be:\n \n$$\\langle V - I \\rangle\\_{\\rm o} \\approx -0.22 + 0.08(\\langle I \\rangle -17)$$ \nwhere we have accounted for the fact that more luminous B-type MS stars tend to be more massive, hotter, and bluer. The dust extinction *A**I* is simply estimated from the observed color ⟨*V* − *I*⟩ and our adopted dust reddening law *E*(*V* − *I*) = ⟨*V* − *I*⟩ − $\\langle V - I \\rangle\\_{\\rm o}$ = 0.7*A**I* (see Fig. 2).\nWe then use the eclipse depths Δ*I*1 and Δ*I*2 to approximate the mass ratio *q* = *M*2/*M*1. For a MS+MS binary in a circular orbit, the ratio of eclipse depths Δ*I*2/Δ*I*1 provides an accurate indicator of the luminosity contrast *L*2/*L*1 (Fig. 2). This luminosity contrast can then be used to infer the mass ratio *q* according to a MS mass-luminosity relation (see Fig. 3 in Paper I). For eccentric orbits, however, the eclipse depth ratio can be modified because the projected distances during primary and secondary eclipses can be different. Nonetheless, deeper eclipses still suggest larger mass ratios. For example, Δ*I*2 > 0.4 mag requires *q* > 0.7, regardless of the eccentricity or whether the secondary is a MS or pre-MS star. We use a linear combination of these methods to estimate the mass ratio:\n \n$$q \\approx 0.6\\Delta I\\_1 + 0.5\\Delta I\\_2 + 0.5 \\frac{\\Delta I\\_2}{\\Delta I\\_1}$$ \nwhere the eclipse depths are in magnitudes.\nWe next use the observed mean magnitude ⟨*I*⟩ and sum of eclipse widths Θ1 + Θ2 to simultaneously measure the primary mass *M*1 and age *τ*. Assuming non-grazing eclipses and standard limb darkening coefficients, the sum of eclipse widths Θ1 + Θ2 directly provides the relative sum of the radii (*R*1+*R*2)/*a*. Our EBs occupy a narrow range of magnitudes 16.0 < ⟨*I*⟩ < 17.6 and therefore span a small interval of total masses *M* = *M*1 + *M*2. The orbital separation *a* ∝ $P^{\\nicefrac{2}{3}} M^{\\nicefrac{1}{3}}$ therefore derives mainly from the known period *P*. We can now use Θ1 + Θ2 and *P* to determine *R*1+*R*2. For EBs with B-type MS primaries, we find the following approximation:\n \n$$R\\_1 + R\\_2 \\approx 7\\,{\\rm R}\\_{\\odot} \\frac{\\Theta\\_1+\\Theta\\_2}{0.01}\\Big(\\frac{P}{30\\,{\\rm days}}\\Big)^{\\nicefrac{2}{3}}$$ \nGiven our estimates for *q* and *A**I* above, we interpolate the stellar evolutionary tracks to determine the primary mass *M*1 and age *τ* that reproduce the sum of the radii *R*1 + *R*2 according to Eqn. 6 and the observed combined magnitude ⟨*I*⟩. Although ⟨*I*⟩ and Θ1 + Θ2 both depend on *M*1 and *τ*, they are sufficiently non-degenerate so that we can calculate a unique solution. Namely, the primary mass *M*1 largely dictates the luminosity and therefore the observed magnitude ⟨*I*⟩, while the age *τ* primarily determines the radii *R*1 + *R*2 and therefore the observed eclipse widths Θ1 + Θ2 (see Fig. 2).\nFinally, we select an inclination *i* that approximately reproduces the observed primary eclipse depth Δ*I*1 (Fig. 2). From our estimates of *M*1, *M*2 = *q**M*1, and *τ*, we interpolate the radii *R*1 and *R*2 and effective temperatures *T*1 and *T*2 from stellar evolutionary tracks. In this step only, we ignore limb darkening and colors of the two stars, and instead assume the stars are uniformly illuminated grey disks (see Paper I). We assume the surface brightnesses of the disks are proportional to the stellar temperatures, i.e. the Rayleigh-Jeans law, because we are observing at relatively long wavelengths in the near-infrared I-band. Using these approximations, we calculate the eclipsed area $A\\_{\\rm o}$ of the primary at the time of primary eclipse $t\\_{\\rm o}$ based on the observed primary eclipse depth:\n \n$$\\Delta I\\_1 \\approx -2.5\\,{\\rm log}\\,\\Big(1 - \\frac{A\\_{\\rm o} T\\_1}{\\pi (R\\_1^2 T\\_1 + R\\_2^2 T\\_2)}\\Big)$$ \nGiven the eclipsed area $A\\_{\\rm o}$ and stellar radii *R*1 and *R*2, we then determine the projected distance $d\\_{\\rm o}$ between the two stars at $t\\_{\\rm o}$. The actual physical separation at primary eclipse is already known via :\n \n$$r\\_{\\rm o} = a\\,\\frac{1-e^2}{1+e\\,{\\rm sin}\\,\\omega}$$ \nwhere *a* derives from our estimates of *M*1, *M*2, and *P* according to Kepler’s third law, and *e* and *ω* are approximated from Eq. 3. Hence, the inclination simply derives from cos*i* = $d\\_{\\rm o}$/$r\\_{\\rm o}$. We limit our initial approximation of the inclination to the interval *i* = 86.5$^{\\rm o}$-89.5$^{\\rm o}$.\nWe now have initial estimates for the nine physical model properties. We emphasize that Eqns. 3-7 are simple approximations, and that the true values of *e*, *ω*, *A**I*, *M*1, *M*2, *τ*, and *i* may substantially differ from the initial values estimated here. We simply use these estimates as initial parameters in our fitting routine in order to minimize the number of iterations and accelarate convergence toward the final solution (see below and §3.2).\nIn the top panel of Fig. 3, we compare the I-band and V-band light curves of ID-2142 to a simulated Nightfall model using the values of the nine physical model properties at the end of Step 1. The model matches key features of the observed light curve, but there are three noticeable differences. First, the simulated phase of the secondary eclipse does not match the observations; recall that Eqn. 3 is an approximation. Second, the simulated color is bluer than the observed ⟨*V* − *I*⟩, indicating we underestimated the dust reddening *A**I* in our initial step. Finally, the simulated eclipses are slightly deeper than the observed because the more accurate Nightfall model accounts for limb darkening and color effects. This suggests the actual inclination is smaller and/or the mass ratio is slightly different. We correct for these visible discrepancies in the following step.\n**Step 2.** Using the nine physical properties from Step 1, we synthesize an I-band light curve with Nightfall. We then fit the simple analytic model of Gaussians (Eqn. 1) as done in §2 to the simulated Nightfall light curve. In this manner, we measure the analytic parameters of the Nightfall model, e.g. $\\Delta I\\_{1, {\\rm mod}}$, $\\Theta\\_{1, {\\rm mod}}$, $\\Phi\\_{2, {\\rm mod}}$, etc.\nWe adjust the properties in our physical models according to the differences between the simulated and observed analytic model parameters. The adjustments are motivated as follows. If the modeled eclipse widths $\\Theta\\_{1, {\\rm mod}}$ + $\\Theta\\_{2, {\\rm mod}}$ are wider than the observed Θ1 + Θ2, we select a slightly younger age *τ* (and vice versa). We increase the dust extinction *A**I* if the simulated color $\\langle V - I \\rangle\\_{\\rm mod}$ is too blue. If the modeled primary eclipse $\\Delta I\\_{1,{\\rm mod}}$ > Δ*I*1 is too deep while the modeled secondary eclipse $\\Delta I\\_{2,{\\rm mod}}$ ≤ Δ*I*2 matches observations or is too shallow, we increase the mass ratio *q* and decrease the inclination *i*. However, if both simulated eclipses are too deep (or both too shallow), we only decrease (increase) the inclination *i*. Finally, we adjust *e* according to the position of and differences in the secondary eclipse phases Φ2 and $\\Phi\\_{2, {\\rm mod}}$. In this step, we fix *P*, $t\\_{\\rm o}$, and *ω* to the values determined in Step 1. Finally, we interpolate *M*1 from the stellar evolutionary tracks based on the observed mean magnitude ⟨*I*⟩ and the revised values for *τ*, *q*, and *A**I*.\nWhen adjusting our physical model properties, we choose step sizes that scale with the differences between the observed and simulated analytic model parameters. After making these adjustments, we synthesize another I-band light curve with Nightfall. We iterate this step until all the analytic model parameters of the simulated and observed light curves match within a small tolerance level. In the middle panel of Fig. 3, we show our solution for ID-2142 at the end of Step 2 after five iterations. We therefore required only six Nightfall light curve simulations during this middle step.\n**Step 3.** This final step is essentially the procedure outlined in Paper II. We calculate the photometric correction factors *f**σ*, *I* and *f**σ*, *V* in both bands. Starting with initial model properties determined at the end of Step 2, we utilize a Levenberg-Marquardt technique to minimize the *χ*2 statistic between the simulated and observed light curves. The Levenberg-Marquardt MPFIT algorithm operates by independently varying each of the nine physical model properties from the previous solution. The routine then measures the resulting deviations between the data and models, and then calculates a new solution. This step therefore requires ten Nightfall simulations per iteration. As in Paper II, we simultaneously fit the I-band and V-band light curves. We clip up to ${\\cal N}\\_{{\\rm c}, I}$ + ${\\cal N}\\_{{\\rm c}, V}$ ≤ 3 data points that exceed 4*σ* from the best-fit model. This results in *ν* = ${\\cal N}\\_I$ + ${\\cal N}\\_V$ − ${\\cal N}\\_{{\\rm c}, I}$ − ${\\cal N}\\_{{\\rm c}, V}$ − 9 degrees of freedom.\nIn the bottom panel of Fig. 3, we display our final solution for ID-2142 after four iterations of the Levenberg-Marquardt MPFIT routine. We therefore simulated light curves with Nightfall a total of 40 times in Step 3. The physical model properties changed only slightly during this final step. In fact, for ID-2142, the variations were all within the uncertainties of the physical model parameters. We emphasize that Steps 1 and 2 were crucial in guarenteeing rapid convergence toward the final solution in Step 3. Without them, this last step would have required many additional iterations or may have converged to a local minimum.\nWe utilize this automated procedure for all 130 detached EBs in our well-defined sample. We present our fitted model parameters, physical properties, and fit statistics for these systems in Table 2. For MS binaries in circular orbits, the deeper primary eclipse Δ*I*1 at time $t\\_{\\rm o}$ always corresponds to the smaller, cooler, less massive secondary passing in front of the larger, hotter, more massive primary. For eccentric orbits, however, the situation can be reversed depending on the combination of *e*, *ω*, and *i*. Indeed, for 18 EBs in our well-defined sample, we determined solutions such that the less massive component was eclipsed at time $t\\_{\\rm o}$. To avoid confusion in nomenclature, we list properties in Table 2 according to the primary “p” and secondary “s” eclipse features. Namely, $M\\_{\\rm p}$, $R\\_{\\rm p}$, and $T\\_{\\rm p}$ correspond to the component that was eclipsed at the epoch of primary eclipse $t\\_{\\rm o}$, and $M\\_{\\rm s}$, $R\\_{\\rm s}$, and $T\\_{\\rm s}$ correspond to the component that was eclipsed at the secondary eclipse phase Φ2. In the text, we refer to primary mass *M*1 = max{$M\\_{\\rm p}$,$M\\_{\\rm s}$}, secondary mass *M*2 = min{$M\\_{\\rm p}$,$M\\_{\\rm s}$}, mass ratio *q* = *M*2/*M*1, etc.\nWe measure primary masses *M*1 = 3.6-13.9, which nearly encompasses the full mass range of B-type MS stars. We determine mass ratios across the interval *q* = 0.20-1.00, which confirms the OGLE-III observations are sensitive to EBs with low-mass companions. Our measured dust extinctions cover *A**I* = 0.10-0.58 mag, which is consistent with the range of extinctions found in Paper II. Finally, we determine ages *τ* = 0.5-190 Myr that span more than two orders of magnitude. We further discuss the EB physical properties, and their interrelations, in §4.\nEleven of the 130 EBs have modest fit statistics *χ*2/*ν* = 1.10-1.14, i.e. probabilities to exceed *χ*2 of *p* ≈ 0.01-0.05 given *ν* ≈ 530 degrees of freedom. Seven of these EBs are extremely young with estimated ages *τ* $\\lesssim$ 0.8 Myr (IDs 5153, 7560, 10422, 13418, 16711, 22691, and 22764). The components in these EBs have small radii, as demonstrated by their narrow eclipses (Eqn. 6), and are therefore consistent with the zero-age MS. The systematic uncertainties in the stellar evolutionary tracks are larger at these young ages, especially considering some of the secondaries may still be pre-MS stars (see Paper II). Three of the 11 EBs with modest fit statistics have primaries at the tip of the MS (IDs 91, 20746, and 21518), as indicated by their wide eclipses. Again, the stellar evolutionary tracks are uncertain at the tip of the MS just prior to the rapid expansion toward the giant phase. The one last EB with a poor physical model fit (ID-17569) has *χ*2/*ν* = 1.11 and *p* ≈ 0.02. Considering our large sample of 130 EBs, we naturally expect 1-3 of these EBs with modest fit statistics. The remaining 119 EBs in our well-defined sample have good fit statistics 0.93 < *χ*2/*ν* < 1.09. This is testament that the nine independent physical model properties can adequately describe detached EBs with known distances and MS primaries.\nComparison between Initial Estimatesand Final Solutions\n-------------------------------------------------------\nIn the following, we compare the initial estimates for *e*, *q*, and *R*1+*R*2 in Step 1 according to Eqns. 3, 5, and 6, respectively, to the final solutions in Step 3 from fitting detailed Nightfall light curve models to the data. We can then address the systematic uncertainties in our initial estimates and further justify the mapping between the basic EB light curve parameters to the physical model properties.\nFor the 130 EBs in our well-defined sample, we compare the initial values of the eccentricities *e* determined from the secondary eclipse phases Φ2 and eclipse widths Θ1 and Θ2 (Eqn. 3) to the final Nightfall solutions (top panel of Fig. 4). The initial estimates agree quite well with the true final values. The rms scatter between the two is only *δ**e* = 0.03. This validates that Eqn. 3 is more than sufficient for starting purposes in our fittng routine. We note that the few systems that change by more than Δ*e* > 0.07 between Steps 1 and 3 have narrow, poorly sampled eclipses so that it is more difficult to precisely measure Θ1 and Θ2.\nSimilarly, in the bottom panel of Fig. 4, we compare the initial estimates of the mass ratios *q* determined from the eclipse depths Δ*I*1 and Δ*I*2 (Eqn. 5) to our final values obtained from Nightfall light curve fittings and *χ*2 minimizations. Although the population as a whole shows rough agreement between the solutions at the ends of Steps 1 and 3, individual systems can substantially deviate from the initial estimates. For example, an EB with an initial estimate of *q* ≈ 0.6 may actually have a mass ratio anywhere in the interval *q* = 0.3-1.0. The rms deviation between the initial and final solutions is *δ**q* = 0.12, or *δ**q*/*q* ≈ 20% the respective values. If we had randomly chosen mass ratios *q* in Step 1 while keeping the other initial estimates unchanged, the Nightfall light curve solutions in Step 3 would still converge to the same final values. We simply find that by adopting Eqn. 5 in Step 1 to provide initial estimates for *q*, the number of iterations in Steps 2 and 3 are dramatically reduced.\nFinally, we evaluate the discrepancies between the values of *R*1+*R*2 estimated from the sum of eclipse widths Θ1+Θ2 and orbital periods *P* according to Eqn. 6 to the final Nightfall solutions. We measure an rms deviation of *δ*(*R*1+*R*2) = 1.2 , or *δ*(*R*1+*R*2)/(*R*1+*R*2) ≈ 20% the respective values. The coefficient in Eqn. 6 should therefore be 7.0 ± 1.2 , valid only for EBs with B-type MS primaries. As with the mass ratios *q*, the approximations for *R*1+*R*2 based on the observed light curve parameters are imprecise but sufficiently accurate to provide initial conditions for our fitting routine.\nAs mentioned in §3.1, *R*1+*R*2 is primarily an indictor of age *τ* of an EB in our sample rather than the component masses *M*1 and/or *M*2. For example, at age *τ* = 5 Myr, the sum of the stellar radii must be contained on the interval *R*1+*R*2 ≈ 4.4-8.7 given any combination of *M*1 and *q* ≥ 0.2 that satisfies our magnitude limits 16.0 < ⟨*I*⟩ < 17.6 and − 0.25 < ⟨*V* − *I*⟩ < 0.20 and measured range of dust extinctions *A**I* ≈ 0.1-0.5 mag. Meanwhile, at age *τ* = 100 Myr, the sum of the radii are systematically larger and confined to the interval *R*1+*R*2 ≈ 5.3-12.3 given the same photometric requirements. Hence, EBs in our sample with *R*1+*R*2 < 5.3 must be relatively young while those with *R*1+*R*2 > 8.7 must be relatively old. We initially estimated *R*1+*R*2 in Step 1 from the observed sum of eclipse widths Θ1+Θ2 according to Eqn. 6. Although not as accurate as the final solutions, Eqn. 6 provides a model-independent measurement of *R*1+*R*2. The sum of radii *R*1+*R*2 estimated from Eqn. 6 is therefore a robust and model-independent indicator of age *τ*.\nIn Fig. 5, we display the eccentricities *e* measured in Step 1 from Eqn. 3 as a function of the approximate sum of stellar radii *R*1+*R*2 estimated in Step 1 from Eqn. 6. Both sets of parameters are model independent and based solely on the observed light curve features. According to a Spearman rank correlation test, we find that the approximate values of *e* and *R*1+*R*2 are anticorrelated (*ρ* = − 0.18) at a statistically significant level (*p* = 0.04). This suggests that EBs with larger components, which are systematically older, favor smaller eccentricities. We also compare the 19 EBs with approximate *R*1+*R*2 < 5.3 , which must be relatively young, to the 27 EBs with *R*1+*R*2 > 8.7 , which must be relatively old. According to a K-S test, we find these young and old populations of EBs have distributions of eccentricities that are discrepant with each other at the *p* = 0.02 significance level. The anticorrelation between *R*1+*R*2, which is an indicator of age *τ*, and *e* is therefore statistically significant, robust, and model independent. In §4, we further investigate this anticorrelation between *e* and *τ* based on the more accurate final solutions obtained from the Nightfall light curve models.\nUncertainties\n-------------\nWe now analyze the uncertainties in the final solutions of our Nightfall light curve models (see also Paper II). For each system, we utilize MPFIT at the end of Step 3 (§3.1) to calculate the measurement uncertainties. For all 130 well-defined EBs, the nine physical model parameters have unique solutions and finite measurement uncertainties. Some of the model parameters, however, have solutions that are correlated with each other. In addition, uncertainties in the dust reddening law, stellar evolutionary tracks, bolometric corrections, and Nightfall light curve models can lead to large systematic uncertainties in the physical model parameters. In the following, we fully investigate the measurement uncertainties, parameter correlations, and systematic uncertainties in the context of a specific example EB, ID-2142. We then determine the median total uncertainties of each model parameter (Eqns. 9-17) for the entire population of 130 well-defined EBs.\n[t!]\nFor our example EB, ID-2142, we explore the physical parameter space via a Markov chain Monte Carlo (MCMC) technique. Starting with our final solution at the end of Step 3, we implement a Metropolis-Hastings “random walk” MCMC algorithm to generate and select steps in our phase space of nine physical model parameters. At each proposed step, we synthesize a Nightfall light curve model given the proposed nine physical model parameters. The probability *p* ∝ *e*− Δ*χ*2/2 of accepting the proposed step is determined by evaluating the difference in the *χ*2 statistic between the proposed step and the current solution. Obviously, if Δ*χ*2 < 0, the proposed step is always taken. If the proposed step is rejected, the step length is effectively zero, i.e., the previous solution is counted again. We generate proposed steps according to a Gaussian distribution with a fixed standard deviation for each of the nine physical model parameters. We choose the standard deviation in the step sizes so that approximately one-third of the proposed steps are accepted. We simulate 32,000 proposed steps and light curves with Nightfall, which exceeds the total number of models generated in §3.1 used to fit solutions for all 130 well-defined EBs! It is therefore quite computationally expensive to calculate robust measurement uncertainties and correlations between model parameters for an individual EB with this MCMC algorithm. The distribution of the ≈ 12,000 accepted steps and ≈ 20,000 repeated solutions provide the nine-dimensional joint probability distribution for the physical models. For each of the nine physical model parameters, we marginalize across the other eight parameters to calculate the one-dimensional probability density function. We also compute the two-dimensional joint probability distributions for each of the 9*C*2 = 36 parameter combinations.\nIn Fig. 6, we display the one-dimensional probability distributions for the nine physical model parameters (diagonal panels) and the two-dimensional joint probability distributions for the 36 parameter combinations (off-diagonal panels). Although some of the parameters are mildly to significantly correlated with each other, the measurement uncertainties are finite for all nine physical model parameters. The MCMC technique confirms the uniqueness and non-degeneracy of the physical model solutions. Moreover, the measurement uncertainties we determined from the robust MCMC algorithm are consistent with the measurement uncertainties we evaluated with the MPFIT routine. We can therefore rely on the MPFIT measurement uncertainties we calculated for all 130 well-defined EBs.\nThe uncertainties in the orbital parameters *P* and $t\\_{\\rm o}$ are solely due to the measurement uncertainties and dictated by the sensitivity and cadence of the OGLE-III LMC observations. The solutions for *P* and $t\\_{\\rm o}$ are therefore independent of the other seven model parameters (note the fairly circular contours in the first and second columns of panels in Fig. 6). For ID-2142, we measure 1*σ* uncertainties of *δ**P* ≈ 0.0001 days and $\\delta t\\_{\\rm o}$ ≈ 0.004 days. We find the median 1*σ* uncertainties for the entire population of 130 well-defined EBs to be:\n \n$$\\langle \\delta P \\rangle \\approx 1.4\\times10^{-5} P \\approx 0.0004~{\\rm days}$$ \n \n$$\\langle \\delta t\\_{\\rm o} \\rangle \\approx 0.007~{\\rm days}$$ \nNote that our example ID-2142 has slightly smaller uncertainties than average because it is relatively bright and its eclipses are well sampled.\nAs with *P* and $t\\_{\\rm o}$, the uncertainties in *e* and *ω* are primarily determined by the sensitivity and cadence of the OGLE-III LMC observations. The solutions for *e* and *ω* are therefore independent of the other parameters, but are slightly correlated with each other (see last two rows in Fig. 6). The eclipses are sufficiently sampled to easily break this degeneracy. For ID-2142, we calculate a 95% confidence interval of *ω* = 175$^{\\rm o}$-180$^{\\rm o}$. Note that we measured eclipse widths Θ1 = 0.0047 $\\lesssim$ Θ2 = 0.0050, also indicating *ω* $\\lesssim$ 180$^{\\rm o}$ according to the approximations in Step 1 (Eqn. 3). Based on the Nightfall light curve models, we calculate formal 1*σ* measurement uncertainties of $\\delta e\\_{\\rm meas}$ ≈ 0.001 and $\\delta \\omega\\_{\\rm meas}$ ≈ 1.4$^{\\rm o}$ for ID-2142.\nFor ID-2142 and some other EBs in our sample, the measurement uncertainties $\\delta e\\_{\\rm meas}$ $\\lesssim$ 0.005 and $\\delta \\omega\\_{\\rm meas}$ $\\lesssim$ 1.5$^{\\rm o}$ are extremely small. Nightfall treats each stellar component as a three-dimensional polyhedral mesh with a finite number of flat surfaces. We suspect this finite resolution limits the true sensitivity to systematic uncertainties of $\\delta e\\_{\\rm sys}$ ≈ 0.005 and $\\delta \\omega\\_{\\rm meas}$ ≈ 1.5$^{\\rm o}$. In any case, the measurement uncertainties $\\delta e\\_{\\rm meas}$ and $\\delta \\omega\\_{\\rm meas}$ increase and dominate the total uncertainties as the eccentricities *e* decrease. We measure median total uncertainties of *δ**e* ≈ 0.02 and *δ**ω* ≈ 4$^{\\rm o}$ for *e* $\\gtrsim$ 0.5, *δ**e* ≈ 0.03 and *δ**ω* ≈ 10$^{\\rm o}$ for *e* ≈ 0.3, and *δ**e* ≈ 0.05 and *δ**ω* ≈ 20$^{\\rm o}$ for *e* ≈ 0.1. Obviously, the periastron angle *ω* is not defined, and therefore not constrained, if the orbits are circular. For the entire population of 130 well-defined EBs, we find the following relations adequately describe the median total uncertainties:\n \n⟨*δ**e*⟩ ≈ 0.06 − 0.07*e* \n \n$$\\langle \\delta \\omega \\rangle \\approx \\frac{2^{\\rm o}}{e}$$ \nSolutions for the remaining five parameters *M*1, *M*2, *τ*, *i*, and *A**I* are all correlated with each other (see Fig. 6). Moreover, unlike *P*, $t\\_{\\rm o}$, *e*, and *ω*, which have relatively symmetric Gaussian errors, the probability density functions of *M*1, *M*2, *τ*, *i*, and *A**I* are mildly to significantly asymmetric. The three parameters *M*1, *τ*, and *A**I* are especially correlated along the observed magnitude ⟨*I*⟩. In other words, solutions with more massive primaries *M*1 require younger ages *τ* and higher extinctions *A**I* to produce the same observed I-band flux. The secondary mass *M*2 is also anticorrelated with *τ*. Finally, the inclination *i* mildly depends on the three parameters *M*1, *τ*, and *A**I* that are significantly correlated with each other.\nAlthough *M*1, *M*2, *τ*, *i* and *A**I* are correlated with each other, there is sufficient information in the observed light curves and our constraints (e.g., distance, evolutionary tracks, dust reddening law) to break the degeneracies and provide unique solutions (see also §3.1). For example, if we were to fix the primary mass at *M*1 = 11.0 (i.e., the 2.5*σ* upper limit according to the probability density function in Fig. 6), the other parameters would converge to *M*2 = 7.1, *τ* = 3.6 Myr, *i* = 86.35$^{\\rm o}$, and *A**I* = 0.45 mag with a fit statistic that is Δ*χ*2 = 6.7 larger than the best-fit solution. For this larger primary mass, there is no combination of *τ* and *A**I* that can satisfactorally reproduce the observed magnitude ⟨*I*⟩ and color ⟨*V* − *I*⟩. Similarly, if we were to fix the primary mass at *M*1 = 9.2 (i.e., the 2.5*σ* lower limit), the other parameters would converge to *M*2 = 6.1 , *τ* = 13.3 Myr, *i* = 86.08$^{\\rm o}$, and *A**I* = 0.39 mag with a fit statistic that is Δ*χ*2 = 5.8 larger than the best-fit solution. In this case, the component masses both decrease by ≈ 12% (to maintain the same ratio of eclipse depths Δ*I*1/Δ*I*2), and so the orbital separation *a* decreases by 4% according to Kepler’s third law. The relative sum of the radii (*R*1 + *R*2)/*a*, which derives directly from the sum of eclipse widths Θ1 + Θ2, is measured to 1% precision in our Nightfall light curve models. If *a* decreases by 4%, then *R*1 + *R*2 must also decrease by ≈ 4%. According to the MS stellar evolutionary tracks, if *M*1 and *M*2 decrease by 12%, then the radii *R*1 and *R*2 decrease by 9% given the same age *τ* = 7.2 Myr. Hence, the age must increase to *τ* = 13.3 Myr so that the sum of radii *R*1 + *R*2 only decreases by 4%. If the masses decrease, the radii decrease, and the age increases, then the temperatures *T*1 and *T*2 both decrease according to the stellar evolutionary constraints. However, if *R*1 and *T*1 both decrease, it is difficult to maintain the same values of ⟨*I*⟩ and ⟨*V* − *I*⟩ with only one free extra paramter *A**I*. Hence, there is no combination of *M*2, *τ*, and *A**I* that can satisfactorially reproduce the observed values of Δ*I*1/Δ*I*2, Θ1 + Θ2, ⟨*I*⟩, and ⟨*V* − *I*⟩ if *M*1 = 9.2 . This line of reasoning holds for all EBs in our sample, and so the physical model parameters will always have unique solutions with finite measurement uncertainties. For ID-2142, we measure formal 1*σ* measurement uncertainties of $\\delta M\\_{1,{\\rm meas}}$ ≈ 0.04*M*1 ≈ 0.5 , $\\delta M\\_{2,{\\rm meas}}$ ≈ 0.07*M*2 ≈ 0.4 , $\\delta \\tau\\_{\\rm meas}$ ≈ 0.25*τ* ≈ 1.8 Myr, $\\delta i\\_{\\rm meas}$ ≈ 0.1$^{\\rm o}$, and $\\delta A\\_{I,{\\rm meas}}$ ≈ 0.01*A**I* ≈ 0.006 mag. We find similar percentage measurement uncertainties in these parameters for the 130 well-defined EBs in our sample.\nThe systematic uncertainties in *M*1, *M*2, *τ*, *i* and *A**I* can be considerably larger and derive from a variety of sources. We first investigate the systematic uncertainties in the adopted bolometric corrections. Our B-type MS primaries and secondaries span a large range of temperatures *T* ≈ 10,000-30,000 K and therefore a broad interval of bolometric corrections *B**C* = M$\\_{\\rm bol}$ − M*V* ≈ − 3.0- − 0.3 mag. For the hottest stars in our sample with *T* ≈ 30,000 K and *B**C* ≈ − 3.0, the bolometric corrections are uncertain by *δ**B**C* ≈ 0.2 mag, i.e. *δ**B**C*/*B**C* ≈ 7%. To propagate this systematic uncertainty into our solution for ID-2142, we decrease the absolute magnitudes of the bolometric corrections by 7% and repeat our fitting routine from §3.1. The Nightfall light curve models now converge to a final solution of *M*1 = 9.3 , *M*2 = 6.3 , *τ* = 7.5 Myr, *i* = 86.35$^{\\rm o}$, and *A**I* = 0.40 mag. The main effect of decreasing ∣*B**C*∣ is to decrease the masses *M*1 and *M*2. This is because more of the flux is radiated in the optical and so the component luminosities need to be reduced to maintain the same observed magnitude ⟨*I*⟩. Fortunately, the mass ratio *q* = *M*2/*M*1 is not significantly affected by the uncertainties in the bolometric corrections. The decrease in masses lead to slightly longer ages (to maintain the observed eclipse widths), higher inclinations (to maintain the observed eclipse depths), and lower extinctions (to maintain the observed color). The systematic uncertainties in the physical model parameters due to the uncertainties in the bolometric corrections are therefore $\\delta M\\_{1,{\\rm BC}}$ = 0.11*M*1 = 1.1 , $\\delta M\\_{2,{\\rm BC}}$ = 0.08*M*2 = 0.5 , $\\delta \\tau\\_{\\rm BC}$ = 0.04*τ* = 0.3 Myr, $\\delta i\\_{\\rm BC}$ = 0.18$^{\\rm o}$, and $\\delta A\\_{I,{\\rm BC}}$ = 0.1*A**I* = 0.04 mag. Because the primary mass *M*1 is mainly dictated by the observed ⟨*I*⟩ and the bolometric corrections, we expect similar percentage systematic uncertainties in *M*1, *M*2, *τ*, *i*, and *A**I* for the other EBs in our sample.\nWe next propagate the uncertainties in the intrinsic colors, observed colors, and dust reddening law. The uncertainty in the intrinsic colors of B-type MS stars are ≈ 0.01-0.02 mag, the color calibrations of stars in the OGLE-III LMC database are also uncertain by ≈ 0.01-0.02 mag, and the coefficient in our adopted dust reddening law *E*(*V* − *I*) = 0.70*A**I* has a ≈ 10% uncertainty. The systematic uncertainty in the dust extinction *A**I* due to dust/color uncertainties is therefore $\\delta A\\_{I,{\\rm dust/color}}$ = max{0.02 mag, 0.1*A**I*}. To confirm this estimate, we replace the dust reddening law with *E*(*V* − *I*) = 0.63*A**I* in our models for ID-2142 and repeat our fitting routine from §3.1. As expected, we measure *A**I* = 0.48 mag, i.e. the dust extinction increased by *δ**A**I* = 0.1*A**I* = 0.04 mag, while the other parameters do not vary beyond the measurement uncertainties.\nWe finally investigate the uncertainties in the stellar evolutionary tracks, including the effects of metallicity and rotation. We replace the Z=0.008 tracks from with the Z=0.006 non-rotating models from. We refit ID-2142 and measure *M*1 = 10.5 , *M*2 = 6.8 , *τ* = 9.0 Myr, *i* = 86.38$^{\\rm o}$, and *A**I* = 0.45 mag. Hence, the systematic uncertainties in the stellar evolutionary models, including our ability to interpolate between the tracks, dominates the uncertainty in the age $\\delta \\tau\\_{\\rm track}$ = 0.26*τ* = 1.9 Myr and inclination $\\delta i\\_{\\rm track}$ = 0.21$^{\\rm o}$. We then replace the evolutionary tracks with the Z=0.006 tracks from that are rotating on the zero-age MS at $v/v\\_{\\rm crit}$ = 50% the critical break-up velocity. We note that ≈ 80% of B-type MS stars are rotating at *v* $\\lesssim$ 0.5$v\\_{\\rm crit}$ ≈ 250 km s− 1, and our EBs with intermediate orbital periods *P* = 20-50 days may have tidally evolved toward slower rotational velocities. B-type MS stars initially rotating at *v*/$v\\_{\\rm crit}$ = 0.5 have equatorial radii that are only ≈ (3-4)% larger than their polar radii, but MS lifetimes $\\tau\\_{\\rm MS}$ that are 20% longer. It is therefore the differences in the evolutionary tracks of stars with rotation, not the distortions in their shapes, that can significantly affect our model solutions. We refit ID-2142 with the rotating non-synchronized stellar models, and measure *M*1 = 10.5 , *M*2 = 6.9 , *τ* = 8.6 Myr, *i* = 86.30$^{\\rm o}$, and *A**I* = 0.45 mag. For ID-2142, the differences between the non-rotating and rotating tracks from are within the measurement uncertainties. This is because the tracks with *v*/$v\\_{\\rm crit}$ = 0.5 do not significantly deviate from their non-rotating counterparts until the ages reach *τ* > 0.8$\\tau\\_{\\rm MS}$ the non-rotating MS lifetimes. For ID-2142 and the majority of EBs in our sample with primary ages *τ* < 0.8$\\tau\\_{\\rm MS}$, the uncertainties due to the effects of rotation are negligible. For the few systems that are extremely young (*τ* < 1 Myr) or old (*τ* > 0.8$\\tau\\_{\\rm MS}$), we expect slightly larger systematic uncertainties in the ages and masses.\nBy adding the measurement uncertainties and various systematic uncertainties above in quadrature, we estimate the total median 1*σ* uncertainties for the 130 well-defined EBs to be:\n \n⟨*δ**M*1⟩ ≈ 0.15*M*1 \n \n⟨*δ**M*2⟩ ≈ 0.15*M*2 \n \n$$\\langle \\delta \\tau \\rangle \\approx {\\rm max}\\{0.5~{\\rm Myr},~0.35 \\tau \\}$$ \n \n$$\\langle \\delta i \\rangle \\approx 0.4^{\\rm o}$$ \n \n$$\\langle \\delta A\\_I \\rangle \\approx {\\rm max}\\{0.03~{\\rm mag},~0.15 A\\_I \\}$$ \nFor these five parameters, the uncertainties are dominated by the systematic uncertainties in the bolometric corrections, dust reddening law, and evolutionary tracks. For older EBs with primary ages $\\gtrsim$80% their MS lifetimes, the total uncertainties in the masses ⟨*δ**M*1⟩ ≈ 0.2*M*1 and ⟨*δ**M*2⟩ ≈ 0.2*M*2 and ages ⟨*δ**τ*⟩ ≈ 0.45*τ* are slightly larger due to the effects of rotation.\nSome of our EB light curve model solutions can be biased due to contamination with a third light source, e.g., a tertiary companion or a background/foreground object along similar lines of sight. In Papers I and II, we estimated that only ≈ 10% of our B-type MS EBs in the LMC can be contaminated by a third light source that is bright enough to significantly contribute to the systematic uncertainties. Unlike the previously discussed sources of systematic uncertainties that contribute to all 130 well-defined EBs, contamination by a third light source affects only a small subset of our sample.\nIn addition to calculating the uncertainties for the nine independent physical model parameters, we also estimate the uncertainties in the dependent physical properties. The total uncertainties in *M*1 and *M*2 are ≈ 15% but slightly correlated with each other (see above). The total median uncertainty in the mass ratio is therefore ⟨*δ**q*⟩ = max{0.03, 0.12*q*}. Because the quantity (*R*1+*R*2)/*a* is precisely constrained from the observed eclipse widths, the uncertainties in *R*1, *R*2, and *a* mainly derive from the uncertainties in *M*1 and *M*2 according to Kepler’s third law. We measure ⟨*δ**R*1⟩ ≈ 0.07*R*1 ≈ 0.3, ⟨*δ**R*2⟩ ≈ 0.07*R*2 ≈ 0.2 , and ⟨*δ**a*⟩ ≈ 0.06*a* ≈ 6 . Finally, given the ≈ (20-30)% uncertainties in the luminosities (primarily due to uncertainties in the bolometric corrections) and the ≈ 7% uncertainties in the radii, the uncertainties in the temperatures are ≈ 8% according to the Stefan-Boltzmann law. Hence, the total median uncertainties are ⟨*δ**T*1⟩ ≈ 0.08*T*1 ≈ 1,500 K and ⟨*δ**T*2⟩ ≈ 0.08*T*2 ≈ 1,100 K.\nEB Trends\n=========\nIn our sample of 130 EBs, several trends and correlations exist among the nine physical model properties. Most of these trends are caused by geometrical and evolutionary selection effects in our magnitude-limited sample of EBs. We correct for these selection effects in the third stage of our pipeline (§5). Two correlations, however, are intrinsic to the population of binaries with B-type MS primaries. In this section, we first discuss these two empirical relations we uncovered from the data, and then we explain the trends that are caused by selection effects.\nIn Fig. 7, we display the measured I-band dust extinctions *A**I* as a function of age *τ* for the 130 well-defined EBs. These two parameters are anticorrelated (Spearman rank correlation coefficient *ρ* = − 0.34) at a statistically significant level (probability of independence *p* = 8 × 10− 5). We fit a log-linear trend to the total population of 130 EBs (green line in Fig. 7):\n \n$$A\\_{I,\\rm{total}}\\,({\\rm mag}) = 0.39 - 0.07\\,{\\rm log}\\Big(\\frac{\\tau}{1\\,{\\rm Myr}}\\Big)$$ \nThe slope in the above relation may be biased toward negative values due to a photometric selection effect in our magnitude-limited sample. Specifically, EBs that are intrinsically bluer and more luminous systematically contain younger, short-lived, more massive primaries. These blue, luminous, younger EBs may therefore require larger dust extinctions and reddenings to satisfy our photometric selection criteria (and vice versa). In Fig. 8, we show the measured absolute magnitudes *M**I* and intrinsic colors $\\langle V-I \\rangle\\_{\\rm o}$ as a function of dust extinction *A**I* for our 130 well-defined EBs. We also display our photometric selection criteria based on the observed magnitudes 16.0 < ⟨*I*⟩ < 17.6 and observed colors − 0.25 < ⟨*V* − *I*⟩ < 0.20 (green lines). Indeed, there are several intrinsically red, low-luminosity, older EBs with *M**I* ≈ − 1.2 that are in our sample only because they have small dust extinctions *A**I* ≈ 0.2 mag. If they were to have slightly higher dust extinctions, they would fall below our selection limit of ⟨*I*⟩ = 17.6 (see Fig. 8).\nIn §5, we account for our photometric selection criteria when analyzing all 130 EBs in our well-defined sample. Here, we correct for photometric selection effects by further culling our sample according to the intrinsic properties of *M**I* and $\\langle V-I \\rangle\\_{\\rm o}$. To obtain an unbiased subsample, we can choose EBs across any interval of *M**I* and $\\langle V-I \\rangle\\_{\\rm o}$ that also satisfies our selection criteria on observed magnitudes and colors. To retain most of the sample, we select the regions enclosed by − 2.63 < *M**I* < − 1.35, − 0.341 < $\\langle V-I \\rangle\\_{\\rm o}$ < − 0.115, and 0.13 < *A**I*(mag) < 0.45 (red lines in Fig. 8). The 98 EBs that satisfy these extra selection criteria (filled blue systems in Fig. 8) represent an unbiased sample relatively free from photometric selection effects.\nEven within this unbiased sample of 98 EBs, the intrinsic colors $\\langle V-I \\rangle\\_{\\rm o}$ and dust extinctions *A**I* are still anticorrelated (*ρ* = − 0.25) at a statistically significant level (*p* = 0.02). As can be observed in the bottom panel of Fig. 8, there are relatively few intrinsically blue systems $\\langle V-I \\rangle\\_{\\rm o}$ ≈ − 0.30 with small dust extinctions *A**I* ≈ 0.2 mag. Similarly, there are few intrinsically redder EBs $\\langle V-I \\rangle\\_{\\rm o}$ ≈ − 0.15 with large dust extinctions *A**I* ≈ 0.4 mag. Intrinsically bluer EBs contain hot primaries that are systematically more massive, short-lived, and younger. Hence, the anticorrelation between age *τ* and dust extinction *A**I* is real.\nIn Fig. 8, we also display *τ* and *A**I* for the 98 EBs (filled blue) in our unbiased subsample. Although not as prominent, the ages *τ* and dust extinctions *A**I* for the 98 EBs in our unbiased sample are still anticorrelated (*ρ* = − 0.23) at a statistically significant level (*p* = 0.02). For example, there is a complete absence of EBs with *A**I* < 0.2 mag at *τ* < 15 Myr. In contrast, there are many EBs in our unbiased sample with *A**I* = 0.1-0.2 mag at *τ* > 15 Myr. The intrinsic anticorrelation between *A**I* and *τ* in our unbiased sample demonstrates a relationship between dust content and ages of stellar environments. Young EBs, and young B-type MS stars in general, with *τ* ≈ 1 Myr are embedded in dusty envelopes and/or molecular clouds with photometric extinctions *A**I* ≈ 0.33 mag. Meanwhile, older EBs with *τ* ≈ 100 Myr reside in less attenuating environments with *A**I* ≈ 0.26 mag. We fit a log-linear trend to the unbiased sample of 98 EBs:\n \n$$A\\_{I,\\rm{unbiased}}\\,({\\rm mag}) = 0.33 - 0.03\\,{\\rm log}\\Big(\\frac{\\tau}{1\\,{\\rm Myr}}\\Big)$$ \nvalid for 0.5 Myr < *τ* < 200 Myr (red line in Fig. 8). Even after accounting for selection effects, the value of and measurement uncertainty in the slope − 0.029 ± 0.011 is still inconsistent with zero at the 2.6*σ* confidence level. This is a similar probability of significance based on the Spearman rank test above (probability of no correlation *p* = 0.02 between *A**I* and $\\langle V - I \\rangle\\_{\\rm o}$). The ≈ 30% systematic uncertainty in the ages *τ* and ≈ 10% systematic uncertainty in the extinctions *A**I* propagate into Eqn. 19. The values of and total uncertainties are therefore − 0.029 ± 0.014 for the slope and 0.33 ± 0.04 mag for the mean dust extinction at *τ* ≈ 1 Myr. The rms in the measured dust extinctions *A**I* around the above relation is *σ* = 0.08 mag.\nIt had been previously known that younger early-type stars in the LMC experience slightly higher dust extinctions than late-type stars. In the present study, we have measured the relationship between age *τ* and dust extinction *A**I*. Quantifying age-dependent dust extinctions is crucial when analyzing the spectral energy distributions of unresolved stellar populations in distant galaxies. Young O- and B-type stars, which dominate the ultraviolet component in star-forming galaxies, will experience systematically higher dust extinctions than the older, redder stars. To accurately constrain the star-formation histories of these galaxies, it is imperative to account for age-dependent dust extinctions. We note that different galaxies and stellar populations will have slightly different dust extinctions as a function of age. Nonetheless, our empirical age-extinction relation (Eqn. 19) can provide insight when calibrating models of unresolved stellar populations.\n found that the dust extinction distribution toward young, hot stars in the LMC peaks at *A**I* ≈ 0.25 mag with a long tail toward higher values. This is consistent with our total population of 130 EBs with B-type MS primaries (see Fig. 9). By dividing our EB population into young (*τ* ≤ 12 Myr) and old (*τ* > 12 Myr) subsamples, we find that both subsamples can be fitted with simple Gaussians centered at *A**I* ≈ 0.34 mag and *A**I* ≈ 0.25 mag, respectively. Hence, the non-Gaussian distribution of dust extinction may be simply due to a selection effect with age. The very young EBs, which represent a small fraction of the total population, occupy the long tail toward large dust extinctions. Meanwhile, the long-lived EBs, which comprise the majority of the sample, form the peak in the distribution at *A**I* ≈ 0.25 mag.\nWe now examine the second physically-genuine trend in our EB population. In Fig. 10, we show the measured eccentricities *e* as a function of age *τ* for the 130 EBs in our well-defined sample. The eccentricities and ages are anticorrelated (Spearman rank correlation coefficient *ρ* = − 0.39) at a statistically significant level (probability of no correlation *p* = 5 × 10− 6).\nThis observed anticorrelation is primarily because eccentricities decrease with time due to tidal evolution. The observed trend may be accentuated by a secondary effect, whereby EBs with more massive, short-lived primaries favor larger eccentricities. However, this relation between primary mass *M*1 and eccentricity *e* cannot fully explain the observed anticorrelation between *τ* and *e*. For example, the eccentricities and ages of the 32 EBs with massive primaries *M*1 ≈ 8.5-13.9 are still anticorrelated (*ρ* = − 0.44) at a statistically significant level (*p* = 0.01). Similarly, the 98 less massive EBs with *M*1 ≈ 3.6-8.5 have eccentricities and ages that are anticorrelated (*ρ* = − 0.28) at a statistically significant level (*p* = 0.005). Although EBs with early-B primaries may be born with systematically larger eccentricities, the anticorrelation between age *τ* and eccentricity *e* is dominated by tidal evolution and is observed in both early-B and late-B MS subsamples.\nFor late-type stars with *M* $\\lesssim$ 1.3, orbital energy is most efficiently dissipated into the interior of the stars via convective eddies in the stellar atmospheres. This equilibrium tide model for convective damping has been tested against observations of late-type binaries in various environments with different ages. For more massive stars *M* > 1.3 with radiative envelopes, such as our B-type MS stars, tides operate dynamically via oscillations in the stellar interiors. By estimating the ages of 130 early-type EBs, we have measured the evolution of binary eccentricities due to dynamical tides with radiative damping.\nThe slope of the observed age-eccentricity anticorrelation provides insight into the tidal evolution of highly eccentric binaries. We fit a log-linear trend to the observations (red line in Fig. 10):\n \n$$e\\_{\\rm fit} = 0.53 - 0.14\\,{\\rm log}\\Big(\\frac{\\tau}{1\\,{\\rm Myr}}\\Big)$$ \nThe value of and measurement uncertainty in the slope is − 0.14 ± 0.03. Hence, the slope is negative at the 5*σ* confidence level, similar to the statistical significance determined from the Spearman correlation test above. Again, systematic uncertainties in the ages *τ* and eccentricities *e* contribute to the uncertainties in the coefficients in Eqn. 20. After calculating the total uncertainties, we find the mean eccentricity at *τ* ≈ 1 Myr is 0.53 ± 0.05 while the slope is − 0.14 ± 0.05. The rms scatter in the measured eccentricities around the above relation is *σ**e* ≈ 0.16.\nThe intercept in Eqn. 20 implies a circularization timescale of $\\tau\\_{\\rm circ}$ ≈ 5 Gyr for our EBs with B-type MS primaries and moderate orbital periods *P* ≈ 20-50 days. However, tidal damping is not as efficient when the orbits become less eccentric. The true circularization timescale may therefore be longer if the age-eccentricity relation flattens beyond *τ* > 200 Myr. Conversely, older EBs have systematically larger components (see Fig. 5), and so tidal damping may become more efficient as the primary fills a larger fraction of its Roche lobe. In any case, these short-lived B-type MS primaries will expand beyond *R*1 $\\gtrsim$ 10 and evolve toward the giant branch long before the orbits are completely circularized.\nOur young EBs with generally large eccentricities experience extreme tidal forces. In fact, a few of the EBs with *e* > 0.6 in our sample have modest Roche-lobe fill-factors *R**L**F**F* ≈ 0.3 at periastron. Tidal evolution of highly eccentric binaries is quite complicated, especially considering second-order effects and non-linear terms can become quite important. A full analysis of tidal evolution in our EB sample is therefore not within the scope of the present study. Nonetheless, the observed age-eccentricity anticorrelation provides a constraint for models of tidal evolution in highly eccentric early-type binaries.\nIn Fig. 11, we display the cumulative distribution function of the eccentricities for the 128 EBs with *e* ≤ 0.68 (green). We do not consider the two EBs with *e* = 0.71 and 0.77 because highly eccentric binaries are not complete in our EB sample (see below and §5). Moreover, as discussed above, binaries with *P* = 20-50 days and *e* = 0.7-0.8 nearly fill their Roche lobes at periastron, and are expected to evolve toward smaller eccentricities on rapid timescales. In Fig. 11, we also divide our sample into the 91 old EBs with *τ* > 10 Myr (red) and 37 young EBs with *τ* ≤ 10 Myr (blue). Using a maximum likelihood method, we fit a power-law eccentricity probability distribution *p**e* ∝ *e**η* to the observed EBs. We measure *η* = 0.1 ± 0.2, − 0.1 ± 0.2, and 0.8 ± 0.3 for the total, old, and young EB samples, respectively. Our total population of EBs (*η* = 0.1 ± 0.2) is consistent with the flat distribution (*η* = 0) observed by for his sample of binaries with B-type MS primaries and intermediate orbital periods.\nIf the orbital velocities and energies of a binary population follow a Maxwellian “thermal” probability distribution, then the eccentricity probability distribution *p**e* = 2*e**d**e* will be weighted toward large eccentricities. Such a population of eccentric and thermalized binaries would suggest the binaries formed through dynamical interactions, either through tidal / disk capture, dynamical perturbations in a dense cluster, three-body exchanges, and/or Kozai cycles with a tertiary companion. Surprisingly, the observed population of 37 young EBs (*η* = 0.8 ± 0.3) is fully consistent with a thermal eccentricity probability distribution (*η* = 1; dashed black line in Fig. 11). This indicates that massive binaries with intermediate orbital periods formed via dynamical interactions on rapid timescales *τ* < 5 Myr.\nPrevious observations of spectroscopic and visual solar-type binaries have indicated a thermal eccentricity distribution. However, these studies recovered the thermal eccentricity distribution only after applying large and uncertain correction factors for incompleteness. In both the spectroscopic and visual binary surveys, the raw samples were weighted significantly toward smaller eccentricities relative to the thermal distribution. In addition, more recent and complete observations of solar-type and early-type binaries at intermediate orbital periods have revealed a uniform eccentricity distribution that is clearly discrepant with a thermal distribution.\nOur raw sample of young early-type EBs is only slightly biased toward small eccentricities. In fact, the small excess of young EBs with *e* ≈ 0.1-0.3 relative to the thermal distribution in Fig. 11 would be reduced after correcting for selection effects. In other words, we expect even better agreement between our sample of young early-type EBs and the thermal eccentricity distribution after considering observational biases (§5). By choosing only the EBs with young ages, we have probed the initial binary properties of massive stars shortly after their formation. For the first time, we have directly observed the theoretical thermal eccentricity distribution before tides have dramatically reduced the eccentricities.\nIn the following, we compare other physical model parameters and examine additional trends that could be caused by observational biases. We use these observed distributions to further justify our selection criteria in §2. We also motivate the necessity for incompleteness corrections and Monte Carlo simulations, which we perform in §5.\nWe display the measured eccentricities *e* as a function of mass ratio *q* in Fig. 12. A Spearman rank test reveals no statistically significant correlation (*p* = 0.25). The mass ratios *q* of early-type binaries are independent of their eccentricities *e* at intermediate orbital periods *P* = 20-50 days.\nIn §2 (see item D), we removed 23 EBs with nearly identical primary and secondary eclipses separated by ≈ 50% in orbital phase. We concluded the majority of these systems have half their listed orbital periods, and therefore exhibit only one eclipse per orbit. If we were to fit physical models to these systems assuming the listed orbital periods, they would all have *q* > 0.84 and *e* < 0.08 (blue diamonds within red region of Fig. 12). A concentration of 23 EBs in this small corner of the parameter space is highly unlikely considering the density of systems in the surrounding phase space is substantially smaller. We expect only 3-5 of the 23 EBs to be twins in nearly circular orbits with the listed orbital periods. Given only the photometric data, however, we cannot easily determine which of the systems are truly twins with small eccentricities and which have half the listed orbital periods. In §2, we simply excluded all 23 EBs with ambiguous periods, and we account for the incidental removal of the 3-5 genuine systems in §5. We emphasize that most of the 23 EBs with ambiguous orbital periods have half the listed values, and therefore it was appropriate to remove these systems.\nIn Fig. 13, we compare the measured eccentricities *e* to the arguments of periastron *ω* for the 130 well-defined EBs. Assuming random orientations, the periastron angle should be uniformly distributed across 0$^{\\rm o}$ ≤ *ω* < 360$^{\\rm o}$. However, the observed systems are not evenly concentrated across all *ω* and *e*. We notice two observational biases in the data, both of which are due to geometrical selection effects.\nFirst, for modest to large eccentricities *e* > 0.4, the EBs cluster near *ω* = 0$^{\\rm o}$ and *ω* = 180$^{\\rm o}$. In fact, the two systems with *e* ≈ 0.7-0.8 have *ω* ≈ 0$^{\\rm o}$. For EBs with *ω* = 90*o* or *ω* = 270*o*, one of the eclipses would occur at periastron while the other at apastron. The eclipse at periaston would be quite narrow according to Kepler’s second law, and may be too narrow to be accurately measured given the cadence of the OGLE-III data (see item B in §2). If the inclination is not sufficiently close to edge-on, e.g. *i* ≈ 87$^{\\rm o}$, then the eclipse at apastron may be too shallow to be accurately measured (again, see item B in §2). If the inclination was even smaller, e.g. *i* ≈ 85$^{\\rm o}$, the projected separation at apastron could be large enough so there would be no secondary eclipse. These systems would exhibit only one eclipse per orbit such as those presented in item A of §2. Considering the above, it is extremely difficult to observe and measure highly eccentric EBs with eclipses that occur near periastron and apastron. As the eccentricity increases, well-defined EBs are only detected as the argument of periastron approaches *ω* ≈ 0$^{\\rm o}$ or *ω* ≈ 180$^{\\rm o}$.\nSecond, there is an overabundance of EBs with *ω* ≈ 90$^{\\rm o}$ relative to those with *ω* ≈ 270$^{\\rm o}$. Quantitatively, there are 90 EBs with 0$^{\\rm o}$ < *ω* < 180$^{\\rm o}$ and only 40 EBs with 180$^{\\rm o}$ < *ω* < 360$^{\\rm o}$. These two values are discrepant at the 4.4*σ* level according to Poisson statistics. This observational bias is due to our definition of the primary eclipse minimum $t\\_{\\rm o}$, which determines the reference frame for *ω*. Recall the primary eclipse Δ*I*1 > Δ*I*2 at $t\\_{\\rm o}$ must be deeper than the secondary eclipse. If *e* $\\gtrsim$ 0.2, *i* $\\lesssim$ 89$^{\\rm o}$, and the primary *M*1 > *M*2 is eclipsed closer to apastron, then the eclipse of the most massive luminous component *M*1 may actually coincide with the secondary eclipse Δ*I*2 < Δ*I*1. Indeed, we found 18 EBs in such a configuration whereby *M*1 is eclipsed at Φ2 and *M*2 is eclipsed at $t\\_{\\rm o}$ (see §3 and Table 2). Sixteen of these 18 EBs have 0$^{\\rm o}$ < *ω* < 180$^{\\rm o}$. If we were to define *ω* according to *M*1 instead of in terms of Δ*I*1, then 74 EBs would have 0$^{\\rm o}$ < *ω* < 180$^{\\rm o}$ and 58 EBs would have 180$^{\\rm o}$ < *ω* < 360$^{\\rm o}$. These two values are now consistent with each other, i.e. they only differ at the 1.4*σ* significance level.\nAs indicated above and discussed in §2, we suspect the majority of the 48 EBs we removed in items A and B of §2 have *e* > 0.4 and either 20$^{\\rm o}$ < *ω* < 160$^{\\rm o}$ or 200$^{\\rm o}$ < *ω* < 340$^{\\rm o}$. We test this hypothesis using the statistics of the measured systems in our well-defined sample. Of the 53 EBs with *e* > 0.4, 22 have *ω* < 20$^{\\rm o}$, 160$^{\\rm o}$ < *ω* < 200$^{\\rm o}$, or *ω* > 340$^{\\rm o}$. If these 22 systems are complete across the specified intervals of *ω*, which total 80$^{\\rm o}$, and if the intrinsic distribution of periastron angles is uniform, then we expect 22 × 360$^{\\rm o}$/80$^{\\rm o}$ = 99 EBs with *e* > 0.4. We detected only 53 EBs with *e* > 0.4, implying 99 − 53 = 46 EBs did not satisfy our selection criteria. These 46 EBs most likley have secondary eclipses that are too narrow, too shallow, or completely absent. This prediction of 46 missing EBs nearly matches the 48 EBs we removed in items A and B of §2. This consistency further demonstrates that geometrical selection effects are understood in our sample and the removal of EBs in §2 were well-motivated.\nWe compare the mass ratios *q* to the ages *τ* of our 130 EBs in Fig. 14. There is a lack of extreme mass-ratio binaries *q* < 0.3 at young (*τ* < 2 Myr) and old (*τ* > 12 Myr) ages. The former is due to geometrical selection effects. At extremely young ages, we have shown early-type binaries with intermediate orbital periods favor large eccentricities. In fact, the median eccentricity of the 16 EBs with *τ* ≤ 2 Myr is ⟨*e*⟩ = 0.59. At these large eccentricities, the eclipse that occurs closest to apastron will have a larger projected distance, and may therefore have a shallower eclipse (see above). Shallow eclipses are easily missed given the sensitvity and cadence of the OGLE-III observations, especially if the EB contains a small, low-mass companion *q* < 0.3.\nThe bias against low-mass companions *q* < 0.3 at older ages is primarily due to an evolutionary selection effect. As the primary evolves and expands, the ratio of radii *R*2/*R*1 decreases and the primary eclipse depth Δ*I*1 becomes shallower (see Fig. 5 in Paper I). At *τ* ≈ 15 Myr, only companions with *q* > 0.3 produce eclipses Δ*I*1 $\\gtrsim$ 0.15 mag that are deep enough to be detected given the sensitivity of the OGLE-III data. If the primary is near the tip of the MS, then *q* > 0.45 is required to produce a visible and well-defined eclipse. Considering the above, only EBs with ages *τ* ≈ 2-12 are sensitive toward low-mass companions with *q* ≈ 0.2-0.3.\nWe compare the primary masses *M*1 to the mass ratios *q* = *M*2/*M*1 in Fig. 15. There is a clear observational bias such that massive primaries *M*1 = 12-14 contain only small mass ratios *q* = 0.2-0.5 while late-B MS primaries with *M*1 = 3.6-4.5 include only large mass ratios *q* = 0.6-1.0. This trend is simply due to the magnitude limits imposed by our photometric selection criteria. Massive MS primaries *M*1 $\\gtrsim$ 12 with luminous *q* $\\gtrsim$ 0.6 MS companions will be brighter than our selection limit of ⟨*I*⟩ = 16.0. Similarly, low-mass primaries with *M*1 $\\lesssim$ 5will be fainter than our detection limit of ⟨*I*⟩ = 17.6 unless there is a bright companion *q* $\\gtrsim$ 0.6 that increases the total luminosity of the system.\nThe precise mass versus mass-ratio cutoffs in our sample also depend on the age of the binary. For example, older primaries with *M*1 ≈ 5-7 on the upper MS will be bright enough ⟨*I*⟩ < 17.6 to satisfy our photometric selection criteria. As stated above, EBs with small mass ratios *q* = 0.2-0.3 produce visible well-defined eclipses with Δ*I*1 $\\gtrsim$ 0.15 mag only when the primary is relatively small and young. However, young modererate-mass primaries *M*1 ≈ 5-7 with low-luminosity companions are fainter than our detection limit of ⟨*I*⟩ = 17.6. Hence, our EB sample is sensitive to extreme mass ratios *q* ≈ = 0.2-0.3 only if *M*1 $\\gtrsim$ 7. To be conservative, we consider only the primary mass interval *M*1 = 8-11 to be sensitive to companions across the entire interval *q* = 0.2-1.0 (distinguished by red and blue lines in Fig. 15).\nConsidering the above, our EB sample is relatively unbiased across the mass-ratio interval *q* = 0.22-0.84 (Fig. 12), age interval *τ* = 2-12 Myr (Fig. 14), and primary mass interval *M*1 = 8-11 (Fig. 15). The 13 EBs that are contained in this cube of the three-dimensional phase space therefore represent a small subsample relatively free from geometrical, evolutionary, and photometric selection effects. In Fig. 16, we display the cumulative distribution of mass ratios *q* for these 13 EBs in our unbiased subsample. Using a maximum liklihood technique, we fit a power-law mass-ratio probability distribution *p**q* ∝ *q**γ* to these 13 EBs. We measure *γ* = − 1.6 ± 0.4, demonstrating binaries with massive primaries favor extreme mass ratios *q* ≈ 0.2-0.3 at intermediate orbital periods *P* = 20-50 days.\nWe emphasize this statistic is based on the small unbiased subsample of the 13 EBs, and therefore valid only for early-B MS primaries with *M*1 ≈ 10. The median primary mass in our total sample of 130 EBs is ⟨*M*1⟩ ≈ 6. We therefore utilize our total sample to derive more accurate statistics as well as probe the companion properties of late-B MS stars. In the following section, we correct for selection effects so that we can make full use of all EBs in our well-defined sample.\nCorrected Binary Statistics (Stage III)\n=======================================\nFor the final stage of our pipeline, we recover the intrinsic binary statistics and distributions by correcting for selection effects. As done in Paper II, we first determine the probability density functions that describe the nine physical model parameters of our EBs (§5.1). We then calculate simple estimates for the detection efficiencies (§5.2), and then synthesize a large population of EBs via a Monte Carlo technique (§5.3). In §5.4, we present our results for the intrinsic binary fraction and mass-ratio distribution.\nProbability Density Functions\n-----------------------------\nWe utilize probability density functions similar to those in Paper II. For example, we assume random epochs of primary eclipse minima $t\\_{\\rm o}$, and that the logarithmic orbital periods log *P* are uniformly distributed across *P* = 20-50 days. We select primary masses *M*1 = 3-30 and ages *τ* = 0-320Myr according to the initial mass function (IMF) and star-formation history, respectively, measured for the OGLE-III LMC footprint in Paper II. In short, we fitted an IMF slope *α* = − 2.4 consistent with the Salpeter value and a star-formation history such that the present-day star-formation rate is approximately double the rate at earlier epochs *τ* = 40-320 Myr. We assume random orientations, i.e. cos*i* = 0-1 and *ω* = 0$^{\\rm o}$-360$^{\\rm o}$ are both uniformly distributed across their respective intervals.\nIn the present study, we account for the empirical age-extinction and age-eccentricity anticorrelations. Given an age *τ*, we select dust extinctions *A**I* according to a Gaussian distribution:\n \n$$p\\_{A\\_I} \\propto {\\rm exp} \\Big(-\\frac{[A\\_I - A\\_{I,\\rm{unbiased}}(\\tau)]^2}{2 \\sigma\\_{A\\_I}^2}\\Big)$$ \nfor 0 < *A**I*(mag) < 1 and where $A\\_{I,\\rm{unbiased}}$(*τ*) and *σ**A**I* = 0.08 mag derive from the fit to the unbiased subsample in Eqn. 19. We also choose eccentricities *e* from an age-dependent Gaussian distribution:\n \n$$p\\_e \\propto {\\rm exp} \\Big(-\\frac{[e - e\\_{\\rm fit}(\\tau)]^2}{2 \\sigma\\_{e}^2}\\Big)$$ \nfor 0.0 < *e* < 0.8 and where $e\\_{\\rm fit}$(*τ*) and *σ**e* = 0.16 derive from Eqn. 20. Finally, we consider the detection efficiencies as a continuous function of mass ratio *q* = 0.2-1.0.\nSimple Estimates\n----------------\nBefore we conduct detailed Monte Carlo simulations, we perform simple calculations to estimate the probabilities of detecting EBs with *P* = 20-50 days. For *q* = 0.8-1.0 companions, the detection efficiencies are primarily dictated by two geometrical selection effects. First, the orientations must be sufficiently close to edge-on. About 90% of our well-defined EBs have *i* > 86.6$^{\\rm o}$, implying the probability of having the necessary inclinations to produce observable eclipses is ${\\cal P}\\_i$ = cos(86.6$^{\\rm o}$) = 0.06.\nSecond, EBs with longer orbital periods are more likely to be missed. Not only do binaries with longer periods require larger inclinations *i* $\\gtrsim$ 87$^{\\rm o}$ to produce eclipses, but the eclipse widths can also become too narrow to be detected given the cadence of the OGLE-III LMC observations. We found 73 well-defined EBs with *P* = 20-30 days. Assuming the intrinsic distribution of log *P* is uniform, then we would expect ≈ 92 EBs with *P* = 30-50 days. Our well-defined sample includes only 57 EBs with *P* = 30-50 days, suggesting 35 systems were missed due to narrow and/or shallow eclipses. Note that this is consistent with the 32 EBs we removed in item B of §2 with uncertain eclipse parameters. The probability that EBs have orbital periods that are sufficiently short is therefore ${\\cal P}\\_P$ = 130 / (130 + 35) ≈ 0.8. Considering these two factors, the probability of detecting well-defined EBs with *q* ≈ 0.8-1.0 is ${\\cal P}\\_i$${\\cal P}\\_P$ = 0.06 × 0.8 ≈ 5% (red line in Fig. 17).\nFor *q* = 0.2-0.3 companions, we must also consider evolutionary and photometric selection effects. As discussed in §4, well-defined EBs satisfy our photometric selection criteria and are sensitive to low-mass companions only if the primaries are relatively young with ages *τ* = 2-12 Myr (Fig. 14) and massive with *M*1 ≈ 8-11 (Fig. 15). Given a typical MS lifetime of $\\tau\\_{\\rm MS}$ ≈ 30 Myr for *M*1 ≈ 8-11 primaries, then the probability of having the necessary ages *τ* = 2-12 Myr is ${\\cal P}\\_{\\tau}$ ≈ (12 − 2)/30 ≈ 0.3. The smallest primary mass in our well-defined EB sample is 3.6. Assuming our adopted IMF, the probability that an EB contains a massive primary *M*1 > 8compared to the probability of having any B-type MS primary with *M*1 > 3.6 is ${\\cal P}\\_{M\\_1}$ = 0.2. Combining these additional factors, then the probability of detecting well-defined EBs with *q* ≈ 0.2-0.3 is ${\\cal P}\\_i$${\\cal P}\\_P$${\\cal P}\\_{\\tau}$${\\cal P}\\_{M\\_1}$ = 0.06 × 0.8 × 0.3 × 0.2 ≈ 0.3% (blue line in Fig. 17).\nMonte Carlo Simulations\n-----------------------\nWe utilize the same technique from Paper II to correct for incompleteness across a continuous function of mass ratios *q*. For a given *q*, we select *M*1, *τ*, and *A**I* from their respective probability density functions. If the simulated binary does not satisfy our photometric selection criteria, we generate a new binary. Otherwise, we keep the binary and consider its contribution toward the total number ${\\cal N}\\_{\\rm sim}$ of simulated binaries. We then select the other physical parameters, i.e. $t\\_{\\rm o}$, *P*, *i*, *e*, and *ω*, from their respective probability density functions.\nWith the nine physical model parameters for our simulated binary, we synthesize an I-band light curve with Nightfall. We match the cadence and sensitivity of the OGLE-III LMC survey. Specifically, we sample the simulated light curve at $\\langle {\\cal N}\\_I \\rangle$ = 470 random epochs and add Gaussian noise according to:\n \n$$\\sigma\\_{I} = \\big[1+10^{(I-17.0)/2}\\big]\\,\\times\\,0.0075~{\\rm mag}$$ \nThis equation derives from fitting the relation between the I-band magnitudes and corrected photometric uncertainties for all 221 OGLE-III LMC EBs with intermediate orbital periods *P* = 20-50 days.\nWe then fit our analytic model of Gaussians (Eqn. 1) to the simulated Nightfall light curve. As in §2, we utilize the MPFIT Levenberg-Marquardt routine to measure the values of and uncertainties in the eight analytic model parameters, e.g. Δ*I*1, Φ2, etc. To be considered well-defined, we impose the same selection criteria adopted in §2. Namely, we require the uncertainties in the eclipse depths Δ*I*1 and Δ*I*2 and eclipse widths Θ1 and Θ2 to be < 20% their respective values. We also require the fitted orbital periods to be unambiguous according to Eqn. 2. If the synthesized binary satisfies these selection criteria, we consider its contribution toward the total number ${\\cal N}\\_{\\rm well}$ of well-defined EBs. Using a Monte Carlo technique, we repeat the above procedure until we simulate ${\\cal N}\\_{\\rm well}$ = 50 well-defined EBs for each value of *q*. The probability of detecting well-defined EBs is simply ${\\cal P}$ = ${\\cal N}\\_{\\rm well}$/${\\cal N}\\_{\\rm sim}$.\nBy creating this mock data set of EBs, we find the measurement uncertainties in the eclipse depths and widths calculated by MPFIT are themselves uncertain by ≈ 20%. Hence, a simulated light curve with ≈ 6*σ* confidence in the analytic light curve parameters may be accidentally rejected, while a system with ≈ 4*σ* detections in the eclipse depths and widths may be included as a well-defined EB. To account for this systematic uncertainty in our selection criteria, we simulate two additional sets of EB populations. We first relax our criteria and consider EBs as well-defined if the MPFIT uncertainties in the eclipse depths Δ*I*1 and Δ*I*2 and eclipse widths Θ1 and Θ2 are < 25% their respective values. For our final set of simulations, we impose a more stringent requirement that the MPFIT relative uncertainties are < 15%. For each value of *q*, we therefore simulate a total of 3 × ${\\cal N}\\_{\\rm well}$ = 150 well-defined EBs. In this manner, we have determined the values of and uncertainties in ${\\cal P}(q)$.\nIn Fig. 17, we display the probabilities ${\\cal P}$ (and their uncertainties) of detecting well-defined EBs as a function of mass ratio *q*. As expected, the ability to detect EBs with extreme mass ratios *q* = 0.2-0.3 is substantially smaller than the ability to observe EBs with mass ratios near unity. At large *q* > 0.6, the relative uncertainties in the probabilities are $\\delta{\\cal P}$/${\\cal P}$ ≈ 11%, which is only slightly larger than that expected from Poisson statistics 150$^{-\\nicefrac{1}{2}}$ = 8%. Essentially, the majority of EBs with *q* > 0.6 have deep and accurately measured eclipse properties, and so the precise definition of our selection criteria does not significantly affect which EBs are considered well-defined. At the smallest mass ratios *q* = 0.2, however, the relative uncertainties more than double to $\\delta{\\cal P}$/${\\cal P}$ ≈ 25%. All extreme mass-ratio EBs have shallow eclipses (see §3) and are close to the detection limit, and so the probabilities ${\\cal P}$ of detecting well-defined EBs are more uncertain. The probabilities ${\\cal P}$ we calculate from our Monte Carlo simulations are consistent with the simple estimates derived in §5.2. This demonstrates the selection effects are well-understood and the probabilities ${\\cal P}$ are reliably measured.\nCorrected Binary Fraction\n-------------------------\nThe intrinsic binary statistics are determined by weighting each well-defined EB by the inverse of their respective probability ${\\cal P}$(*q*) of detection as displayed in Fig. 17. The total number of B-type MS stars with companions *q* = 0.2-1.0 at *P* = 20-50 days is simply ${\\cal N}\\_{\\rm comp}$ = $\\sum\\_{j=1}^{130} [{\\cal P}(q\\_j)]^{-1}$ ≈ 6,500. Given ${\\cal N}\\_{\\rm B}$ = 96,000 B-type MS primaries in our photometric sample, then ${\\cal F}$ = ${\\cal N}\\_{\\rm comp}$/${\\cal N}\\_{\\rm B}$ = 6,500/96,000 ≈ 6.7% of B-type MS stars have companions with *P* = 20-50 days and *q* = 0.2-1.0. The uncertainty in this fraction derives from a variety of sources. First, the predicted number ${\\cal N}\\_{0.2<q<0.3}$ ≈ 2,000 of low-mass companions with *q* = 0.2-0.3 is relatively large but also uncertain. In our sample of 130 well-defined EBs, only 8 systems have mass ratios *q* = 0.2-0.3, and so the measurement uncertainty from Poisson statistics is 8$^{-\\nicefrac{1}{2}}$ ≈ 35%. The systematic uncertainty at *q* = 0.2-0.3 is ≈ 25% due to the uncertainty in the probabilities ${\\cal P}$ of detection (see above). The total relative uncertainty in the number of companions with *q* = 0.2-0.3 is therefore ≈ 43%, and so ${\\cal N}\\_{0.2<q<0.3}$ = 2,000 ± 900. We repeat this calculation for the other mass ratio intervals, and find the total relative uncertainty in the number of companions is ≈ 31%, i.e. ${\\cal N}\\_{\\rm comp}$ = 6,500 ± 2,000. Finally, in our Monte Carlo simulations, we account for the removal of the 74 systems represented in panels A-D of Fig. 1 from our total initial sample of 221 EBs. We did not, however, account for the 17 EBs represented in panels E-F that exhibited variable or peculiar eclipse properties. These systems contribute a small relative uncertainty of 17/221 = 8%. Hence, the total relative uncertainty in the number of companions is ≈ 33%. The fraction of B-type MS stars that have companions with *P* = 20-50 days and *q* = 0.2-1.0 is therefore ${\\cal F}$ = (6.7 ± 2.2)%.\nSurveys for double-lined spectroscopic binaries (SB2s) with early-type primaries are generally complete for modest mass ratios *q* > 0.25 and short orbital periods *P* < 20 days. In a sample of 109 B-type MS stars, found seven SB2s with *q* > 0.25 and *P* = 2-20 days. Similarly, in a sample of 71 O-type stars, identified 18 SB2s across the same mass-ratio and period intervals. These statistics imply $f\\_{\\rm logP}$ = 7/109 = 0.06 ± 0.02 and $f\\_{\\rm logP}$ = 18/71 = 0.25 ± 0.06 companions with *q* > 0.25 per decade of orbital period at log*P*(days) = 0.8 for B-type and O-type stars, respectively. As discussed in Paper II and in, the close binary fraction dramatically increases with primary mass.\nBased on our B-type MS EBs, we measure a (5.6 ± 1.4)% corrected binary fraction across *P* = 20-50 days and *q* > 0.25. This results in $f\\_{\\rm logP}$ = (0.056 ± 0.014)/(log50 − log20) = 0.14 ± 0.04 companions with *q* > 0.25 per decade of orbital period centered at log*P*(days) = 1.5. This value is consistent with the early-type spectroscopic binary fraction measured at short orbital periods *P* < 20 days, implying the intrinsic period distribution of early-type binaries closely resembles Öpik’s law.\nCorrected Mass-ratio Distribution\n---------------------------------\nIn Fig. 18, we display the cumulative distribution of mass ratios *q* after weighting each well-defined EB by the inverse of their respective probability ${\\cal P}$ of producing observable eclipses. By fitting a power-law probability distribution *p**q* ∝ *q**γ* to the corrected mass-ratio distribution, we measure *γ* = − 1.1 ± 0.3. This is consistent with our estimate in §4 (Fig. 16) of *γ* = − 1.4 ± 0.3 based on a relatively unbiased subsample of 13 young EBs with early-B MS primaries *M*1 ≈ 10. In both cases, binaries with B-type MS primaries and orbital periods *P* = 20-50 days favor small mass ratios *q* = 0.2-0.3.\nObservations of early-type spectroscopic and eclipsing binaries with short orbital periods *P* = 2-20 days reveal a mass-ratio probability distribution that is only slightly weighted toward small values, e.g. *γ* ≈ − 0.9- − 0.2. In addition to the power-law component, close massive binaries with *P* < 20 days exhibit a small excess of twins with *q* $\\gtrsim$ 0.9. The preponderance of close binaries with moderate mass ratios and excess of twins suggest early-type binaries with *P* < 20 days coevolved via fragmentation and competitive accretion in the circumbinary disk.\nIn contrast to close massive binaries with *P* < 20 days, we find early-type binaries at moderate orbital periods *P* = 20-50 days are even further weighted toward extreme mass ratios, i.e. *γ* = − 1.1 ± 0.3 for our total sample and *γ* = − 1.6 ± 0.4 for early-B MS primaries. In addition, there is no evidence for an excess population of twins at intermediate orbital periods. Previous spectroscopic surveys have indicated that the mass-ratio probability distribution becomes weighted toward smaller values with increasing orbital period. However, this result is primarily based on the smaller frequency of SB2s at intermediate orbital periods, especially when compared to the frequency of single-lined spectroscopic binaries. In the present study, we have measured the mass-ratio probability distribution at intermediate orbital periods. Our results indicate that early-type binaries at slightly longer orbital periods *P* = 20-50 days have experienced less coevolution. In a future paper (Moe et al., in prep.), we will analyze early-type binaries discovered through other observational techniques, e.g. long-baseline interferometry, adaptive optics, common proper motion, etc., and investigate this anticorrelation between *P* and *q* in a more thorough and self-consistent manner.\nSummary\n=======\n*Eclipsing Binary Sample (§2)*. We analyzed the 221 EBs in the OGLE-III LMC database with B-type MS primaries and orbital periods *P* = 20-50 days. After fitting analytic models of Gaussians to the observed light curves, we identified 130 detached EBs that exhibit two well-defined eclipses per orbit. The remaining 90 EBs have uncertain, peculiar, and/or variable eclipse properties, including 12 systems that displayed changes in the secondary eclipse parameters most likely due to orbital motion with a tertiary companion.\n*Physical Models (§3)*. We developed an automated procedure to robustly and quickly fit detailed physical models to the EB light curves. Our algorithm can be adapted for any population of detached EBs with known distances and MS primaries. We implemented our procedure on our 130 detached well-defined EBs to measure their intrinsic physical properties, including their ages *τ*, component masses *M*1 and *M*2, dust extinctions *A**I*, and eccentricities *e*. We incorporated various techniques to demonstrate the uniqueness and robustness of the model solutions as well as the accuracy of the model parameters.\n*Age-Extinction Anticorrelation (§4)*. Even after considering selection effects, we find the ages *τ* and dust extinctions *A**I* are anticorrelated (*ρ* = − 0.23) at a statistically significant level (*p* = 0.02). This suggests young stars with *τ* < 10 Myr are embedded in dusty envelopes and/or molecular clouds with *A**I* ≈ 0.35 mag, while older stars with *τ* > 100 Myr reside in less attenuating environments with *A**I* ≈ 0.25 mag. This empirical relation between *τ* and *A**I* should prove beneficial when modeling stellar populations.\n*Age-Eccentricity Anticorrelation (§4)*. We also discover the ages *τ* and eccentricities *e* are anticorrelated (*ρ* = − 0.39) at a statistically significant level (*p* = 5 × 10− 6) due to tidal evolution. The slope in the observed trend provides a diagnostic for the radiative damping constant via dynamical tides in highly eccentric binaries with hot MS components. We note the tidal circularization timescales *e*/$\\dot{e}$ in highly eccentric binaries with *e* ≈ 0.5-0.8 may be orders of magnitude shorter than the circularization timescales when the eccentricities *e* $\\lesssim$ 0.4 are already small.\n*Initial Eccentricity Distribution (§4)*. We find that massive binaries at *P* = 20-50 days are initially born with larger eccentricities ⟨*e*⟩ ≈ 0.6. Assuming a power-law eccentricity probability distribution *p**e* ∝ *e**η*, we measure *η* = 0.8 ± 0.3 for our young early-type EBs with *τ* ≤ 10 Myr. This is consistent with a Maxwellian “thermal” eccentricity distribution (*η* = 1), which indicates massive binaries with intermediate orbital periods formed via dynamical interactions, either through tidal / disk capture, dynamical perturbations in a dense cluster, three-body exchanges, and/or Kozai cycles with a tertiary companion.\n*Binary Fraction (§5)*. After utilizing a Monte Carlo technique to correct for selection effects, we measure that (6.7 ± 2.2)% of B-type MS stars have companions with *P* = 20-50 days and *q* = 0.2-1.0. The frequency of companions per decade of orbital period at log *P*(days) = 1.5 is consistent with spectroscopic observations of close massive binaries at log *P*(days) = 0.8. This suggests the intrinsic period distribution of binary companions to B-type MS stars closely resembles Öpik’s law for *P* < 50 days.\n*Mass-ratio Distribution (§5)*. In our corrected binary sample with B-type MS primaries ⟨*M*1⟩ = 6, we measure a mass-ratio probability distribution *p**q* ∝ *q**γ* weighted toward small values (*γ* = − 1.1 ± 0.3). There is a slight indication that binaries with early-B MS primaries ⟨*M*1⟩ = 10 are even further skewed toward extreme mass ratios (*γ* = − 1.6 ± 0.4). Close massive binaries with *P* < 20 days favor moderate mass ratios and exhibit a small excess of twin components *q* $\\gtrsim$ 0.9. This indicates our early-type MS binaries with intermediate orbital periods *P* = 20-50 days have experienced substantially less coevolution via fragmentation and competitive accretion in the circumbinary disk.\nThis work was supported in part by support from NSF AST-1211843, AST-0708924 and AST-0908878 and NASA NNX12AE39G and AR-13243.01-A. We thank the anonymous referee for the useful comments that improved the quality of the manuscript. We also acknowledge use of publicly available data from the Optical Gravitational Lensing Experiment.\n---\n1. http://www.hs.uni-hamburg.de/DE/Ins/Per/Wichmann/ \nNightfall.html[↩](#fnref1)\n2. The three steps discussed in this section are not to be confused with the three full stages of our automated pipeline, which classifies EBs (Stage I - §2), fits physical models to the light curves (Stage II - §3), and corrects for selection effects (Stage III - §5). The three steps regarding physical models are all included in Stage II.[↩](#fnref2)\n\nEarly-type Eclipsing Binaries with Intermediate Orbital Periods\n===============================================================\nWe analyze 221 eclipsing binaries (EBs) in the Large Magellanic Cloud with B-type main-sequence (MS) primaries (*M*1 ≈ 4-14 ) and orbital periods *P* = 20-50 days that were photometrically monitored by the Optical Gravitational Lensing Experiment. We utilize our three-stage automated pipeline to (1) classify all 221 EBs, (2) fit physical models to the light curves of 130 detached well-defined EBs from which unique parameters can be determined, and (3) recover the intrinsic binary statistics by correcting for selection effects. We uncover two statistically significant trends with age. First, younger EBs tend to reside in dustier environments with larger photometric extinctions, an empirical relation that can be implemented when modeling stellar populations. Second, younger EBs generally have large eccentricities. This demonstrates that massive binaries at moderate orbital periods are born with a Maxwellian “thermal” orbital velocity distribution, which indicates they formed via dynamical interactions. In addition, the age-eccentricity anticorrelation provides a direct constraint for tidal evolution in highly eccentric binaries containing hot MS stars with radiative envelopes. The intrinsic fraction of B-type MS stars with stellar companions *q* = *M*2/*M*1 > 0.2 and orbital periods *P*=20-50 days is (7 ± 2)%. We find early-type binaries at *P* = 20-50 days are weighted significantly toward small mass ratios *q* ≈ 0.2-0.3, which is different than the results from previous observations of closer binaries with *P* < 20 days. This indicates that early-type binaries at slightly wider orbital separations have experienced substantially less competitive accretion and coevolution during their formation in the circumbinary disk.\nIntroduction\n============\nIt has long been understood that the main-sequence (MS) binary star fraction increases with primary mass. Indeed, most massive stars with *M*1 > 10 will interact with a stellar companion before they explode as core-collapse supernovae. Throughout the decades, there have been significant advances in the detection of close and wide companions to massive stars. However, the intrinsic properties of binary companions to early-type primaries, e.g. their eccentricity and mass-ratio distributions, remain elusive at intermediate orbital periods. The major goal of this work is to help fill this particular portion of the parameter space.\nEclipsing binaries (EBs) offer a key to the accurate measurement of the binary properties of early-type stars. Large photometric surveys, such as the third phase of the Optical Gravitational Lensing Experiment (OGLE-III), have discovered tens of thousands of EBs. These populations of EBs are orders of magnitude larger than previous binary samples. Despite the geometrical selection effects, we can still achieve large sample statistics to reliably infer the intrinsic binary fraction and properties at intermediate orbital periods. We emphasize that EBs can probe a unique portion of the binary parameter space unavailable to other observational techniques.\nIn, we incorporated OGLE catalogs of EBs in the Large and Small Magellanic Clouds (LMC and SMC, respectively) as well as Hipparcos observations of EBs in the Milky Way. We compared the close binary properties (*P* < 20 days) of early-B MS primaries in the three different galaxies. The Milky Way and SMC EB samples are too small to warrant an analysis of period-dependent binary properties. The OGLE-III LMC EB catalog, on the other hand, contains ≈ 5-40 times more systems, is relatively complete toward shallow eclipse depths, and includes the full I-band and V-band light curves.\nIn, we developed a three-stage automated pipeline to analyze EBs with short orbital periods in the OGLE-III LMC database. This pipeline (1) classifies EBs according to their light curve characteristics, (2) measures the intrinsic physical properties of detached EBs, e.g. ages and component masses, based on the observed radii and temperatures, and (3) recovers the intrinsic binary statistics by correcting for selection effects.\nIn the present study, we utilize EBs in the OGLE-III LMC database to measure the binary fraction, mass-ratio distribution, and eccentricity distribution of B-type MS stars with intermediate orbital periods *P* = 20-50 days. We organize the rest of this paper as follows. In §2, we define our selection criteria for identifying EBs with B-type MS primaries, intermediate orbital periods, and well-defined eclipse parameters. We next describe an automated procedure we developed to fit detailed physical models to the observed EB light curves, and we present our results for the physical properties of the individual EBs (§3). In §4, we explain the observed trends in the measured EB parameters, paying special attention to the empirical age-extinction and age-eccentricity anticorrelations. We then perform Monte Carlo simulations to quantify selection effects, and present our results for the corrected binary statistics (§5). We summarize our main results and conclusions in §6.\nEB Selection and Classification (Stage I)\n=========================================\nIn Paper II, we developed a three-stage automated pipeline to fully analyze short-period EBs in the OGLE-III LMC database. In the present study, we adapt our routine to identify intermediate-period EBs with well-defined light curves (Stage I - this section), measure their physical properties (Stage II - §3), and correct for selection effects (Stage III - §5). EBs with intermediate orbital periods exhibit two major differences that must be considered. First, the eclipse widths Θ1 and Θ2, which are expressed as a fraction of the orbital period *P*, become narrower with increasing orbital separation. Given the average number $\\langle {\\cal N}\\_{I} \\rangle$ ≈ 470 of I-band measurements in the OGLE-III LMC survey, the light curves are not sufficiently sampled if either of the eclipse widths Θ < $\\langle {\\cal N}\\_{I} \\rangle^{-1}$ ≈ 0.0021 are too narrow. EBs with small MS components and long orbital periods *P* $\\gtrsim$ 50 days have narrow eclipses Θ $\\lesssim$ 0.002, and are therefore not Nyquist sampled. This subsampling leads to detection incompleteness, issues with aliasing, and the inability to fully characterize their intrinsic physical properties. Hence, it is the finite cadence of the OGLE-III observations, *not* geometrical selection effects, that limits our present study of EBs to *P* = 20-50 days (see also ).\nSecond, the majority of early-type EBs at *P* > 20 days are in eccentric orbits. We must therefore adapt our physical models to simultaneously fit the eccentricity *e* and argument of periastron *ω* (§3). In addition, it is possible for an eccentric binary to have a certain combination of eccentricity, periastron angle, and inclination that is sufficiently offset from edge-on (e.g., *i* $\\lesssim$ 86$^{\\rm o}$) so that there is only one eclipse per orbit. Indeed, there are many EBs with single eclipses in the OGLE-III LMC database (see below). Unfortunately, we cannot measure the physical properties of these systems. We therefore remove single-eclipse EBs from our well-defined sample, and we account for their removal when we correct for selection effects (see §5). In the following, we review our methods from Paper II, where we pay special attention to the nuances of EBs with intermediate orbital periods.\nIn this study, we select the ${\\cal N}\\_{\\rm B}$ ≈ 96,000 systems in the OGLE-III LMC catalog with mean magnitudes 16.0 < ⟨*I*⟩ < 17.6 and observed colors − 0.25 < ⟨*V* − *I*⟩ < 0.20. Given the distance modulus *μ* = 18.5 to the LMC and typical dust reddenings *E*(*V* − *I*) ≈ 0.1-0.3 mag toward hot young stars in the LMC, these stars have luminosities and surface temperatures that correspond to B-type MS primaries. From this sample, we analyze the 221 systems that were identified as EBs with orbital periods *P* = 20-50 days. In Table 1, we list the OGLE-III LMC EB identification numbers, observed colors ⟨*V* − *I*⟩, and numbers of I-band measurements ${\\cal N}\\_I$ for each of these 221 EBs.\nAs in Paper II, we measure the intrinsic rms scatter in the *I*-band light curve outside of eclipses for each EB. We then calculate the correction factor *f**σ*, *I* ≥ 1.0, i.e. the ratio between the actual rms scatter and photometric uncertainties reported in the catalog. For each *I*-band measurement in an EB light curve, we multiply the listed photometric uncertainties by the correction factor *f**σ*, *I* to determine the corrected uncertainties.\nWe classify EBs based on an analytic light curve model of two Gaussians with eight total free parameters. The orbital phase 0 ≤ *ϕ* < 1 is determined by the time of observation and two model parameters: the orbital period *P* (in days) and epoch of primary eclipse minimum $t\\_{\\rm o}$ (Julian date − 2450000). The six remaining analytic model parameters are the average I-band magnitude outside of eclipses ⟨*I*⟩, primary and secondary eclipse depths Δ*I*1 and Δ*I*2, primary and secondary eclipse widths Θ1 and Θ2, and the phase of secondary eclipse Φ2. The analytic model of Gaussians is:\n \n$$\\begin{aligned}\nI\\_{\\rm G}(\\phi) = \\langle I \\rangle &+\\Delta I\\_1 \\Big[{\\rm exp}\\Big(\\frac{-\\phi^2}{2\\Theta\\_1^2}\\Big)+\n \\rm{exp}\\Big(\\frac{-(\\phi-1)^2}{2\\Theta\\_1^2}\\Big)\\Big]\\nonumber \\\\\n &+ \\Delta I\\_2\\, \\rm{exp} \\Big(\\frac{-(\\phi-\\Phi\\_2)^2}{2\\Theta\\_2^2}\\Big)\\end{aligned}$$ \nWe fit this analytic model to each EB I-band light curve. Specifically, we utilize an automated Levenberg-Marquardt algorithm to minimize the $\\chi^2\\_{\\rm G}$ statistic. The MPFIT routine provides robust best-fit solutions and measurement uncertainties for the eight analytic model parameters. Some of the photometric measurements are clear outliers, so we clip up to ${\\cal N}\\_{\\rm c}$ ≤ 2 data points per light curve that exceed 4*σ* from the model. This results in *ν* = ${\\cal N}\\_I$ − ${\\cal N}\\_{\\rm c}$ − 8 degrees of freedom. For each EB, we report in Table 1 the eight fitted analytic model parameters and the fit statistics. Excluding the few EBs that exhibit variability or are evolved Roche-lobe filling systems (see below), the goodness-of-fit statistics $\\chi^2\\_{\\rm G}$/*ν* = 0.87-1.16 indicate the analytic models can adequately describe the EB light curves.\nWe can measure the physical"},"data_id":{"kind":"string","value":"arxiv_0183889"}}},{"rowIdx":45998,"cells":{"content":{"kind":"string","value":"The QED four-photon amplitudes off-shell: part 2\n================================================\nThis is the second one of a series of four papers devoted to a first calculation of the scalar and spinor QED four-photon amplitudes completely off-shell. We use the worldline formalism which provides a gauge-invariant decomposition for these amplitudes as well as compact integral representations. It also makes it straightforward to integrate out any given photon leg in the low-energy limit, and in the present sequel we do this with two of the four photons. For the special case where the two unrestricted photon momenta are equal and opposite the information on these amplitudes is also contained in the constant-field vacuum polarisation tensors, which provides a check on our results. Although these amplitudes are finite, for possible use as higher-loop building blocks we evaluate all integrals in dimensional regularisation. As an example, we use them to construct the two-loop vacuum polarisation tensors in the low-energy approximation, rederive from those the two-loop *β*-function coefficients and analyse their anatomy with respect to the gauge-invariant decomposition. As an application to an external-field problem, we provide a streamlined calculation of the Delbrück scattering amplitudes in the low-energy limit. All calculations are done in parallel for scalar and spinor QED.\nIntroduction\n============\nThe present paper is the second one in a series of four devoted to a first calculation of the scalar and spinor QED one-loop four-photon amplitudes fully off-shell, using the worldline representation of these amplitudes. In part I we already derived these representations, discussed their properties, and calculated them for the simple limiting case where all four photons are taken in the low-energy (or “Euler-Heisenberg” limit). In the present sequel, we calculate them explicitly for the case where only photons 3 and 4 are taken in that limit, but 1 and 2 have arbitrary off-shell momenta, see Fig. [43low].\n[43low]Four-photon box diagram with two low-energy legs *k*4 and *k*3 indicated by empty bullets at their ends. A sum over permutations is understood, as well as the inclusion of seagull diagrams in the scalar QED case. \nThe results are given in compact form in terms of the hypergeometric function 2*F*1 for the dimensionally continued case, and in trigonometric form for *D* = 4.\nAlthough to our knowledge this special case of the four-photon amplitudes has not been considered by other authors, for *k*1 = − *k*2, *k*3 = − *k*4 it is straightforward to extract them from the well-studied photon polarization tensors in a constant field, and we use this fact for a check on our calculations.\nOff-shell legs can be used for creating internal propagators by sewing, or for connecting them to external fields. As an example for the former, we use our results to construct, by sewing of the legs 1 and 2 that carry full momentum, the two-loop scalar and spinor QED vacuum polarization tensors in the low-energy limit, see Fig. [beta].\n[beta] Construction of the two-loop photon propagator from the four-photon amplitude by sewing.\nIn this limit the vacuum polarization tensors just reduce to the induced Maxwell terms, from which we can rederive the two-loop *β*-function coefficients. Apart from providing another check, this also improves on previous work where the worldline formalism had already been applied to the calculation of these coefficients. Although these coefficients were already known for decades, that work was motivated by the fact that the formalism allows one to obtain parameter-integral representations that unify all the Feynman diagrams contributing to the quenched (single fermion loop) photon propagator at any loop order, see., e.g., Fig. [fig-3loopbetadiag] for the three-loop case.\n[fig-3loopbetadiag]\nThe interesting feature of the worldline formalism is that the sewing results in parameter integrals that represent not some particular Feynman diagram, but the whole set of Feynman diagrams shown in Fig. [fig-3loopbetadiag]. And it is precisely this type of sums of diagrams that are known for particularly extensive cancellations between diagrams. In 1967 Johnson, Willey and Baker showed that the quenched vacuum polarisation has, at any loop order, only a single overall UV divergence. And the coefficient of this pole, which is essentially the *β*-function, turned out to be rational up to the four-loop level (see and refs. therein). The role of gauge invariance in these cancellations is not transparent, and remains an object of active investigation even at the two-loop level. The worldline formalism was applied to the recalculation of the two-loop scalar and spinor QED *β*-functions by M.G. Schmidt and one of the authors in (see also ) to achieve significant cancellations already at the integrand level. However, those calculations were done using a different approach based on two-loop worldline Green’s functions, which provides a shortcut but obscures the role of gauge invariance. Only after that work a refinement of the usual homogeneising Bern-Kosower integration-by-parts procedure was found that led to a decomposition of the four-photon amplitudes into sixteen individually gauge invariant contributions (see and part I). This gives us a chance to improve on by asking the following two questions: first, can we identify parts of that decomposition that drop out in the construction of the two-loop *β* functions? Second, are there cancellations in their computation beyond what is expected from gauge invariance, that is, not only inside each gauge invariant structure, but also between them? Thus we will study here the distribution of the *β*-function coefficients, as well as the cancellation of the double pole in the $\\frac{1}{\\epsilon}$ expansion, in the gauge-invariant decomposition derived in part I.\nAs an application to an external-field problem, among the many processes related to the four-photon amplitudes we have chosen here to reanalyse low-energy Delbrück scattering, the deflection of photons in the Coulomb field of nuclei due to the vacuum polarization (partially motivated also by the fact that the worldline formalism has already been applied extensively to calculations involving constant, plane-wave and combinations of the two types of fields, but not as yet to Coulomb fields). Delbrück introduced this scattering in 1933 in order to explain the discrepancies Meitner and Kösters had found in their experiment for the Compton scattering on heavy atoms. Later Bethe and Rohrlich computed the angular distribution for small angles and the total cross section for Delbrück scattering. In 1973 DESY reported a first observation of this scattering in the high-energy, small-angle limit which was in agreement with the prediction made in a series of papers by *Cheng* and *Wu* and later confirmed in. In 1975 the Göttingen group performed another experiment which was the first one where exact predictions based on Feynman diagrams were confirmed with high precision after considering other background phenomena like atomic and nuclear Rayleigh scattering. The most accurate high energy experiment done so far on Delbrück scattering is the one carried out at BINP. Here we will use our results for a short and efficient calculation of the Delbrück scattering cross section in the low-energy limit.\nThis paper is organized as follows. In Section [WLrep] we shortly review the worldline representation of the scalar and spinor QED four-photon amplitudes, and discuss our general computational strategy in this paper. In particular, we provide all the integral formulas required to integrate out low-energy photons without having to split these multi-photon integrals into ordered sectors. The central Section is [2-low], where we list our explicit final results for the amplitudes, in *D* as well as in four dimensions, and for both scalar and spinor loops. Section [check] contains the above-mentioned comparison with the amplitudes extracted from the vacuum polarization in a constant field. In Section [section-beta] we recalculate the two-loop *β*-functions, while Section [delbruecksection] is devoted to low-energy Delbrück scattering. Finally, Section [summary] gives a summary and outline of future work. Supplementary information and formulas are provided in the appendices.\nWorldline representation of the four-photon amplitudes\n======================================================\nIn part I, we presented in detail the derivation and structure of the worldline representation of the *N*-photon amplitudes. Due to the inherent freedom in the integration-by-parts procedure, this representation comes in various slightly different versions. For easy reference, let us summarize here the representation of the four-photon amplitudes that we will actually use in the present sequel [1](#fn1):\n \n$$\\label{expand4point-Q}\n \\Gamma\\_{\\rm scal}(k\\_1,\\varepsilon\\_{1};\\cdots;k\\_4,\\varepsilon\\_4)\n =\\frac{(-ie)^{4}}{(4\\pi)^{\\frac{D}{2}}}\\int\\_{0}^{\\infty} \\frac{dT}{T}\\, T^{4-\\frac{D}{2}} \\e^{-m^2 T}\\, \\int\\_0^1\\prod\\_{i=1}^4du\\_i\\, Q\\_{{{\\rm scal}}}\\,\\e^{(\\cdot)}\\,.$$ \nHere we have already done the usual rescaling *τ**i* = *T**u**i* such that the exponential part is\n \n$$\\label{i-q4}\n \\e^{(\\cdot)} \\equiv \\e^{T\\sum\\_{i<j=1}^4G\\_{ij}k\\_i\\cdot k\\_j}\\,,$$ \nthe bosonic Green’s functions are\n \n*G**i**j* ≡ *G*(*u**i*, *u**j*) = |*u**i* − *u**j*| − (*u**i* − *u**j*)2 , \nand the polynomial $Q\\_{{{\\rm scal}}}$ is given by [2](#fn2) Qscal&=&Q4scal+Q3scal+Q2scal+Q22scal,Q4scal&=&(1234)+(2314)+(3124),Q3scal&=&(123)T(4)+(234)T(1)+(341)T(2)+(412)T(3),Q2scal&=&(12)Tsh(34)+(13)Tsh(24)+(14)Tsh(23) +(23)Tsh(14)+(24)Tsh(13)+(34)Tsh(12),Q22scal&=&(12)(34)+(13)(24)+(14)(23).[q4-scal] The extraordinary compactness of this representation is made possible through the introduction of the “Lorentz-cycle” *Z**n*(*i*1*i*2…*i**n*) as\n \n$$\\begin{aligned}\n\\begin{split}\nZ\\_2(ij)&\\equiv \n\\half {\\rm tr}\\bigl(f\\_if\\_j\\bigr) = \\varepsilon\\_i\\cdot k\\_j\\varepsilon\\_j\\cdot k\\_i - \\varepsilon\\_i\\cdot\\varepsilon\\_jk\\_i\\cdot k\\_j \\,,\n\\\\\nZ\\_n(i\\_1i\\_2\\ldots i\\_n)&\\equiv {\\rm tr}\n\\Bigl(\n\\prod\\_{j=1}^n\nf\\_{i\\_j}\\Bigr)\\,, \n\\quad (n\\geq 3) \\,,\n\\label{defZn}\n\\end{split}\\end{aligned}$$ \nwhere $f\\_i^{\\mu\\nu}=k\\_i^\\mu \\veps i^\\nu-k\\_i^\\nu\\veps i^\\mu$ is the field strength tensor of photon *i*, and the “bicycle” (i1i2in)&& i1i2i2i3ini1Zn(i1in). It is the “tails” that exist in various versions. For the present computation, we use the one-photon tail *T*(*i*) of the original *Q*-representation and the “short tail” *T**s**h*(*i**j*), introduced in part I, as the two-photon tail: T(i)&&riirir, \nTsh(ij) && r,si,j ri js . [two-tails] The spinor-loop result is obtained by employing the *Bern-Kosower replacement rule*, i.e. replacing simultaneously every closed (full) cycle as $\\Gd\\_{i\\_1i\\_2}\\Gd\\_{i\\_2i\\_3}\\cdots \\Gd\\_{i\\_ni\\_1}$ appearing in the integrand of the scalar-loop with i1i2i2i3ini1-GFi1i2GFi2i3GFini1 [replacement] where $G\\_{Fij}={\\rm sgn}(u\\_i-u\\_j)$ is the fermionic Green function (here it is understood that $\\dot G\\_{ij}=- \\dot G\\_{ji}$ may have to be used to achieve the cycle form). We write the spinor-loop amplitude as spin(k1,1;;k4,4) &=&-20 T4- -m2 T 01i=14dui Qspin (). [expand4point-Q-spin] Thus, apart from a global factor of − 2, the only difference to the scalar QED formula is the replacement of $Q\\_{\\rm scal}$ by $Q\\_{\\rm spin }$ according to the rule ([replacement]). Let us also emphasize once more that equations ([expand4point-Q]), ([expand4point-Q-spin]) are valid off-shell, and that the right-hand sides are manifestly finite term-by-term. The well-known spurious UV-divergences of the four-photon diagrams that usually cancel only in the sum of diagrams would show up here as logarithmic divergences of the *T*-integration at *T* = 0, but have been eliminated already at the beginning by the IBP procedure that led from the P-representation to the Q-representation, see part I.\nTo avoid carrying common prefactors we define\n \n$$\\hat{\\Gamma}\\_{\\left\\{{{\\rm scal}}\\atop{{\\rm spin}}\\right\\}}\n \\equiv \\int\\_{0}^{\\infty} \\frac{dT}{T}\\, T^{4-\\frac{D}{2}} \\e^{-m^2 T}\\, \\int\\_0^1\\prod\\_{i=1}^4du\\_i\\, Q\\_{\\left\\{{{\\rm scal}}\\atop{{\\rm spin}}\\right\\}} \\,\\e^{(\\cdot)}\\,.\n \\label{defGammahat}$$ \nCase distinction for the low energy limit of photons 3 and 4\n------------------------------------------------------------\nAs has been mentioned above, in the present part II the photons number 3 and 4 are taken in low-energy limit which means that in the expressions $Q\\_{\\rm scal/spin} \\,e^{(\\cdot)}$ we only consider those contributions which are linear in both *k*3 and *k*4 (for more details about the low-energy limit see part I). For the sake of clarity, in this subsection we list the different cases that appear in our calculations:\n* **Case 1**: Terms in $Q\\_{\\rm scal}$ which are already linear in both *k*3 and *k*4 do not need any further factors of *k*3, 4 from the exponential, so that we can simply replace $\\e^{(\\cdot)}\\rightarrow \\e^{(\\cdot)\\vert\\_{k\\_3,k\\_4\\rightarrow 0}}=e^{TG\\_{12}\\kk12}$. This includes all the “pure cycle” terms, defined by the absence of tails. For instance, the four-cycle $Q\\_{{\\rm scal}}^4$\n \n$$\\dot{G}(1234)\\,\\e^{(\\cdot)}\\rightarrow \\dot{G}(1234)\\e^{TG\\_{12}\\kk12}.$$ \nOr the two-two cycles $Q\\_{{\\rm scal}}^{22}$, where it is possible to have both low-energy photons in the same cycle\n \n$$\\Gd(12) \\Gd(34)\\, \\e^{(\\cdot)}\\rightarrow \\Gd(12) \\Gd(34)\\,\\e^{TG\\_{12}\\kk12},$$ \nor distributed among different cycles (13) (24) ()(13) (24)TG1212.\n* **Case 2**: Terms in $Q\\_{{\\rm scal}}^3$ which contain one of the low-energy momenta, say, *k*3, in the three-cycle, but lack *k*4. Those require an expansion of the exponential factor to linear order in *k*4. For instance, the one-tail term $\\Gd(123)T(4)\\,\\e^{(\\cdot)}$ gives (123)T(4) ()(123)TG1212.\n* **Case 3**: Terms in $Q\\_{\\rm scal}^3$ which have two low-energy momenta in the three-cycle, for those we neglect terms in the one-tail that are linear in any of the two low-energy momenta. For instance, the one-tail term $\\Gd(234)T(1)\\,\\e^{(\\cdot)}$ gives (234)T(1) ()(234)1212 TG1212. However, it turns out that the contributions of these particular terms vanish after integration over *u*3 and *u*4.\n* **Case 4**: Terms in $Q\\_{{\\rm scal}}^2$ which lack both *k*3 and *k*4. Such terms occur only in the two-tail term $\\Gd(12)T\\_{\\rm sh}(34)\\,\\e^{(\\cdot)}$. Here the exponential factor must be expanded to linear order in both *k*3 and *k*4, (12)Tsh(34) ()(12)Tsh(34) ( TG3434 + T2i,j=12 Gi44 Gj33 ) TG1212.\n* **Case 5**: Terms in $Q\\_{{\\rm scal}}^2$ which have only one of the low-energy momenta in the two-cycle. For instance, the two-tail term $\\Gd(13)T\\_{\\rm sh}(24)\\,\\e^{(\\cdot)}$ gives (13)Tsh(24) ()(13) 1241 TG1212 where we took the terms linear in *k*4 from the exponential.\n* **Case 6**: Terms in $Q\\_{{\\rm scal}}^2$ which are already quadratic in at least one of the low-energy momenta can be discarded. For instance, in the two-tail term $\\Gd(34)T\\_{\\rm sh}(12)\\,\\e^{(\\cdot)}$ we have terms like (34)3124()0. In particular, all terms in $\\Gd(34)T\\_{\\rm sh}(12)\\e^{(\\cdot)}$ can be neglected.\nThus we see that, in the limit of two low-energy photons which is our object of interest in this paper, there are many terms that drop out of the amplitude at the integrand level, and others that drop out after integration. The transition to spinor QED does not lead to any new considerations.\nLow-energy limit for two of the photons: computational examples\n---------------------------------------------------------------\nNext, let us explain our strategy for computing the four-photon amplitude with two low-energy photons, using some sample terms from both $Q\\_{{{\\rm scal}}}$ and $Q\\_{{{\\rm spin}}}$.\n**Example 1**: Let us consider the following one-tail term:\n \n$$\\label{q43,123}\n\\begin{split}\nQ\\_{{{\\rm scal}}}^3(123;4) =&\\textbf{ }\\dot{G}\\_{12} \\dot{G}\\_{23} \\dot{G}\\_{31}Z\\_3(123) \\left( \\dot{G}\\_{41} \\varepsilon\\_4 \\cdot k\\_1+\\dot{G}\\_{42} \\varepsilon\\_4 \\cdot k\\_2 + \\dot{G}\\_{43} \\varepsilon\\_4 \\cdot k\\_3 \\right).\n\\end{split}$$ \nIn order to take the low-energy limit for leg 4, we define\n \n$$\\label{Q3(4)}\nQ^{3}\\_{{{\\rm scal}}(4)} (123;4) \\equiv \\left. \\int\\_0^1 du\\_4 ~Q\\_{{{\\rm scal}}}^3(123;4)~ \\e^{(\\cdot 4)} \\right|\\_{{\\rm lin}~ k\\_4}\\,,$$ \nwhere in the full exponent we single out those terms which have a *k*4 [e(.4)] (4) T(G1414+G24 24+G3434), such that () = (4)T(G1212+G1313+ G2323). Here, for convenience, we set the following convention for low-energy legs: on the left-hand side of expression ([Q3(4)]) the subscript ‘(4)’ indicates that the leg number 4 with momentum *k*4 is to be taken in the low energy limit and integrated out. In [app2] we give a list of all the occurring integrals (up to permutations). It allows us to perform the integral over *u*4 without fixing an ordering for the remaining photon legs (for efficient techniques to derive such formulas see, e.g., ).\nSince in this example we do not have linear terms in *k*4, we take them from the exponential part and we use Eq. ([intGs]) to integrate\n \n$$\\begin{split}\nQ\\_{{{\\rm scal}}(4)}^3(123;4) & = \\frac{T}{3} Z\\_3(123)\\, \\dot{G}\\_{12} \\dot{G}\\_{23} \\dot{G}\\_{31}\\left(\\sum\\_{i=1}^3 \\dot{G}\\_{1i}G\\_{i1}\\, k\\_i \\cdot k\\_4 \\varepsilon\\_4 \\cdot k\\_1\\right.\\\\\n& \\quad +\\sum\\_{i=1}^3 \\left.\\dot{G}\\_{2i}G\\_{i2} \\, k\\_i \\cdot k\\_4 \\varepsilon\\_4 \\cdot k\\_2 + \\sum\\_{i=1}^3 \\dot{G}\\_{3i}G\\_{i3} \\, k\\_i \\cdot k\\_4 \\varepsilon\\_4 \\cdot k\\_3\\right) \\,.\n\\end{split}$$ \nCombining terms, we can write the result in a manifestly gauge invariant form\n \n$$\\begin{split}\nQ\\_{{{\\rm scal}}(4)}^3(123;4) &= \\frac{T}{3} Z\\_3(123) \\dot{G}\\_{12} \\dot{G}\\_{23} \\dot{G}\\_{31}\\Big(\\dot{G}\\_{12}G\\_{21} \\, k\\_2 \\cdot f\\_4 \\cdot k\\_1 \\\\\n&\\quad + \\dot{G}\\_{23}G\\_{32}\\, k\\_3 \\cdot f\\_4 \\cdot k\\_2 + \\dot{G}\\_{31}G\\_{13}\\,k\\_1 \\cdot f\\_4 \\cdot k\\_3 \\Big) \\,.\\\\\n\\end{split}\n\\label{q43scalar}$$ \nNext we repeat all this with photon number 3. We define\n \n$$\\begin{split}\n \\label{Q3(34)}\n {Q\\_{{\\rm scal}(34)}}^{3} (123;4) &\\equiv \\left. \\int\\_0^1 du\\_3 \\int\\_0^1 du\\_4 ~Q\\_{{{\\rm scal}}}^3(123;4) ~ \\e^{(\\cdot 3)}\\e^{(\\cdot 4)} \\right|\\_{{\\rm lin}~ k\\_4, k\\_3}\\\\\n &= \\left. \\int\\_0^1 du\\_3 ~Q\\_{{{\\rm scal}}(4)}^3(123;4) ~ \\e^{(\\cdot 3)} \\right|\\_{{\\rm lin}~ k\\_3}\\,,\n \\end{split}$$ \nwhere\n \n$$\\label{.3}\n \\e^{(\\cdot 3)} \\equiv \\e^{T(G\\_{13}\\kk13+G\\_{23}\\kk23)}\\,.$$ \nSimilarly to the above, the subscript ‘(34)’ in equation ([Q3(34)]) indicates that legs number 3 and 4 are both projected on their low energy limits. Note that some terms in ([q43scalar]) are quadratic in *k*3 and can be neglected. We use ([intGdots]) to perform the integral over *u*3 and, with the aid of the identity $\\dot G\\_{ij}^2 = 1-4G\\_{ij}$, write the result as\n \n$${Q\\_{{\\rm scal}(34)}}^3(123;4) = -\\frac{T}{9} Z\\_3(123) k\\_2 \\cdot f\\_4 \\cdot k\\_1 \\left(G\\_{12} - 10 G\\_{12}^2 + 24 G\\_{12}^3\\right).\n\\label{example1}$$ \nTherefore at this stage the contribution of $Q\\_{{{\\rm scal}}}^3(123;4)$ to the four-photon amplitude with legs 3 and 4 at low energy is given by\n \n$$\\label{ynl1}\n\\hat{\\Gamma}\\_{{\\rm scal}(34)}^3(123;4) =\\int\\_{0}^{\\infty} \\frac{dT}{T}\\, T^{4-\\frac{D}{2}}\\e^{-m^2 T}\\,\\int\\_0^1 du\\_1 du\\_2\\, {Q\\_{{\\rm scal}(34)}}^3(123;4)\\,\\e^{TG\\_{12} k\\_1\\cdot k\\_2}\\,.$$ \nThis leads us to define\n \n$$\\label{defYnl}\n\\begin{split}\nY\\_{nl} \\equiv \\int\\_0^{\\infty} \\frac{dT}{T}\\, T^{n-D/2} ~\\int\\_0^1 du\\_1\\int\\_0^1 du\\_2 ~ G\\_{12}^l \\e^{-T[m^2 - G\\_{12} k\\_1 \\cdot k\\_2]} \\,.\n\\end{split}$$ \nThe proper-time integral *T* is elementary and, due to the unbroken translation invariance along the loop, one of the two parameters *u*1, *u*2 can be fixed at some arbitrary value, such as *u*2 = 0 and *u*1 = *u*. To the remaining *u* integral we apply a tensor reduction procedure, explained in [B.2], to arrive at the following expression,\n \n$$\\label{Ynl}\n\\begin{split}\nY\\_{nl} = \\frac{\\Gamma\\left(n-\\frac{D}{2}-l\\right)}{m^{2n-D} } ~\\frac{d^l}{d\\hat{k}\\_{12}^l} ~ {}\\_2F\\_1 \\left(1,n-l-\\frac{D}{2}; \\frac{3}{2};\\frac{\\hat{k}\\_{12}}{4} \\right),\n\\end{split}$$ \nwhere $\\hat{k}\\_{12} \\equiv \\frac{k\\_1\\cdot k\\_2}{m^2}$ and 2*F*1(*a*, *b*; *c*; *x*) is the Gauss hypergeometric function. With these definitions, and ([ynl1]) transform into\n \n$$\\hat{\\Gamma}\\_{{\\rm scal}(34)}^3(123;4) = -\\frac{1}{9}Z\\_3(123)\\, k\\_2 \\cdot f\\_4 \\cdot k\\_1 \\left(Y\\_{51} - 10 Y\\_{52} + 24 Y\\_{53}\\right) \\,.\n\\label{example1scal}$$ \nProceeding to the spinor QED case, since for the term at hand the integration over *u*4 does not interfere with the application of the Bern-Kosower replacement rule, the term corresponding to ([q43scalar]) reads\nQspin(4)3(123;4) &= (12 23 31-GF12GF23GF31)Z3(123) \n&(12G21k2 f4 k1 + 23G32k3 f4 k2 + 31G13k1 f4 k3 ). [example1spinor]\nIt would seem that we now have to extend our table of integrals in [app2] to integrals involving both $\\dot G\\_{ij}$ and *G**F**i**j*, but we can avoid this by eliminating the latter in favor of the former. This can be done using the basic identities GFij2 =1, GFijGFjkGFki = -ij-jk-ki [idbasic] which transforms into\nQspin(4)3(123;4) &=(12 23 31+12+23+31)Z3(123) \n&(12G21k2 f4 k1 + 23G32k3 f4 k2 + 31G13k1 f4 k3 ). [q43spin]\nProceeding as in the scalar case, one finds that the spinor-QED equivalent of becomes spin(34)3(123;4) &=& Z3(123) k2 f4 k1(Y51 -Y52 - 12Y53), as the reader can easily verify using formulas of [app2].\n**Example 2**: Let us consider another one-tail term,\n \n$$\\label{qf1}\n\\begin{split}\nQ\\_{{{\\rm scal}}}^3(234;1) & = Z\\_3(234)\\, \\dot{G}\\_{23} \\dot{G}\\_{34} \\dot{G}\\_{42} \\left( \\dot{G}\\_{12} \\varepsilon\\_1 \\cdot k\\_2+\\dot{G}\\_{13} \\varepsilon\\_1 \\cdot k\\_3 + \\dot{G}\\_{14} \\varepsilon\\_1 \\cdot k\\_4 \\right) \\,.\n\\end{split}$$ \nNotice that it has a quadratic term in *k*4 and two linear terms in *k*4. Omitting the former, and integration the latter over *u*4 using ([intGdots]), we obtain\n \n$$Q\\_{{{\\rm scal}}(4)}^3(234;1) = Z\\_3(234)\\, \\dot{G}\\_{23} \\left( \\frac{1}{6}-\\frac{1}{2}\\dot{G}\\_{32}^2 \\right) \\left( \\dot{G}\\_{12} \\varepsilon\\_1 \\cdot k\\_2+\\dot{G}\\_{13} \\varepsilon\\_1 \\cdot k\\_3 \\right)\\,.$$ \nThus we have a term quadratic in *k*3 and a term linear in *k*3, and for the latter the integral over *u*3 turns out to vanish. Therefore we get nothing,\n \n$${{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^3(234;1) = 0\\,.$$ \nWe leave it to the reader to check that the additional terms generated by applying the Bern-Kosower replacement rule to the (234) cycle vanish, too. Therefore\n \n$${{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^3(234;1) = 0\\,.$$ \n**Example 3**: As our final example, we consider the following 4-cycle term in the spinor QED integrand,\n \n$$Q\\_{{{\\rm spin}}}^4(1234) = Z\\_4(1234) \\Big(\\dot{G}\\_{12} \\dot{G}\\_{23} \\dot{G}\\_{34} \\dot{G}\\_{41} -G\\_{F12} G\\_{F23} G\\_{F34} G\\_{F41} \\Big) \\,.$$ \nTo reexpress the spin contribution *G**F*12*G**F*23*G**F*34*G**F*41, we insert a factor 1 = − *G**F*13*G**F*31, and once more use the identity $G\\_{Fij}G\\_{Fjk}G\\_{Fki} = -\\dot{G}\\_{ij}-\\dot{G}\\_{jk}-\\dot{G}\\_{ki}$. This leads to\n \n$$\\begin{split}\nQ\\_{{{\\rm spin}}(4)}^4(1234) &= Z\\_4(1234) \\int\\_0^1 du\\_4 \\Big[\\dot{G}\\_{12} \\dot{G}\\_{23} \\dot{G}\\_{34} \\dot{G}\\_{41} \n+ \\left(\\dot{G}\\_{12} + \\dot{G}\\_{23} + \\dot{G}\\_{31} \\right)\\left( \\dot{G}\\_{13} + \\dot{G}\\_{34} + \\dot{G}\\_{41} \\right) \\Big] \\,.\n\\end{split}$$ \nUpon integration we find, after some cancellations,\n \n$$\\begin{split}\nQ\\_{{{\\rm spin}}(4)}^4(1234) = Z\\_4(1234) \\left[\\dot{G}\\_{12} \\dot{G}\\_{23}\\left( \\frac{1}{6}-\\frac{1}{2}\\dot{G}\\_{31}^2 \\right) + \\left(\\dot{G}\\_{12} + \\dot{G}\\_{23} + \\dot{G}\\_{31} \\right)\\dot{G}\\_{13}\\right]\\,.\n\\end{split}$$ \nIntegrating over *u*3 we obtain\n \n$$Q\\_{{{\\rm spin}}(34)}^4(1234) = - \\frac{4}{3} Z\\_4(1234) \\left(G\\_{12}+2G\\_{12}^2 \\right) \\,.$$ \nThus with the notation ([defYnl]), the contribution to the amplitude of this term reads\n \n$${{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^4(1234) = - \\frac{4}{3} Z\\_4(1234) \\left(Y\\_{41}+2Y\\_{42} \\right) \\,.$$ \nResults: low-energy limit for two of the photons\n================================================\nIn this section, we present our results for the complete four-photon amplitudes in both scalar and spinor QED, off-shell but under the restriction that photons 3 and 4 are taken in the low-energy limit i. e., we consider only contributions that are linear in *k*3 and *k*4 (see part I). These amplitudes represent the main result of the present article. In the worldline formalism, they appear naturally decomposed as\n \n$$\\hat \\Gamma\\_{\\left\\{{{\\rm scal}}\\atop{{\\rm spin}}\\right\\}(34)} = \n \\hat \\Gamma\\_{\\left\\{{{\\rm scal}}\\atop{{\\rm spin}}\\right\\}(34)}^4 \n +\n \\hat \\Gamma\\_{\\left\\{{{\\rm scal}}\\atop{{\\rm spin}}\\right\\}(34)}^3\n +\n \\hat \\Gamma\\_{\\left\\{{{\\rm scal}}\\atop{{\\rm spin}}\\right\\}(34)}^2\n +\n \\hat \\Gamma\\_{\\left\\{{{\\rm scal}}\\atop{{\\rm spin}}\\right\\}(34)}^{22}\n \\label{decomp}$$ \nwith\n \n$$\\label{Intu1u2T}\n \\hat \\Gamma\\_{\\left\\{{{\\rm scal}}\\atop{{\\rm spin}}\\right\\}(34)}^i\n = \\int\\_0^1 du\\_1 \\int\\_0^1 du\\_2 \\int\\_0^{\\infty} \\frac{dT}{T}\\, T^{4-\\frac{D}{2}}\\e^{-T(m^2 - G\\_{12} k\\_1 \\cdot k\\_2)} \n Q\\_{\\left\\{{{\\rm scal}}\\atop{{\\rm spin}}\\right\\}(34)}^i\n ~~,~~i=4,3,2,22\\,.~~~$$ \nRecall that the subscript (34) in $\\hat\\Gamma\\_{(34)}^i$ indicates that the integrals over *u*3 and *u*4 were performed taking the low-energy limit for the corresponding photons. The absolute normalizations of the amplitudes are given by scal(34)(k1,1; ;k4,4) &=& scal(34), \nspin(34)(k1,1; ;k4,4) &=&-2 spin(34). [expand4point-34]\nWe give them first in dimensionally regularized form, and then in *D* = 4. Since the worldline representation of these amplitudes is manifestly finite, taking this limit is trivial and will not involve a cancellation of spurious 1/*ε* poles as would be the case in a Feynman diagram calculation (in our conventions *ε* = *D* − 4).\nDimensionally regularized amplitudes\n------------------------------------\nThe integrals appearing in ([Intu1u2T]) are all of the form of ([defYnl]) so that we choose to express them in terms of the functions *Y**n**l* that have been given in terms of 2*F*1 in ([Ynl]). We note that this implies some redundancy since, as shown in [B.2], there exist recursion relations between the *Y**n**l* functions that would also allow us to rewrite all our results entirely in terms of the *Y**n*0. However, the resulting representation would be less compact than the one given in the following.\n### Scalar QED\nFor $Q\\_{{{\\rm scal}}}^4$ scal(34)4(1234) &=& Z4(1234) (Y41 -4 Y42 ), \nscal(34)4(2314) &=& Z4(2314) (Y40-12 Y41 + 36 Y42 ), \nscal(34)4(3124) &=& Z4(3124) (Y41-4 Y42 ). For $Q\\_{{{\\rm scal}}}^3$ scal(34)3(123;4) &=& -Z3(123) k2 f4 k1 (Y51 - 10 Y52 + 24 Y53), \nscal(34)3(234;1) &=& 0, \nscal(34)3(341;2) &=& 0, \nscal(34)3(412;3) &=&- Z3(412) k2 f3 k1 (Y51 - 10 Y52 + 24 Y53). For $Q\\_{{{\\rm scal}}}^2$\n \n$$\\begin{split}\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^2(12;34) &= -\\frac{1}{18} Z\\_2(12) \\Bigg\\{ \\left[\\frac{1}{5}\\left(Y\\_{50}- 4Y\\_{51} \\right) - 6 \\left(Y\\_{52}- 4Y\\_{53} \\right)\\right] k\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot k\\_2 \\\\\n & \\quad + \\frac{1}{5}\\left(Y\\_{50}- 4Y\\_{51} \\right) k\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot k\\_1 +(1\\leftrightarrow 2) \\Bigg\\}\\\\\n & \\quad -\\frac{1}{9} Z\\_2(12) \\left(Y\\_{62}- 8Y\\_{63} + 16 Y\\_{64} \\right) k\\_1 \\cdot f\\_3 \\cdot k\\_2 k\\_1 \\cdot f\\_4 \\cdot k\\_2 \\,,\n \\end{split}$$ \n \n$${{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^2(13;24) + {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^2(23;14) = -\\frac{1}{9} Z\\_2(13) k\\_1 \\cdot f\\_2 \\cdot f\\_4 \\cdot k\\_1 \\, \\left(Y\\_{51}- 4Y\\_{52} \\right) + (1\\leftrightarrow 2)\\,,$$ \n \n$${{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^2(14;23) + {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^2(24;13)= - \\frac{1}{9} Z\\_2(14) k\\_1 \\cdot f\\_2 \\cdot f\\_3 \\cdot k\\_1 \\, \\left(Y\\_{51}- 4Y\\_{52} \\right) + (1\\leftrightarrow 2)\\,,$$ \n \n$${{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^2(34;12) = 0\\,.$$ \nAnd finally for $Q\\_{{{\\rm scal}}}^{22}$ scal(34)22(12,34) &=& Z2(12) Z2(34) (Y40-4Y41 ), \nscal(34)22(13,24) &=&Z2(13)Z2(24) Y40, \nscal(34)22(14,23) &=&Z2(14)Z2(23) Y40.\n### Spinor QED\nFor $Q\\_{{{\\rm spin}}}^4$ spin(34)4(1234) &=& - Z4(1234) (Y41+2Y42 ), \nspin(34)4(2314) &=& - Z4(2314) (2Y40- 6Y41 - 9 Y42 ), \nspin(34)4(3124) &=& - Z4(3124) (Y41+2Y42 ). For $Q\\_{{{\\rm spin}}}^3$ spin(34)3(123;4) &=& Z3(123) k2 f4 k1(Y51 -Y52 - 12Y53), \nspin(34)3(234;1) &=& 0, \nspin(34)3(341;2) &=& 0, \nspin(34)3(412;3) &=& Z3(412) k2 f3 k1 (Y51 - Y52 - 12Y53). For $Q\\_{{{\\rm spin}}}^2$\n \n$$\\begin{split}\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^2(12;34) &= \\frac{2}{9} Z\\_2(12) \\, \\Bigg\\{\\left[\\frac{1}{5}Y\\_{51}k\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot k\\_1 + \\left(\\frac{1}{5}Y\\_{51} - 6 Y\\_{53}\\right)k\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot k\\_2 + (1\\leftrightarrow 2) \\right]\\\\\n & \\quad + 2\\, \\left(Y\\_{63}- 4Y\\_{64} \\right) k\\_1 \\cdot f\\_3 \\cdot k\\_2 k\\_1 \\cdot f\\_4 \\cdot k\\_2 \\Bigg\\}\\,,\n \\end{split}$$ \n \n$${{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^2(13;24) + {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^2(23;14) = \\frac{2}{9} Z\\_2(13) k\\_1 \\cdot f\\_2 \\cdot f\\_4 \\cdot k\\_1 \\, \\left(Y\\_{51}- 4Y\\_{52} \\right) + (1\\leftrightarrow 2) \\,,$$ \n \n$${{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^2(14;23) + {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^2(24;13)= \\frac{2}{9} Z\\_2(14) k\\_1 \\cdot f\\_2 \\cdot f\\_3 \\cdot k\\_1 \\, \\left(Y\\_{51}- 4Y\\_{52} \\right) + (1\\leftrightarrow 2) \\,,$$ \n \n$${{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^2(34;12) = 0\\,.$$ \nAnd finally for $Q\\_{{{\\rm spin}}}^{22}$\n \n$${{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^{22}(12,34) = \\frac{8}{3} Z\\_2(12) Z\\_2(34) Y\\_{41}\\,,$$ \n \n$${{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^{22}(13,24) =\\frac{4}{9} Z\\_2(13)Z\\_2(24)Y\\_{40}\\,,$$ \n \n$${{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^{22}(14,23) =\\frac{4}{9}Z\\_2(14)Z\\_2(23) Y\\_{40}\\,.$$ \nFour-dimensional amplitudes\n---------------------------\nIn this subsection we specialize the previous results to four space-time dimensions (*D* = 4), after which it becomes possible to write the amplitudes completely in terms of elementary and trigonometric functions. In the interest of compactness, it will now be useful to introduce the following dimensionless variables, [4-dim] 12=, p0.\n### Scalar QED\nFor $Q\\_{{{\\rm scal}}}^4$\n \n$$\\begin{split}\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^4(1234) &= -\\frac{12 + 8({{\\hat{k}\\_{12}}}-6)p\\_0}{3m^4 {{\\hat{k}\\_{12}}}^2} Z\\_4(1234)\\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^4(2314) &= -\\frac{2({{\\hat{k}\\_{12}}}^2 -30{{\\hat{k}\\_{12}}}+108)-8p\\_0(5{{\\hat{k}\\_{12}}}^2 - 48{{\\hat{k}\\_{12}}}+ 108)}{9m^4({{\\hat{k}\\_{12}}}-4) {{\\hat{k}\\_{12}}}^2} Z\\_4(2314)\\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^4(3124) &= -\\frac{12 + 8({{\\hat{k}\\_{12}}}-6)p\\_0}{3m^4 {{\\hat{k}\\_{12}}}^2} Z\\_4(3124)\\,. \n \\end{split}$$ \nFor $Q\\_{{{\\rm scal}}}^3$\n \n$$\\begin{split}\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^3(123;4) &= -\\frac{2({{\\hat{k}\\_{12}}}^2 -48{{\\hat{k}\\_{12}}}+180)-16p\\_0(4{{\\hat{k}\\_{12}}}^2 - 39{{\\hat{k}\\_{12}}}+ 90)}{9m^6({{\\hat{k}\\_{12}}}-4) {{\\hat{k}\\_{12}}}^3} Z\\_3(123) k\\_2 \\cdot f\\_4 \\cdot k\\_1 \\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^3(412;3) &= -\\frac{2({{\\hat{k}\\_{12}}}^2 -48{{\\hat{k}\\_{12}}}+180)-16p\\_0(4{{\\hat{k}\\_{12}}}^2 - 39{{\\hat{k}\\_{12}}}+ 90)}{9m^6({{\\hat{k}\\_{12}}}-4) {{\\hat{k}\\_{12}}}^3} Z\\_3(412) k\\_2 \\cdot f\\_3 \\cdot k\\_1\\,.\n \\end{split}$$ \nFor $Q\\_{{{\\rm scal}}}^2$\n \n$$\\begin{split}\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^2(12;34) &= \\frac{-1}{90} Z\\_2(12)\\Bigg\\{\\Bigg[ \\frac{2(-{{\\hat{k}\\_{12}}}+ 2)-16p\\_0}{m^6({{\\hat{k}\\_{12}}}-4) {{\\hat{k}\\_{12}}}} \\, k\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot k\\_1 \\\\\n & + P\\_1 \\, k\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot k\\_2 + (1\\leftrightarrow 2)\\Bigg] + 10 P\\_2 ~ k\\_2 \\cdot f\\_4 \\cdot k\\_1 k\\_2 \\cdot f\\_3 \\cdot k\\_1 \\Bigg\\}\\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^2(13;24) &= -\\frac{2({{\\hat{k}\\_{12}}}-6)-16p\\_0({{\\hat{k}\\_{12}}}-3)}{9m^6 ({{\\hat{k}\\_{12}}}-4){{\\hat{k}\\_{12}}}^2} Z\\_2(13) k\\_1 \\cdot f\\_2 \\cdot f\\_4 \\cdot k\\_1\\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^2(23;14) &= -\\frac{2({{\\hat{k}\\_{12}}}-6)-16p\\_0({{\\hat{k}\\_{12}}}-3)}{9m^6 ({{\\hat{k}\\_{12}}}-4){{\\hat{k}\\_{12}}}^2} Z\\_2(23) k\\_2 \\cdot f\\_1 \\cdot f\\_4 \\cdot k\\_2 \\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^2(14;23) &= -\\frac{2({{\\hat{k}\\_{12}}}-6)-16p\\_0({{\\hat{k}\\_{12}}}-3)}{9m^6 ({{\\hat{k}\\_{12}}}-4){{\\hat{k}\\_{12}}}^2} Z\\_2(14) k\\_1 \\cdot f\\_2 \\cdot f\\_3 \\cdot k\\_1 \\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^2(24;31) &= -\\frac{2({{\\hat{k}\\_{12}}}-6)-16p\\_0({{\\hat{k}\\_{12}}}-3)}{9m^6 ({{\\hat{k}\\_{12}}}-4){{\\hat{k}\\_{12}}}^2} Z\\_2(24) k\\_2 \\cdot f\\_1 \\cdot f\\_3 \\cdot k\\_2 \\,.\n \\end{split}$$ \nAnd finally, for $Q\\_{{{\\rm scal}}}^{22}$\n \n$$\\begin{split}\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^{22}(12,34) &= -\\frac{2-8p\\_0}{3m^4 {{\\hat{k}\\_{12}}}} Z\\_2(12) Z\\_2(34)\\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^{22}(13,24) &= -\\frac{2+8p\\_0}{9m^4({{\\hat{k}\\_{12}}}-4)} Z\\_2(13) Z\\_2(24)\\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm scal}(34)}}}^{22}(14,23) &= -\\frac{2+8p\\_0}{9m^4({{\\hat{k}\\_{12}}}-4)} Z\\_2(14) Z\\_2(23)\\,.\n \\end{split}$$ \nHere we have introduced two more functions, *P**i* = *P**i*(*k̂*12) with *i* = 1, 2, which are defined as\n \n$$P\\_1 \\equiv \\frac{2(- {{\\hat{k}\\_{12}}}^3 +2 {{\\hat{k}\\_{12}}}^2 -210{{\\hat{k}\\_{12}}}+ 900 ) -32p\\_0(8 {{\\hat{k}\\_{12}}}^2 -90{{\\hat{k}\\_{12}}}+225)}{m^6({{\\hat{k}\\_{12}}}-4) {{\\hat{k}\\_{12}}}^3}\\,,$$ \n \n$$P\\_2 \\equiv \\frac{4(-{{\\hat{k}\\_{12}}}^2 + 55{{\\hat{k}\\_{12}}}- 210) + 48p\\_0(3{{\\hat{k}\\_{12}}}^2 - 30{{\\hat{k}\\_{12}}}+ 70)}{m^8({{\\hat{k}\\_{12}}}-4) {{\\hat{k}\\_{12}}}^4}\\,.$$ \n### Spinor QED\nFor $Q\\_{{{\\rm spin}}}^4$\n \n$$\\begin{split}\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^4(1234) &= \\frac{16[3+p\\_0({{\\hat{k}\\_{12}}}^2 + 2{{\\hat{k}\\_{12}}}- 12 )]}{3m^4({{\\hat{k}\\_{12}}}-4) {{\\hat{k}\\_{12}}}^2} Z\\_4(1234)\\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^4(2314) &= \\frac{4[4{{\\hat{k}\\_{12}}}^2 -3{{\\hat{k}\\_{12}}}-54-8p\\_0({{\\hat{k}\\_{12}}}^2 + 3{{\\hat{k}\\_{12}}}- 27 )]}{9m^4({{\\hat{k}\\_{12}}}-4) {{\\hat{k}\\_{12}}}^2} Z\\_4(2314)\\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^4(3124) &= \\frac{16[3+p\\_0({{\\hat{k}\\_{12}}}^2 + 2{{\\hat{k}\\_{12}}}- 12 )]}{3m^4({{\\hat{k}\\_{12}}}-4) {{\\hat{k}\\_{12}}}^2} Z\\_4(3124)\\,. \n \\end{split}$$ \nFor $Q\\_{{{\\rm spin}}}^3$\n \n$$\\begin{split}\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^3(123;4) &= \\frac{4({{\\hat{k}\\_{12}}}^2 +15{{\\hat{k}\\_{12}}}-90)+16p\\_0({{\\hat{k}\\_{12}}}^2 - 30{{\\hat{k}\\_{12}}}+90 )}{9m^6({{\\hat{k}\\_{12}}}-4){{\\hat{k}\\_{12}}}^3} Z\\_3(123) k\\_2 \\cdot f\\_4 \\cdot k\\_1 \\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^3(412;3) &= \\frac{4({{\\hat{k}\\_{12}}}^2 +15{{\\hat{k}\\_{12}}}-90)+16p\\_0({{\\hat{k}\\_{12}}}^2 - 30{{\\hat{k}\\_{12}}}+90 )}{9m^6({{\\hat{k}\\_{12}}}-4){{\\hat{k}\\_{12}}}^3} Z\\_3(412) k\\_2 \\cdot f\\_3 \\cdot k\\_1\\,.\n \\end{split}$$ \nFor $Q\\_{{{\\rm spin}}}^2$\n \n$$\\begin{split}\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^2(12;34) &= \\frac{1}{3} Z\\_2(12)\\Bigg\\{\\Bigg[ \\frac{4({{\\hat{k}\\_{12}}}+2) + 32p\\_0({{\\hat{k}\\_{12}}}-1)}{15m^6(4-{{\\hat{k}\\_{12}}})^2{{\\hat{k}\\_{12}}}}\\, k\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot k\\_1 \\\\ \n &+\\frac{2}{15}\\tilde{P}\\_1 \\, k\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot k\\_2 + (1\\leftrightarrow 2) \\Bigg] + \\frac{4}{3}\\tilde{P}\\_2\\, k\\_2 \\cdot f\\_4 \\cdot k\\_1 k\\_2 \\cdot f\\_3 \\cdot k\\_1 \\Bigg\\}\\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^2(13;24) &= \\frac{4({{\\hat{k}\\_{12}}}-6)-32p\\_0({{\\hat{k}\\_{12}}}-3)}{9m^6 ({{\\hat{k}\\_{12}}}-4){{\\hat{k}\\_{12}}}^2} Z\\_2(13) k\\_1 \\cdot f\\_2 \\cdot f\\_4 \\cdot k\\_1\\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^2(23;14) &= \\frac{4({{\\hat{k}\\_{12}}}-6)-32p\\_0({{\\hat{k}\\_{12}}}-3)}{9m^6 ({{\\hat{k}\\_{12}}}-4){{\\hat{k}\\_{12}}}^2} Z\\_2(23) k\\_2 \\cdot f\\_1 \\cdot f\\_4 \\cdot k\\_2 \\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^2(14;23) &= \\frac{4({{\\hat{k}\\_{12}}}-6)-32p\\_0({{\\hat{k}\\_{12}}}-3)}{9m^6 ({{\\hat{k}\\_{12}}}-4){{\\hat{k}\\_{12}}}^2} Z\\_2(14) k\\_1 \\cdot f\\_2 \\cdot f\\_3 \\cdot k\\_1\\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^2(24;31) &= \\frac{4({{\\hat{k}\\_{12}}}-6)-32p\\_0({{\\hat{k}\\_{12}}}-3)}{9m^6 ({{\\hat{k}\\_{12}}}-4){{\\hat{k}\\_{12}}}^2} Z\\_2(24) k\\_2 \\cdot f\\_1 \\cdot f\\_3 \\cdot k\\_2\\,.\n \\end{split}$$ \nAnd finally, for $Q\\_{{{\\rm spin}}}^{22}$\n \n$$\\begin{split}\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^{22}(12,34)&= -\\frac{16[1+2p\\_0({{\\hat{k}\\_{12}}}-2)]}{3m^4 ({{\\hat{k}\\_{12}}}-4){{\\hat{k}\\_{12}}}} Z\\_2(12) Z\\_2(34)\\,,\\\\\n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^{22}(13,24) &=-\\frac{8(1+4p\\_0)}{9m^4({{\\hat{k}\\_{12}}}-4)} Z\\_2(13)Z\\_2(24)\\,,\\\\ \n {{\\hat{\\Gamma}\\_{{\\rm spin}(34)}}}^{22}(14,23) &=-\\frac{8(1+4p\\_0)}{9m^4({{\\hat{k}\\_{12}}}-4)} Z\\_2(14)Z\\_2(23)\\,,\n \\end{split}$$ \nHere, we have defined *P̃**i* = *P̃**i*(*k̂*12) as 1, 2.\nCheck: the photon propagator in a constant field\n================================================\nThe kinematical regime of the four-photon amplitudes that we are studying in this paper has, to the best of our knowledge, not been treated in the literature before. However, for the special case where *k*1 = − *k*2 (which then also implies *k*3 = − *k*4) they can be easily extracted from the photon polarization tensor in a constant background field, a quantity that has been intensely studied in spinor and to a lesser degree in scalar QED. The field can be treated non-perturbatively using the exact Dirac or Klein-Gordon propagator in the field, indicated by the double line on the left-hand side of Fig. [vacuum] as is customary.\n[vacuum] The first three terms in the diagrammatic expansion of the photon polarization tensor in powers of the constant background field. The second diagram on the right-hand side represents the four-photon amplitude with two low-energy photons and *k*1 = − *k*2.\nExpanding out the full non-perturbative polarization tensor in powers of the background field one obtains the series depicted on the right-hand side of Fig. [vacuum], containing any even number of interactions with the field. Since a constant field cannot inject energy or momentum into the loop, these interactions are mediated by zero-momentum photons. Thus the second diagram on the right-hand side of Fig. [vacuum] corresponds to our case of two arbitrary and two low-energy photons, the two low-energy legs corresponding to the two photons taken from the background field, however under the restriction *k*1 = − *k*2 = *k* imposed by the energy-momentum conservation in the constant field.\nWe would now like to use this correspondence for a check on our above results. While such a comparison could be made using the results of any of the references cited above, it will be convenient to compare with, where the worldline formalism was already used to compute the constant-field photon polarization tensors for both scalar and spinor QED. Although that calculation is still quite different from our present one (mainly because the integrating out of the low-energy photon legs there was effectively done already at the level of the construction of the generalized worldline Green’s functions, shown below in, ), it is still close enough to our present one to make it possible to verify the equivalence at an early stage, without the need to perform all the integrals. It will be sufficient to work with the intermediate results of our above calculations after integrating out *u*3 and *u*4, which we collect in [app4]. Using suitable integrations-by-part, we identify the resulting set of parameter integrals with the ones obtained by expanding the integral representations given in for the scalar and spinor QED photon propagators in a constant background field to second order in the field strength tensor *F**μ**ν*.\nScalar QED\n----------\nThe above replacement for the momenta (*k*1 = − *k*2 = *k*) actually makes some terms in $Q\\_{{{\\rm scal}}}$ vanish. The remaining terms in the scalar loop case can be compactly written as:\n \n$$\\label{Q(34)in}\n {Q\\_{{\\rm scal}(34)}}^4(1234) = \\frac{2}{3}Z\\_4(1234) \\left(G\\_{12}-4G\\_{12}^2\\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^4(2314) = \\frac{1}{9}Z\\_4(2314) \\left(1- 12G\\_{12} + 36 G\\_{12}^2 \\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^4(3124) = \\frac{2}{3} Z\\_4(3124) \\left(G\\_{12} -4G\\_{12}^2\\right),$$ \n \n$$\\begin{aligned}\n {Q\\_{{\\rm scal}(34)}}^3(123;4) = 0, \\\\\n {Q\\_{{\\rm scal}(34)}}^3(234;1) = 0, \\\\\n {Q\\_{{\\rm scal}(34)}}^3(314;2) = 0, \\\\\n {Q\\_{{\\rm scal}(34)}}^3(124;3) = 0,\n \\end{aligned}$$ \n \n$${Q\\_{{\\rm scal}(34)}}^2(12;34) = - \\frac{2T}{3}Z\\_2(12) k\\cdot f\\_4 \\cdot f\\_3 \\cdot k \\left(G\\_{12}^2 - 4G\\_{12}^3\\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^2(13;24) = - \\frac{T}{9}Z\\_2(13) k\\cdot f\\_4 \\cdot f\\_2 \\cdot k \\left(G\\_{12} - 4G\\_{12}^2\\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^2(23;14) = - \\frac{T}{9}Z\\_2(23) k\\cdot f\\_4 \\cdot f\\_1 \\cdot k \\left(G\\_{12} - 4G\\_{12}^2\\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^2(14;23) = -\\frac{T}{9}Z\\_2(14) k\\cdot f\\_2 \\cdot f\\_3 \\cdot k \\left(G\\_{12} - 4G\\_{12}^2\\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^2(24;13) = -\\frac{T}{9}Z\\_2(24) k\\cdot f\\_1 \\cdot f\\_3 \\cdot k \\left(G\\_{12} - 4G\\_{12}^2\\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^2(34;12) = 0,$$ \n \n$${Q\\_{{\\rm scal}(34)}}^{22}(12,34) = \\frac{1}{3} Z\\_2(12) Z\\_2(34) \\left(1-4G\\_{12}\\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^{22}(13,24) =\\frac{1}{9} Z\\_2(13)Z\\_2(24),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^{22}(14,23) = \\frac{1}{9} Z\\_2(14) Z\\_2(23). \n \\label{Q(34)fin}$$ \nNote that with the above replacement $f\\_{1}^{\\mu\\nu}=k^\\mu\\veps1^\\nu-k^\\nu \\veps1^\\mu$ and $f\\_{2}^{\\mu\\nu}=-k^\\mu\\veps2^\\nu+k^\\nu \\veps2^\\mu$. Rewriting the remaining integrals we have scal(k,1;-k,2;k3,3;k4,4)=0 T4--m2T01duQscal(34)-Tk2G(u,0),[equalk-int] where we have also set *u*2 = 0 and *u*1 = *u* so that now *G*12 = *G*(*u*, 0) = *u*(1 − *u*). In order to be able to make the comparison at the integrand level is important to use IBP or equivalently the identity\n \n$$\\label{idGs}\n T \\left(G\\_{12} - 4G\\_{12}^2 \\right) k^2\\, \\e^{-TG\\_{12} k^2}= \\left(1-6 G\\_{12}\\right) \\e^{-TG\\_{12} k^2} -\\partial\\_u \\left(\\Gd\\_{12} G\\_{12}\\e^{-TG\\_{12} k^2}\\right)$$ \nfor many of the terms in ${Q\\_{{\\rm scal}(34)}}^2$. Note that this identity can also be expressed in terms of the *Y**n**l* as (Y51 - 4 Y52)k2 = Y40- 6 Y41 [idYnl] which will be useful in Section [section-beta].\nTurning our attention now to the photon polarization tensors in a constant background field, for scalar QED the worldline calculation of yielded the following presentation:\n \n$$\\Pi^{\\mu\\nu}\\_{\\rm scal}=-\\frac{e^2}{(4\\pi)^{\\frac{D}{2}}}\\int\\_0^\\infty dT \\,T^{1-\\frac{D}{2}}\\e^{-m^2T}\\,{\\rm det}^{-\\half}\\left( \\frac{\\sin \\mathcal{Z}}{\\mathcal{Z}}\\right) \\int\\_0^1 du \\, I^{\\mu\\nu}\\_{\\rm scal}\\e^{-Tk\\cdot \\Phi\\_{12}\\cdot k}\\,,\n \\label{equal-k-constant-sc}$$ \nwhere\n \n$$I\\_{\\rm scal}^{\\mu\\nu}=\\dot{\\mathcal{G}}\\_{B12}^{\\mu\\nu}\\,k\\cdot\\dot{\\mathcal{G}}\\_{B12}\\cdot k-\\Big(\\dot{\\mathcal{G}}\\_{B11}-\\dot{\\mathcal{G}}\\_{B12}\\Big)^{\\mu\\lambda}\\Big(\\dot{\\mathcal{G}}\\_{B21}-\\dot{\\mathcal{G}}\\_{B22}\\Big)^{\\nu\\kappa}k^{\\lambda}k^\\kappa \\,,\n \\label{equal-k-constant}$$ \nand G*B**i**j* is the bosonic worldline Green’s function in a constant field background,\n \n$$\\begin{split}\n \\mathcal{G}\\_{Bij}&=\\frac{T}{2\\mathcal{Z}^2}\\Big(\\frac{\\mathcal{Z}}{\\sin \\mathcal{Z}} \\e^{-i\\mathcal{Z}\\dot{G}\\_{ij}}+i\\mathcal{Z}\\dot{G}\\_{ij}-1\\Big)\\,,\\\\\n \\mathcal{G}\\_{Bii}&=\\frac{T}{2\\mathcal{Z}^2}\\Big(\\mathcal{Z}\\cot \\mathcal{Z}-1\\Big)\\,,\n \\end{split}\n \\label{defcalGB}$$ \nwith Z*μ**ν* = *e**T**F**μ**ν*. For the exponent of ([equal-k-constant-sc]), we have\n \n$$-Tk\\cdot \\Phi\\_{ij}\\cdot k=-k\\cdot\\left[\\frac{T}{2\\mathcal{Z}}\\Big(\\frac{\\e^{-i\\mathcal{Z}\\dot{G}\\_{ij}}-\\cos\\mathcal{Z}}{\\sin \\mathcal{Z}}+i\\dot{G}\\_{ij}\\Big)\\right]\\cdot k \\,.$$ \nIn order to compare with the results in the present work, we choose *F**μ**ν* = *i*(*f*3*μ**ν* + *f*4*μ**ν*) and expand the whole polarization tensor to second order in *F**μ**ν* so as to obtain the first two diagrams on the right-hand side of Fig. [vacuum]. After some straightforward algebra, we can express the result as\n \n$$\\begin{split}\n \\varepsilon\\_1\\cdot \\Pi\\_{\\rm scal}\\cdot\\veps2 &= - \\frac{e^4}{(4\\pi)^{D/2}} \\int\\_0^{\\infty} dT \\, T^{3-D/2} \\e^{-m^2 T} \\int\\_0^1 du \\, \\e^{-TG\\_{12}k^2}\\\\\n &\\hspace{-1cm}\\times\\bigg\\{ (1-4G\\_{12}) \\Big(\\varepsilon\\_1 \\cdot \\varepsilon\\_2 k^2 - \\varepsilon\\_1 \\cdot k \\varepsilon\\_2 \\cdot k\\Big) \\Big[(eT)^{-2} -\\frac{2T}{3}G\\_{12}^2 k\\cdot f\\_3\\cdot f\\_4 \\cdot k + \\frac{1}{6} tr(f\\_3f\\_4) \\Big] \\\\\n &\\hspace{-0.8cm}+ \\frac{2}{3} \\Big(G\\_{12} -4G\\_{12}^2\\Big) \\Big[2\\varepsilon\\_1 \\cdot \\varepsilon\\_2 k\\cdot f\\_3\\cdot f\\_4 \\cdot k + k^2 \\left(\\varepsilon\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot \\varepsilon\\_2 + \\varepsilon\\_1 \\cdot f\\_4 \\cdot f\\_3\\cdot \\varepsilon\\_2 \\right)\\\\\n &\\hspace{-0.8cm}- \\varepsilon\\_2 \\cdot k\\Big( \\varepsilon\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot k + \\varepsilon\\_1 \\cdot f\\_4 \\cdot f\\_3 \\cdot k\\Big) - \\varepsilon\\_1 \\cdot k\\Big( \\varepsilon\\_2 \\cdot f\\_3 \\cdot f\\_4 \\cdot k + \\varepsilon\\_2 \\cdot f\\_4 \\cdot f\\_3 \\cdot k\\Big)\\Big] \\\\\n &\\hspace{-0.8cm} - 4 G\\_{12}^2 \\Big( \\varepsilon\\_1 \\cdot f\\_3 \\cdot k \\varepsilon\\_2 \\cdot f\\_4 \\cdot k + \\varepsilon\\_1 \\cdot f\\_4 \\cdot k \\varepsilon\\_2 \\cdot f\\_3 \\cdot k\\Big) \\bigg\\}\\,,\n \\end{split}$$ \nNote that the term proportional to (*e**T*)− 2 corresponds to the two-point diagram or the vacuum polarization diagram in the absence of the background field. The remaining terms give the sought after four-photon amplitude, and are found to be in complete agreement with ([equalk-int]).\nSpinor QED\n----------\nFor the spinor QED case we find, in complete analogy,\n \n$${Q\\_{{\\rm spin}(34)}}^4(1234) = -\\frac{4}{3} Z\\_4(1234) \\left( G\\_{12}+ 2G\\_{12}^2\\right),$$ \n \n$${Q\\_{{\\rm spin}(34)}}^4(2314) = -\\frac{4}{9} Z\\_4(2314) \\left(2- 6 G\\_{12}- 9G\\_{12}^2 \\right),$$ \n \n$${Q\\_{{\\rm spin}(34)}}^4(3124) = -\\frac{4}{3} Z\\_4(3124) \\left( G\\_{12}+ 2G\\_{12}^2\\right),$$ \n \n$$\\begin{aligned}\n {Q\\_{{\\rm spin}(34)}}^3(123;4) = 0, \\\\\n {Q\\_{{\\rm spin}(34)}}^3(234;1) = 0, \\\\\n {Q\\_{{\\rm spin}(34)}}^3(314;2) = 0, \\\\\n {Q\\_{{\\rm spin}(34)}}^3(124;3) = 0, \n \\end{aligned}$$ \n \n$${Q\\_{{\\rm spin}(34)}}^2(12;34) = \\frac{8T}{3} Z\\_2(12) k\\cdot f\\_4 \\cdot f\\_3 \\cdot k \\, G\\_{12}^3 \\,,$$ \n \n$${Q\\_{{\\rm spin}(34)}}^2(13;24) = \\frac{2T}{9}Z\\_2(13) k\\cdot f\\_4 \\cdot f\\_2 \\cdot k \\left(G\\_{12} - 4G\\_{12}^2\\right),$$ \n \n$${Q\\_{{\\rm spin}(34)}}^2(23;14) = \\frac{2T}{9}Z\\_2(23) k\\cdot f\\_4 \\cdot f\\_1 \\cdot k \\left(G\\_{12} - 4G\\_{12}^2\\right),$$ \n \n$${Q\\_{{\\rm spin}(34)}}^2(14;23) = \\frac{2T}{9}Z\\_2(14) k\\cdot f\\_2 \\cdot f\\_3 \\cdot k \\left(G\\_{12} - 4G\\_{12}^2\\right),$$ \n \n$${Q\\_{{\\rm spin}(34)}}^2(24;13) = \\frac{2T}{9}Z\\_2(24) k\\cdot f\\_1 \\cdot f\\_3 \\cdot k \\left(G\\_{12} - 4G\\_{12}^2\\right),$$ \n \n$${Q\\_{{\\rm spin}(34)}}^2(34;12) = 0,$$ \n \n$${Q\\_{{\\rm spin}(34)}}^{22}(12,34) = \\frac{8}{3} Z\\_2(12) Z\\_2(34) G\\_{12}\\,,$$ \n \n$${Q\\_{{\\rm spin}(34)}}^{22}(13,24) =\\frac{4}{9} Z\\_2(13)Z\\_2(24),$$ \n \n$${Q\\_{{\\rm spin}(34)}}^{22}(14,23) = \\frac{4}{9} Z\\_2(14) Z\\_2(23),$$ \nand the following integral representation\n \n$$\\Gamma\\_{\\rm spin}(k,\\veps1;-k,\\veps2;k\\_3,\\veps3;k\\_4,\\veps4)=-2\\frac{e^4}{(4\\pi)^{\\frac{D}{2}}}\\int\\_0^\\infty \\frac{dT}{T}\\, T^{4-\\frac{D}{2}}\\e^{-m^2T}\\int\\_0^1du\\,{Q\\_{{\\rm spin}(34)}}\\e^{-Tk^2G(u,0)}\\,.\\\\\n \\label{equalk-int-sp}$$ \nAs in the scalar case, to be able to compare at the integrand level it is important to make repeated use of ([idGs]).\nThe spinor-loop result of the vacuum polarization tensor (from ) is\n \n$$\\Pi^{\\mu\\nu}\\_{\\rm spin} = 2\\frac{e^2}{(4\\pi)^{\\frac{D}{2}}}\\int\\_0^\\infty dTT^{1-\\frac{D}{2}}\\e^{-m^2T}\\, {\\rm det}^{-\\half}\\left(\\frac{\\tan \\mathcal{Z}}{\\mathcal{Z}}\\right) \\int\\_0^1 du\\,I^{\\mu\\nu}\\_{\\rm spin}\\,\\e^{-Tk\\cdot \\Phi\\_{12}\\cdot k}\\,,\n \\label{equal-k-constant-sp}$$ \nwhere\n \n$$\\begin{split}\n I\\_{\\rm spin}^{\\mu\\nu}&=\\dot{\\mathcal{G}}\\_{B12}^{\\mu\\nu}\\,k\\cdot\\dot{\\mathcal{G}}\\_{B12}\\cdot k-\\mathcal{G}\\_{F12}^{\\mu\\nu}\\,k\\cdot\\mathcal{G}\\_{F12}\\cdot k\\\\\n &-\\Big[\\Big(\\dot{\\mathcal{G}}\\_{B11}-\\mathcal{G}\\_{F11}- \\dot{\\mathcal{G}}\\_{B12}\\Big)^{\\mu\\lambda} \\Big(\\dot{\\mathcal{G}}\\_{B21}-\\dot{\\mathcal{G}}\\_{B22} +\\mathcal{G}\\_{F22}\\Big)^{\\nu\\kappa} +\\mathcal{G}\\_{F12}^{\\mu\\lambda} \\mathcal{G}\\_{F21}^{\\nu\\kappa}\\Big]k^{\\lambda} k^\\kappa\\,,\n \\end{split}$$ \nwith the fermionic Green’s function in a constant field and its coincidence limit \n \n$$\\begin{split}\n \\mathcal{G}\\_{Fij}&=G\\_{Fij}\\frac{\\e^{-i\\mathcal{Z}\\dot{G}\\_{ij}}}{\\cos\\mathcal{Z}}\\,,\\\\\n \\mathcal{G}\\_{Fii}&=-i\\tan\\mathcal{Z}\\,. \n \\end{split}\n \\label{defcalGF}$$ \nAfter the expansion, choosing Z as in the scalar case, the spinor-loop gives\n \n$$\\begin{split}\n \\varepsilon\\_{1}\\cdot\\Pi\\_{\\rm spin}\\cdot\\veps2& = \\frac{2e^4}{(4\\pi)^{\\frac{D}{2}}} \\int\\_0^{\\infty} dT \\, T^{3-\\frac{D}{2}} \\e^{-m^2 T} \\int\\_0^1 du \\, \\e^{-TG\\_{12}k^2}\\\\\n &\\hspace{-1cm}\\times\\bigg\\{ 4 G\\_{12} \\Big(\\varepsilon\\_1 \\cdot \\varepsilon\\_2 k^2 - \\varepsilon\\_1 \\cdot k \\varepsilon\\_2 \\cdot k\\Big)\\Big[-(eT)^{-2} +\\frac{2T}{3} G\\_{12}^2 k\\cdot f\\_3\\cdot f\\_4 \\cdot k+ \\frac{1}{3} {\\rm tr}(f\\_3f\\_4) \\Big] \\\\\n &\\hspace{-0.8cm}- \\frac{4}{3} \\Big(G\\_{12} + 2 G\\_{12}^2\\Big) \\Big[2\\varepsilon\\_1 \\cdot \\varepsilon\\_2 k\\cdot f\\_3\\cdot f\\_4 \\cdot k + k^2 (\\varepsilon\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot \\varepsilon\\_2 + \\varepsilon\\_1 \\cdot f\\_4 \\cdot f\\_3 \\cdot \\varepsilon\\_2)\\\\\n &\\hspace{-0.8cm}- \\varepsilon\\_2 \\cdot k\\Big( \\varepsilon\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot k + \\varepsilon\\_1 \\cdot f\\_4 \\cdot f\\_3 \\cdot k\\Big) - \\varepsilon\\_1 \\cdot k\\Big( \\varepsilon\\_2 \\cdot f\\_3 \\cdot f\\_4 \\cdot k + \\varepsilon\\_2 \\cdot f\\_4 \\cdot f\\_3 \\cdot k\\Big) \\Big]\\\\\n &\\hspace{-0.8cm}-4 G\\_{12}^2 \\Big( \\varepsilon\\_1 \\cdot f\\_3 \\cdot k \\varepsilon\\_2 \\cdot f\\_4 \\cdot k + \\varepsilon\\_1 \\cdot f\\_4 \\cdot k \\varepsilon\\_2 \\cdot f\\_3 \\cdot k\\Big) \\bigg\\}\\,,\n \\end{split}$$ \nwhich again after removing the term proportional to (*e**T*)− 2 is in perfect agreement with ([equal-k-constant-sp]). Despite of the restriction *k*1 = − *k*2, this provides an excellent check on many of the results presented in Section [2-low] above.\nGauge-invariant decomposition of the two-loop *β*-functions\n===========================================================\nIn this section, we use the results presented in Section [2-low] to go to two loops by sewing together the unrestricted photons 1 and 2 (Fig. [beta]). This yields the two-loop photon propagators in the low-energy limit, that is, the induced Maxwell terms, from whose 1/*ε*-poles we can (up to a contribution from mass renormalization) compute the two-loop *β*-function coefficients for scalar and spinor QED, recuperating the known results. As explained in the introduction, besides providing another check this is motivated by the following two questions suggested by the highly structured worldline integrand : first, are there terms that drop out in the sewing procedure? Second, does the fact that the sixteen terms in this decomposition are (differently from the usual decomposition into Feynman diagrams) individually gauge invariant, have a bearing on the issue of gauge cancellations? Specifically, will the vanishing of the double poles involve only cancellations inside each gauge-invariant structure, or between them?\nWe use Feynman gauge in the sewing, so that it is done by the replacement\n \n$$\\veps1^\\mu\\veps2^\\nu\\rightarrow \\frac{\\eta^{\\mu\\nu}}{k^2}\\,,$$ \n(*k*1 = *k* = − *k*2) with *k* to be integrated over. The induced Maxwell term in our present set-up will appear as ${\\rm tr}(f\\_3f\\_4)$. The sewing will now change the argument of *Y**n**l* functions, such that in this section it is understood that *Y**n**l* = *Y**n**l*( − *k*2, *m*2, *D*). Going back to the results in *D*-dimensions in Section [2-low] for the scalar/spinor-loop, we have the following expressions which will be used in both cases:\n \n$$\\begin{split}\n Z\\_4(1234)\\vert\\_{\\rm sewing}=Z\\_4(3124)\\vert\\_{\\rm sewing}&\\Rightarrow\\frac{2}{D}(D-1){\\rm tr}(f\\_3f\\_4)\\,,\\\\\n Z\\_4(2314)\\vert\\_{\\rm sewing}&\\Rightarrow\\frac{2}{D}{\\rm tr}(f\\_3f\\_4)\\,,\\\\\n Z\\_3(123)\\vert\\_{\\rm sewing}\\, k\\cdot f\\_4\\cdot k &\\Rightarrow0\\,,\\\\\n Z\\_2(12)\\vert\\_{\\rm sewing}\\, k\\cdot f\\_3\\cdot f\\_4\\cdot k &\\Rightarrow\\frac{1}{D}(D-1)k^2{\\rm tr}(f\\_3f\\_4)\\,,\\\\\n Z\\_2(13)(k\\cdot f\\_4\\cdot f\\_2\\cdot k)\\vert\\_{\\rm sewing}&\\Rightarrow\\frac{1}{D}{\\rm tr}(f\\_3f\\_4)k^2\\,,\\\\\n Z\\_2(12)Z\\_2(34)\\vert\\_{\\rm sewing}&\\Rightarrow\\frac{(D-1)}{2}{\\rm tr}(f\\_3f\\_4)\\,,\\\\\n Z\\_2(13)Z\\_2(24)\\vert\\_{\\rm sewing}=Z\\_2(14) Z\\_2(23)\\vert\\_{\\rm sewing}&\\Rightarrow\\frac{1}{D}{\\rm tr}(f\\_3f\\_4)\\,.\\\\\n \\label{sewing-I}\n \\end{split}$$ \nHere we have used the Passarino-Veltman type identity\n \n$$\\int d^D k\\,(k\\cdot f\\_3\\cdot f\\_4\\cdot k)\\, F\\left(k^2\\right)\\rightarrow \\int d^D k\\,\\frac{k^2}{D}{\\rm tr}(f\\_3f\\_4) \\, F\\left(k^2\\right)\\,,\n \\label{idPV}$$ \nwhere *F* is an arbitrary scalar function.\nNote that the three-cycle *Z*3(123) drops out in the sewing, together with its three permutations. Therefore $Q^3\\_{\\rm scal/spin}$ does not contribute here, and the total amplitudes can be written as (using a similar convention as in )\n(2)scal(f3,f4)&= ( (2)4scal + (2)2scal + (2)22scal ), \n(2)spin(f3,f4)&= - ( (2)4spin + (2)2spin + (2)22spin ). [defGammahat2loop]\n(note that a symmetry factor of $\\frac{1}{2}$ has to be included).\nScalar QED *β*-function\n-----------------------\nIn this way, we find the following result for the Maxwell term of the two-loop vacuum polarization tensor in scalar QED,\n \n$$\\begin{split}\n \\Gamma^{(2)}\\_{\\rm scal}\\big(f\\_3,f\\_4\\big) &=\\frac{e^4}{2(4\\pi)^{\\frac{D}{2}}} \\int\\frac{d^Dk}{(2\\pi)^D} \\Bigg\\{\\left(\\frac{4}{9D} + \\frac{D-1}{6} \\right)Y\\_{40} - \\frac{2}{3D}\\big[(D-4)(D-1)+4\\big]Y\\_{41}\\\\\n & -\\frac{8}{3D}(4D-7)Y\\_{42}-\\frac{4}{9D}(Y\\_{51}-4Y\\_{52})k^2 -\\frac{2(D-1)}{3D}(Y\\_{52}-4Y\\_{53})k^2\\Bigg\\}{\\rm tr}(f\\_3f\\_4) \\,.\n \\end{split}$$ \nUsing the identity this can be somewhat simplified,\n \n$$\\begin{split}\n \\Gamma^{(2)}\\_{\\rm scal}(f\\_3,f\\_4)&=\\frac{e^4}{2(4\\pi)^{\\frac{D}{2}}}\\int\\frac{d^Dk}{(2\\pi)^D}\\bigg[\\frac{D-1}{6}Y\\_{40}-\\frac{2}{3D}(D-4)(D-1)Y\\_{41}\\\\\n &\\hspace{1cm}-\\frac{8}{3D}(4D-7)Y\\_{42}-\\frac{2(D-1)}{3D}\\, k^2(Y\\_{52}-4Y\\_{53})\\bigg]{\\rm tr}\n (f\\_3f\\_4)\\,.\n \\end{split}$$ \nNext, we rewrite this as (2)scal(f3,f4) scal(D,m2)tr (f3f4). [2loopbeta] We then use ([u-integrals]) to eliminate the *k* - integration, which leads to\n \n$$\\begin{split}\n \\hspace{-0.3cm}\\gamma\\_{{{\\rm scal}}} (D,m^2) &= \\frac{ (4 \\pi)^{4} \\Gamma (4-D)}{4 (4 \\pi)^{D}m^{2(4-D)}} \\, \\int\\_0^1 \\,du \\, \\frac{1}{3D}\n \\Big\\{2 (D-1) D \\Big[G(u,0)\\Big]^{-\\frac{D}{2}} \\\\\n & \\quad -4(D-1) (3 D-8) \\Big[G(u,0)\\Big]^{1-\\frac{D}{2}} + 16 (D-7) (D-2) \\Big[G(u,0)\\Big]^{2-\\frac{D}{2}}\\Big\\}.\n \\end{split}$$ \nThe remaining *u*-integral can be expressed in terms of the Euler Beta-function as\n \n$$\\int\\_0^1du\\Big[G(u,0)\\Big]^x=B(x+1,x+1), \\qquad B(x,y) \\equiv \\frac{\\Gamma(x)\\Gamma(y)}{\\Gamma(x+y)}\\,,\n \\label{betfn}$$ \nwhich leads to our final result for the coefficient of the induced Maxwell term,\n \n$$\\gamma\\_{{{\\rm scal}}} (D,m^2) = - \\frac{(4\\pi)^{4}\\, \\Gamma\\left(4-D\\right)}{(4\\pi)^D \\, m^{2(4-D)}} \\, B\\left(1-\\frac{D}{2},1-\\frac{D}{2}\\right) \\frac{(D-4) [(D-3) D+8]}{12 (D-5) (D-3) D} \\,.$$ \nNote the global factor of (*D* − 4), which makes it clear that, as expected, the 1/*ε*-expansion has only a single pole, scal(,m2)&=& + (0). [beta-reg] After substituting ([beta-reg]) into ([2loopbeta]) one arrives at (2)scal(f3,f4)= tr (f3f4) + (0), which is in agreement with previous calculations of the two-loop dimensionally regularized effective action in Scalar QED [3](#fn3).\nFor the full two-loop renormalization of the photon propagator, one still needs to add an equal contribution $\\Delta\\Gamma^{(2)}\\_{\\rm scal}(f\\_3,f\\_4)$ from mass renormalization, (2)scal(f3,f4)+(2)scal(f3,f4)~(f3f4). This then gives the two-loop photon wave-function renormalization factor and leads to the standard result for the two-loop *β*-function coefficient in Scalar QED (2)scal()= , =.\nMore interesting is the distribution of the double and single - pole coefficients $a\\_{\\rm scal}$ and $b\\_{\\rm scal}$ in the decomposition, shown in Table [table:ta2].\n[h!]\n |c|c|c||c|c|c||c|c|c| ${{\\gamma\\_{{\\rm scal}}^{}}}$ & $a\\_{\\rm scal}$ & $b\\_{\\rm scal}$ & ${{\\gamma\\_{{\\rm scal}}^{}}}$ & $a\\_{\\rm scal}$ & $b\\_{\\rm scal}$ &${{\\gamma\\_{{\\rm scal}}^{}}}$ & $a\\_{\\rm scal}$ & $b\\_{\\rm scal}$ \n${{\\gamma\\_{{\\rm scal}}^{4}}}(1234)$ & 4 & $\\frac{13}{3} $ & ${{\\gamma\\_{{\\rm scal}}^{2}}}(13;24)$ & $\\frac{4}{9}$ & $-\\frac{2}{9}$ & ${{\\gamma\\_{{\\rm scal}}^{22}}}(12,34)$ & − 4 & $ \\frac{2}{3}$ \n${{\\gamma\\_{{\\rm scal}}^{4}}}(1243)$ & 4 & $\\frac{13}{3} $ & ${{\\gamma\\_{{\\rm scal}}^{2}}}(14;23)$ & $\\frac{4}{9}$ & $-\\frac{2}{9}$ & ${{\\gamma\\_{{\\rm scal}}^{22}}}(13,24)$ & $\\frac{2}{9} $ & $\\frac{1}{18}$ \n${{\\gamma\\_{{\\rm scal}}^{4}}}(1324)$ & $-\\frac{20}{9} $ & $-\\frac{11}{9} $ & ${{\\gamma\\_{{\\rm scal}}^{2}}}(24;13)$ & $\\frac{4}{9}$ & $-\\frac{2}{9}$ & ${{\\gamma\\_{{\\rm scal}}^{22}}}(14,23)$ & $\\frac{2}{9} $ & $\\frac{1}{18}$ \n${{\\gamma\\_{{\\rm scal}}^{2}}}(12;34)$ & − 4 & $-\\frac{16}{3}$ & ${{\\gamma\\_{{\\rm scal}}^{2}}}(23;14)$ & $\\frac{4}{9}$ & $-\\frac{2}{9}$ & & & \n[table:ta2]\nAlthough all terms here are already individually gauge invariant, the vanishing of the double pole still happens through an intricate cancellation between them. Similarly, apart from the vanishing of the three-cycle contributions the other terms all contribute to the single pole, without any clear pattern emerging as one might have hoped.\nSpinor QED *β*-function\n-----------------------\nFor the more relevant spinor QED case, let us also write down the individual contributions in terms of the *Y**n**l*:\n \n$$\\begin{split}\n \\hat{\\Gamma}^{(2)4}\\_{\\rm spin}(1234) = \\hat{\\Gamma}^{(2)4}\\_{\\rm spin}(3124) &= - \\frac{8(D-1)}{3D} \\left(Y\\_{41} + 2Y\\_{42}\\right){\\rm tr}(f\\_3f\\_4)\\,,\\\\\n \\hat{\\Gamma}^{(2)4}\\_{\\rm spin}(2314) & = - \\frac{8}{9D} \\left(2Y\\_{40} - 6Y\\_{41} - 9Y\\_{42}\\right){\\rm tr}(f\\_3f\\_4)\\,,\\\\\n \\hat{\\Gamma}^{(2)2}\\_{\\rm spin}(12;34) & = \\frac{8}{3D} (D-1)\\, k^2\\, Y\\_{53}\\,{\\rm tr}(f\\_3f\\_4)\\,,\\\\\n \\hat{\\Gamma}^{(2)2}\\_{\\rm spin}(13;24) = \\hat{\\Gamma}^{(2)2}\\_{\\rm spin}(23;14) & = \\frac{2}{9D} \\left(Y\\_{40} - 6Y\\_{41}\\right){\\rm tr}(f\\_3f\\_4)\\,,\\\\\n \\hat{\\Gamma}^{(2)2}\\_{\\rm spin}(14;23) = \\hat{\\Gamma}^{(2)2}\\_{\\rm spin}(24;13) & = \\frac{2}{9D} \\left(Y\\_{40} - 6Y\\_{41}\\right){\\rm tr}(f\\_3f\\_4)\\,,\\\\\n \\hat{\\Gamma}^{(2)22}\\_{\\rm spin}(12,34) & = \\frac{4}{3}(D-1)\\,Y\\_{41} \\, {\\rm tr}(f\\_3f\\_4)\\,,\\\\\n \\hat{\\Gamma}^{(2)22}\\_{\\rm spin}(13,24) = \\hat{\\Gamma}^{(2)22}\\_{\\rm spin}(14,23) & = \\frac{4}{9D} \\, Y\\_{40}\\,{\\rm tr}(f\\_3f\\_4)\\,.\n \\end{split}\n \\label{gamma-contri}$$ \nCollecting all terms, we obtain\n \n$$\\begin{split}\n \\Gamma^{(2)}\\_{\\rm spin}(f\\_3,f\\_4)&=-\\frac{e^4}{(4\\pi)^{\\frac{D}{2}}}\\int\\frac{d^Dk}{(2\\pi)^D}\\bigg[\\frac{4}{3D}(D-4)(D-1)Y\\_{41} -\\frac{8}{3D}(4D-7)Y\\_{42}\\\\\n &\\hspace{3cm}+\\frac{8}{3D}(D-1)\\,k^2\\, Y\\_{53}\\bigg]{\\rm tr}\n (f\\_3f\\_4)\\,.\n \\end{split}\n \\label{gammas-beta}$$ \nAs in the scalar case, we define\n \n$$\\Gamma^{(2)}\\_{\\rm spin}(f\\_3,f\\_4)\\equiv -\\frac{e^4}{(4\\pi)^{4}}\\, \\gamma\\_{{{\\rm spin}}} (D,m^2) \\,{\\rm tr} (f\\_3f\\_4)\\,.\n \\label{spin-beta1}$$ \nApplication of leads to the following representation for $\\gamma\\_{\\rm spin} (D,m^2)$,\n \n$$\\begin{split}\n \\gamma\\_{\\rm spin} (D,m^2) &= \\frac{(4 \\pi)^{4} \\Gamma (4-D)}{4(4 \\pi)^{D}m^{2(4-D)}} \\, \\int\\_0^1 \\,du \\, \\frac{16}{3D}\n \\Big\\{ (D -4) (D -1) \\Big[G(u,0)\\Big]^{1-\\frac{D}{2}} \\\\\n & \\quad + (D -7) (D -2) \\Big[G(u,0)\\Big]^{2-\\frac{D}{2}}\\Big\\}.\n \\end{split}\n \\label{2loopuint}$$ \nThis result agrees already at the integrand level with Eq. (36) of, providing another excellent check on many of the equations of the present paper. Performing the *u* - integral brings us to our final result for the induced Maxwell term,\n \n$$\\begin{split}\n \\gamma\\_{{{\\rm spin}}} (D,m^2)&= \\frac{(4\\pi)^{4}\\Gamma\\left(4-D\\right)}{(4\\pi)^D \\, m^{2(4-D)}} \\, B\\left(2-\\frac{D}{2},2-\\frac{D}{2}\\right) \\frac{(D-4)[34+D(5D-33)]}{3D(D-5)}\\,.\n \\end{split}$$ \nNote once more the global factor of (*D* − 4), which reduces the pole in dimensional regularization to a single one,\n \n$$\\label{coef-beta}\n \\begin{split}\n \\gamma\\_{{{\\rm spin}}} (\\epsilon,m^2)= \\frac{6}{\\epsilon} +\\mathcal{O}(\\epsilon^0)\\,.\n \\end{split}$$ \nAdding the appropriate term from mass renormalization (2)spin(f3,f4)~(f3f4), we get the total pole of the induced Maxwell term, (2)spin(f3,f4)+(2)spin(f3,f4)~(f3f4),\nwhich finally gives the correct two-loop spinor QED *β*-function, (2)spin()=. Finally, in Table [table:ta1] we list the double and single pole contributions of the individual terms in the worldline integrand. The picture that emerges is similar to the scalar QED case above.\n[h!]\n |c|c|c||c|c|c||c|c|c| ${{\\gamma\\_{{\\rm spin}}^{}}}$ & $a\\_{\\rm spin}$ & $b\\_{\\rm spin}$ &${{\\gamma\\_{{\\rm spin}}^{}}}$ & $a\\_{\\rm spin}$ & $b\\_{\\rm spin}$ &${{\\gamma\\_{{\\rm spin}}^{}}}$ & $a\\_{\\rm spin}$ & $b\\_{\\rm spin}$ \n${{\\gamma\\_{{\\rm spin}}^{4}}}(1234)$ & − 8 & $\\frac{10}{3} $ & ${{\\gamma\\_{{\\rm spin}}^{2}}}(13;24)$ & $-\\frac{8}{9}$ & $\\frac{4}{9}$ & ${{\\gamma\\_{{\\rm spin}}^{22}}}(12,34)$ & 16 & $ \\frac{16}{3}$ \n${{\\gamma\\_{{\\rm spin}}^{4}}}(1243)$ & − 8 & $\\frac{10}{3} $ & ${{\\gamma\\_{{\\rm spin}}^{2}}}(14;23)$ & $-\\frac{8}{9}$ & $\\frac{4}{9}$ & ${{\\gamma\\_{{\\rm spin}}^{22}}}(13,24)$ & $\\frac{8}{9} $ & $\\frac{2}{9}$ \n${{\\gamma\\_{{\\rm spin}}^{4}}}(1324)$ & $\\frac{16}{9} $ & $-\\frac{38}{9} $ & ${{\\gamma\\_{{\\rm spin}}^{2}}}(24;13)$ & $-\\frac{8}{9}$ & $\\frac{4}{9}$ & ${{\\gamma\\_{{\\rm spin}}^{22}}}(14,23)$ & $\\frac{8}{9} $ & $\\frac{2}{9}$ \n${{\\gamma\\_{{\\rm spin}}^{2}}}(12;34)$ & 0 & − 4 & ${{\\gamma\\_{{\\rm spin}}^{2}}}(23;14)$ & $-\\frac{8}{9}$ & $\\frac{4}{9}$ & & & \n[table:ta1]\nFour-dimensional computation of the Spinor QED *β*-function\n-----------------------------------------------------------\nAlthough the calculation of corresponding quantities in Scalar and Spinor QED in the worldline formalism usually proceeds with only minor differences, a surprising finding of had been that, in the case of the two-loop QED *β* functions, this holds true when employing dimensional regularization but not for strictly four-dimensional regularization schemes such as the use of a proper-time cutoff. In, this observation was explained as a consequence of the above-mentioned theorem by Johnson et al. that the total induced effective action can have only a global divergence. Since, unlike Scalar QED, Spinor QED does not possess true quadratic divergences, the required cancellation of subdivergences here leads to two constraint equations. And since, by the structure of the worldline representation of the induced Maxwell term, the integrand before the final integration over *u* in *D* = 4 must be of the form (compare )\n \n$$\\left(\\frac{A}{G\\_{12}^2}+\\frac{B}{G\\_{12}}+C\\right){\\rm tr}(F^2)\\,,$$ \none can first conclude from the absence of a quadratic subdivergence that *A* = 0, and then from the absence of a logarithmic subdivergence that *B* = 0, leaving only the coefficient *C* non-vanishing and making the final *u*-integral trivial (this argument does not work in dimensional regularization due to the principal suppression of quadratic subdivergences by that scheme).\nThus one would expect that also the present recalculation should simplify in the Spinor QED case when a four-dimensional scheme is used. And indeed, if we return to and put *D* = 4 there, we find spin (D,m2)&&(4)2, and then using any four-dimensional method for the regularization of the global divergence that is still contained in the global *T*-integration will lead to a trivial *u*-integration.\nDelbrück scattering at low energies\n===================================\nIn this section, we compute the differential cross section of Delbrück scattering in scalar and spinor QED under the assumption that the photon that interacts with the Coulomb field has low energy. For the spinor QED case, this quantity was computed in detail in, therefore we will follow their conventions for easy comparison.\nFeynman diagram for low-energy Delbrück scattering.\nHere we use our above results for the four-photon amplitudes with two low-energy photons, and replace the two unrestricted legs 1 and 2 with two Coulomb photons. Furthermore, we now take the two low-energy photons (3 and 4) on-shell.\nThe vector potential for the photons from the Coulomb field is given by\n \n$$A\\_\\mu(x)= \\left(-\\frac{Ze}{4\\pi r},\\,{\\bf 0}\\right)\\,,$$ \nand has the following Fourier representation,\n \n$$A\\_\\mu(x)=\\varepsilon\\_\\mu \\int\\frac{d^4k}{(2\\pi)^{4}}\\, \\frac{Ze}{k^2}\\, 2\\pi\\, \\delta(k\\_0)\\,\\e^{ik\\cdot x}\\,.$$ \nFrom here the new vertex operator for photons from the Coulomb field read as\n \n$$\\begin{split}\n V^\\gamma\\_{\\rm Nuc}[k,\\varepsilon]&=\\frac{Ze}{(2\\pi)^3}\\int d^4k\\frac{\\delta(k\\_{0})}{k^2}\\int\\_0^Td\\tau\\, \\varepsilon\\cdot\\dot{x}\\e^{ik\\cdot x}\\\\\n &=\\frac{Ze}{(2\\pi)^3}\\int \\frac{d^{3} {\\bf k}}{{\\bf k}^2}\\int\\_0^Td\\tau\\, \\varepsilon\\cdot\\dot{x}\\,\\e^{i{\\bf k}\\cdot {\\bf x}}\\,.\n \\end{split}$$ \nThus in the present formalism, we have\n \n$$\\label{DSscal}\n \\Gamma\\_{\\left\\{{{\\rm scal}}(34)\\atop{{\\rm spin}}(34)\\right\\} } \n = \\left\\{1\\atop -2\\right\\} \n \\frac{1}{2}\n \\frac{e^4 (Ze)^2}{(4\\pi)^2(2\\pi)^{6}} \\int\\frac{ (d^{3}{{\\bf k}\\_1}) (d^{3}{{\\bf k}\\_2})}{{\\bf k}\\_1^2 {\\bf k}\\_2^2} (2\\pi)^4 \\delta^4(k\\_1 + k\\_2 + k\\_3 + k\\_4) \\hat{\\Gamma}\\_{\\left\\{{{\\rm scal}}(34)\\atop{{\\rm spin}}(34)\\right\\}}\\,,$$ \nwith $ \\hat{\\Gamma}\\_{\\left\\{{{\\rm scal}}(34)\\atop{{\\rm spin}}(34)\\right\\}}$ as defined in,. Note the factor of $\\frac{1}{2}$ that takes the symmetry between legs 1 and 2 into account [4](#fn4). In order to reproduce the result given in for the differential cross section, we employ the following kinematics\n \n$$\\label{k1k2k3k4}\n\\begin{split}\nk\\_1&=(0,{\\bf q}-{\\bf q}')~,~~~~~~k\\_2=(0,-{\\bf q}-{\\bf q}'), \\\\\nk\\_3&=(i\\omega,{\\bf k}+{\\bf q}')~,~~~~~k\\_4=(-i\\omega,{\\bf q}'-{\\bf k}),\n\\end{split}$$ \nwhere\n \n$$\\begin{split}\n{\\bf q'}&=(\\omega\\sin\\theta/2,0,0),\\\\\n{\\bf k}&=(0,0,\\omega\\cos\\theta/2),\\\\\n{\\bf q}&=(q\\_1,q\\_2,q\\_3),\n\\end{split}$$ \nwith *ω* as the energy and *θ* the scattering angle. The polarizations are chosen as\n \n$$\\begin{split}\n\\varepsilon\\_1^\\mu &= \\varepsilon\\_2^\\mu = (i,0,0,0),\\\\\n\\varepsilon\\_3^\\mu &= \\frac{1}{\\sqrt{2}}(0,-i\\lambda\\_3\\cos\\theta/2,1,+i\\lambda\\_3\\sin\\theta/2),\\\\\n\\varepsilon\\_4^\\mu &= \\frac{1}{\\sqrt{2}}(0,-i\\lambda\\_4\\cos\\theta/2,1,-i\\lambda\\_4\\sin\\theta/2),\n\\end{split}$$ \nwhere *λ**i* = ± 1 for right-and left-handed circular polarization, respectively. In the following it is understood that $\\hat{\\Gamma}\\_{{{\\rm scal}}(34)}^{+-} = \\hat{\\Gamma}\\_{{{\\rm scal}}(34)}|\\_{\\lambda\\_3 = 1,\\, \\lambda\\_4 =-1}$ etc. and we will also use the abbreviations\n \n$$P\\_0 \\equiv \\frac{{\\rm arcsinh}\\left(\\frac{q}{2m}\\right) }{q\\sqrt{4m^2 + q^2}}, \\qquad S \\equiv \\sin\\frac{\\theta}{2}, \\qquad C \\equiv \\cos\\frac{\\theta}{2}\\,.$$ \nWith the kinematics of ([k1k2k3k4]) and using conservation of momentum we can write ([DSscal]) as\n \n$$\\Gamma\\_{\\left\\{{{\\rm scal}}(34)\\atop{{\\rm spin}}(34)\\right\\}}\n= \\left\\{1\\atop -2\\right\\}\n\\frac{1}{2}\n \\frac{e^4 (Ze)^2}{(4\\pi)^2(2\\pi)^{6}} (2\\pi)^4 \\delta(k\\_3^0 + k\\_4^0) \\int\\frac{ d^{3}{\\bf q}}{\\vert{\\bf q}-{\\bf q'}\\vert^2\\vert{\\bf q}+{\\bf q}'\\vert^2}\\hat{\\Gamma}\\_{\\left\\{{{\\rm scal}}(34)\\atop{{\\rm spin}}(34)\\right\\}} \\,.$$ \nFor convenience, let us further define\n \n$$\\tilde{\\Gamma}\\_{\\left\\{{{\\rm scal}}\\atop{{\\rm spin}}\\right\\}}\n \\equiv \\int\\frac{ d^{3}{\\bf q}}{\\vert{\\bf q}-{\\bf q'}\\vert^2\\vert{\\bf q}+{\\bf q}'\\vert^2} \\hat{\\Gamma}\\_{\\left\\{{{\\rm scal}}(34)\\atop{{\\rm spin}}(34)\\right\\}}\\,.$$ \nSince we are considering the low-energy case, *ω* ≪ *m*, we neglect contributions of order superior to *ω*2. We notice that\n \n$$\\begin{split}\n \\int \\frac{ d^{3}{\\bf q}}{\\vert{\\bf q}-{\\bf q'}\\vert^2\\vert{\\bf q}+{\\bf q}'\\vert^2} & = \\int\\_0^\\infty \\int\\_0^\\pi \\int\\_0^{2\\pi} \\frac{q^2 \\sin \\theta' dq \\, d\\theta' \\, d\\phi'}{\\left( q^2 + \\omega^2 \\sin^2 \\frac{\\theta}{2}\\right)^2 - 4q\\_1^2 \\omega^2 \\sin^2 \\frac{\\theta}{2}}\\\\\n & = \\int\\_0^\\infty \\int\\_0^\\pi \\int\\_0^{2\\pi} \\frac{\\sin \\theta' dq \\, d\\theta' \\, d\\phi'}{q^2} + \\mathcal{O}(\\omega)\\,.\n \\end{split}$$ \nUsing the kinematics of ([k1k2k3k4]), we find\n \n$$\\begin{split}\n \\hat{\\Gamma}\\_{{{\\rm scal}}(34)}^{+-} &= \\frac{4 \\omega^2}{3m^2 q^4 (4m^2 + q^2)}\\Big\\{3m^2[(6m^2 + q^2) - 8m^2(3m^2 + q^2)P\\_0](q\\_2^2 - q\\_1^2)\\\\\n &\\quad + [q^4 (2m^2 + q^2) -3m^2(6m^2+q^2)q\\_2^2] S^2 - 8m^4[ q^4 - 3(3m^2+q^2)q\\_2^2]S^2P\\_0\\Big\\} + \\mathcal{O}(\\omega^3)\\,,\n \\end{split}$$ \n \n$$\\begin{split}\n \\hat{\\Gamma}\\_{{{\\rm spin}}(34)}^{+-} &= \\frac{4 \\omega^2}{3 q^4 (4m^2 + q^2)^2}\\Big\\{3(4m^2 + q^2)[(6m^2+q^2)- 8m^2 (3 m^2 + q^2)P\\_0](q\\_2^2 - q\\_1^2) \\\\\n &\\quad + [ q^4(2m^2 - q^2) -3(4m^2+q^2)(6m^2+q^2)q\\_2^2]S^2 \\\\\n &\\quad -8m^2 [ q^4(m^2 + q^2)- 3(4 m^2 + q^2) (3 m^2 + q^2)q\\_2^2]S^2 P\\_0\\Big\\} + \\mathcal{O}(\\omega^3)\\,,\n \\end{split}$$ \nfor the helicity non-conserving component, and\n \n$$\\begin{split}\n \\hat{\\Gamma}\\_{{{\\rm scal}}(34)}^{++} &= \\frac{4 \\omega^2}{3 q^4 (4m^2 + q^2)}\\Big\\{4[(6m^2 + q^2)- 8m^2 (3m^2 +q^2)P\\_0](q\\_2^2 - q\\_3^2) \\\\\n &\\quad- [3q^2(2m^2 + q^2) + 4(6m^2+q^2)q\\_2^2] C^2 \\\\\n &\\quad +4[q^2(6m^4 + 4m^2 q^2 + q^4)+ 8m^2(3m^2+ q^2)q\\_2^2]C^2P\\_0\\Big\\} + \\mathcal{O}(\\omega^3)\\,,\n \\end{split}$$ \n \n$$\\begin{split}\n \\hat{\\Gamma}\\_{{{\\rm spin}}(34)}^{++} &= \\frac{4 \\omega^2}{3 q^4 (4m^2 + q^2)^2}\\Big\\{(6 m^2 + q^2) (16 m^2 + 7 q^2)(q\\_2^2 - q\\_3^2) \\\\\n &\\quad+8[(3 m^2 + q^2) (4 m^2 + q^2)^2 - 3 m^4 q^2](q\\_3^2 - q\\_2^2)P\\_0 \\\\\n &\\quad- [3q^2(8m^4 -2m^2q^2 - q^4) + (6 m^2 + q^2) (16 m^2 + 7 q^2)q\\_2^2]C^2 \\\\\n &\\quad+ 8q^2 (12 m^6 - m^4 q^2 - 5 m^2 q^4 - q^6 ) C^2 P\\_0\\\\\n &\\quad+ 8 [(3 m^2 + q^2) (4 m^2 + q^2)^2 - 3 m^4 q^2]q\\_2^2 C^2 P\\_0\\Big\\} + \\mathcal{O}(\\omega^3)\\,,\n \\end{split}$$ \nfor the conserving one. To perform the integral over ${\\bf q}$ we use spherical coordinates:\n \n*q*1 = *q*cos*θ*ʹ, *q*2 = *q*sin*θ*ʹcos*ϕ*ʹ, *q*3 = *q*sin*θ*ʹsin*ϕ*ʹ . \nThe integrals over *θ*ʹ and *ϕ*ʹ are trivial, and what remains to be calculated is\n \n$$\\tilde{\\Gamma}\\_{{{\\rm scal}}}^{+-} = \\frac{16 \\pi S^2 \\omega^2 }{3}\n \\int\\_0^\\infty \\frac{ dq}{q^2} \\, \\frac{-6m^4 + m^2 q^2 + q^4 + 24 m^6 P\\_0}{ m^2 q^2 (4m^2 + q^2)}\\,,$$ \n \n$$\\tilde{\\Gamma}\\_{{{\\rm spin}}}^{ +-} = \\frac{32 \\pi S^2 \\omega^2 }{3}\n \\int\\_0^\\infty \\frac{ dq}{q^2} \\, \\frac{-12 m^4 - 4 m^2 q^2 - q^4 + 24 m^4(2m^2 + q^2) P\\_0}{ q^2 (4m^2 + q^2)^2}\\,,$$ \n \n$$\\tilde{\\Gamma}\\_{{{\\rm scal}}}^{ ++} = \\frac{16 \\pi C^2 \\omega^2 }{9}\n \\int\\_0^\\infty \\frac{ dq}{q^2} \\, \\frac{-42m^2 -13 q^2 + 4(42 m^4 + 20 m^2 q^2 + 3 q^4) P\\_0}{q^2 (4m^2 + q^2)}\\,,$$ \n \n$$\\tilde{\\Gamma}\\_{{{\\rm spin}}}^{++} = \\frac{32 \\pi C^2 \\omega^2 }{9}\n \\int\\_0^\\infty \\frac{ dq}{q^2} \\, \\frac{-84 m^4 -20 m^2 q^2 +q^4 + 8(42 m^6 + 17 m^4 q^2 -2m^2 q^4 - q^6) P\\_0}{q^2 (4m^2 + q^2)^2}\\,.$$ \nPerforming the integral over *q*, we get\n \n$$\\tilde{\\Gamma}\\_{{{\\rm scal}}}^{ +-} = \\frac{15 \\pi^3 S^2 \\omega^2 }{32 m^3}, \\qquad\n \\tilde{\\Gamma}\\_{{{\\rm spin}}}^{ +-} = -\\frac{5 \\pi^3 S^2 \\omega^2 }{32 m^3}\\,,$$ \n \n$$\\tilde{\\Gamma}\\_{{{\\rm scal}}}^{ ++} = \\frac{3 \\pi^3 C^2 \\omega^2 }{32 m^3}, \\qquad\n \\tilde{\\Gamma}\\_{{{\\rm spin}}}^{ ++} = -\\frac{73 \\pi^3 C^2 \\omega^2 }{288 m^3}\\,.$$ \nFinally, the differential cross section is\n \n$$d\\sigma\\_{{{\\rm scal}}(\\lambda\\_3 \\lambda\\_4)} = \\frac{(Z\\alpha)^4 \\alpha^2}{4(2\\pi)^{6}} \\, \\vert \\tilde{\\Gamma}\\_{{{\\rm scal}}}^{\\lambda\\_3 \\lambda\\_4} \\vert^2 \\, d\\Omega\\,,$$ \n \n$$d\\sigma\\_{{{\\rm spin}}(\\lambda\\_3 \\lambda\\_4)} = \\frac{(Z\\alpha)^4 \\alpha^2}{(2\\pi)^{6}} \\, \\vert \\tilde{\\Gamma}\\_{{{\\rm spin}}}^{\\lambda\\_3 \\lambda\\_4} \\vert^2 \\, d\\Omega\\,.$$ \nFor scalar QED, we have\n \n$$\\begin{aligned}\n d\\sigma\\_{{{\\rm scal}}(++)} &=& d\\sigma\\_{{{\\rm scal}}(--)} = (Z\\alpha)^4 \\left(\\frac{\\alpha}{m}\\right)^2 \\left(\\frac{3}{16}\\right)^2 \\left(\\frac{1}{32}\\right)^2 \\left(\\frac{\\omega}{m}\\right)^4 \\cos^4 \\frac{\\theta}{2} d\\Omega\\,,\\\\\n d\\sigma\\_{{{\\rm scal}}(+-)} &=& d\\sigma\\_{{{\\rm scal}}(-+)} = (Z\\alpha)^4 \\left(\\frac{\\alpha}{m}\\right)^2 \\left(\\frac{15}{16}\\right)^2 \\left(\\frac{1}{32}\\right)^2 \\left(\\frac{\\omega}{m}\\right)^4 \\sin^4 \\frac{\\theta}{2} d\\Omega\\,.\\end{aligned}$$ \nFor spinor QED, we find\n \n$$\\begin{aligned}\n d\\sigma\\_{{{\\rm spin}}(++)} &=& d\\sigma\\_{{{\\rm spin}}(--)} = (Z\\alpha)^4 \\left(\\frac{\\alpha}{m}\\right)^2 \\left(\\frac{73}{72}\\right)^2 \\left(\\frac{1}{32}\\right)^2 \\left(\\frac{\\omega}{m}\\right)^4 \\cos^4 \\frac{\\theta}{2} d\\Omega\\,,\\\\\n d\\sigma\\_{{{\\rm spin}}(+-)} &=& d\\sigma\\_{{{\\rm spin}}(-+)} = (Z\\alpha)^4 \\left(\\frac{\\alpha}{m}\\right)^2 \\left(\\frac{5}{8}\\right)^2 \\left(\\frac{1}{32}\\right)^2 \\left(\\frac{\\omega}{m}\\right)^4 \\sin^4 \\frac{\\theta}{2} d\\Omega\\,,\\end{aligned}$$ \nin agreement with.\nSummary and outlook\n===================\nIn this second part of our series of papers on the off-shell four-photon amplitudes in scalar and spinor QED we have obtained these amplitudes for the case where two of the photon momenta were taken in the low-energy limit, as defined in part I. We have used the worldline representation of these amplitudes which, as outlined in part I, allows one to treat the scalar and spinor cases in parallel, and to arrive at a permutation and gauge-invariant decomposition of these amplitudes that is closely related to the one previously obtained by. The coefficient functions in this decomposition are written in terms of Feynman-Schwinger type parameter integrals, and as our main result we have explicitly evaluated them both in four and in *D* dimensions. Since in our formalism the four-photon amplitudes are manifestly free of UV divergences, the former results are sufficient for one-loop applications. The latter formulas serve the double purpose of making the off-shell four-photon amplitudes useful as input for higher-loop calculations in dimensional regularization, but also giving the correct results for them in other physical dimensions (arbitrary *D* for scalar QED, even *D* for spinor QED). After the integrating out of the two low-energy legs these integrals are of two-point type, and can therefore be written in terms of 2*F*1 for general *D*, and for *D* = 4 in terms of trigonometric functions.\nTo the best of our knowledge, for this momentum configuration the four-photon amplitudes have not been obtained before. However, for the special case *k*1 + *k*2 = *k*3 + *k*4 = 0 they can be extracted from the low-energy limit of the known vacuum polarization tensors in a constant field, and performing this comparison provided an excellent check on our results for both scalar and spinor QED.\nFrom a technical point of view, probably the most important aspect of our calculation is that it demonstrates how to integrate out a low-energy photon leg *without fixing an ordering for the remaining legs*. As discussed in part I, when using the light-by-light diagram as a subdiagram in higher-loop calculation this property effectively allows one to unify the calculation of Feynman diagrams of different topologies, and considering the proliferation of diagrams in higher-loop QED calculations it is obviously of great interest to explore what level of simplification can be achieved along these lines.\nOff-shell photon legs can be used for creating internal photons by sewing, or for connecting to external fields. As an example for sewing, we have used our results for a construction of the two-loop vacuum polarisation tensors in the low-energy limit, which allowed us to recover the two-loop *β*-function coefficients for scalar and spinor QED. Although by present-days standards this is easy enough to do using Feynman diagrams, the worldline calculation in the version presented here has the advantage that it replaces the non-gauge invariant decomposition into diagrams by a gauge-invariant decomposition into sixteen substructures. This leads not only to a more compact parameter integral representation, but also has given us an opportunity to probe into the nature of the extensive cancellations that one generally finds in perturbative QED calculations. If, as is sometimes assumed, these were entirely due to gauge invariance, one might have expected the cancellation of the double poles in the scalar and spinor QED *β*-function calculations to have happened already at the level of each of the gauge-invariant partial structures; instead, we have seen an intricate pattern of cancellations between the various structures. Another interesting result of this calculation has been that four of the sixteen partial structures, namely the one involving a three-cycle, drop out in the sewing. It remains to be seen whether this fact admits a generalization to larger numbers of external photons and multiple sewing.\nAs an example of an external-field calculation, we have applied our formulas to a calculation of the low-energy Delbrück scattering cross sections for both scalar and spinor QED. For the spinor QED case this served just as another check of efficiency and correctness, while the scalar QED result is new, to the best of our knowledge. Both the scalar and spinor QED calculations have been presented in a way that would be easy to adapt to other external fields.\nThe forthcoming third part of this series is devoted to the explicit, *D*-dimensional calculation of the off-shell four-photon amplitudes with only one leg taken in the low-energy limit and its applications.\n#### Acknowledgments:\nWe would like to thank D. Broadhurst, F. Karbstein and D. Kreimer for discussions and correspondence. C. Lopez-Arcos and M. A. Lopez-Lopez thank CONACYT for financial support. N. Ahmadiniaz and M. A. Lopez-Lopez would like to thank R. Schützhold for his support.\nCollection of integral formulas\n===============================\nHere we collect a number of results for the parameter integrals appearing in one-loop worldline calculations, mostly taken from. All Green’s functions have been rescaled to the unit circle, *T* = 1.\nIntegrating out a low-energy leg\n--------------------------------\nAll integrals encountered in the integrating out of a low-energy photon can, using the identities, be reduced to integrals over polynomials in $\\dot G\\_{ij}$. For this type of integrals a closed-form master formula is available even for the most general (abelian) case, integrating an arbitrary monomial in the $\\dot G\\_{ij}$’s involving an arbitrary number of variables in one of the variables, and giving the result as a polynomial in the remaining $\\dot G\\_{ij}$’s : 01 du G(u,u1)k1 G(u,u2)k2 G(u,un)kn &=& 12n i=1n ji lj=0kj kjlj Gijkj-lj li=0kikili \n&&\n( jiGij +1 ) 1+j=1n lj - (-1)ki-li ( jiGij -1 )1+j=1n lj . \n[intmaster] At the four-photon level, apart from the single four-point integral\n \n$$\\label{int6}\n\\int\\_0^1 du\\_4\\, \\dot{G}\\_{41} \\dot{G}\\_{42} \\dot{G}\\_{43} = - \\frac{1}{6}(\\dot{G}\\_{12}-\\dot{G}\\_{23})(\\dot{G}\\_{23}-\\dot{G}\\_{31})(\\dot{G}\\_{31}-\\dot{G}\\_{12})\\,,$$ \nonly three-point integrals appear, for which the master formula reduces to\n \n$$\\label{int7}\n\\int\\_0^1 du\\_3 \\,\\dot{G}\\_{13}^k \\dot{G}\\_{32}^l= \\frac{k!l!}{2}\\sum\\_{m=0}^l \\frac{(1-(-1)^{k+l-m+1})\\dot{G}\\_{12}^m - (1-(-1)^m)\\dot{G}\\_{12}^{k+l-m+1} }{m!(k+l-m+1)!}\\,.$$ \nHowever, the latter formula generally does not provide the most compact way of writing the result in terms of the $\\dot G\\_{ij}$, therefore for easy reference we provide below a list of optimized versions for all the integrals that appeared in our calculations.\n \n$$\\begin{split}\\label{intGdots}\n &\\int\\_0^1 du\\_3 \\, \\dot{G}\\_{13} = 0\\,, \\\\\n &\\int\\_0^1 du\\_3 \\, \\dot{G}\\_{13}^2 = \\frac{1}{3}\\,, \\\\\n &\\int\\_0^1 du\\_3 \\, \\dot{G}\\_{13}^4 = \\frac{1}{5}\\,, \\\\\n &\\int\\_0^1 du\\_3 \\dot{G}\\_{13} \\dot{G}\\_{32} = \\frac{1}{6} - \\frac{1}{2}\\dot{G}\\_{12}^2 \\,, \\\\\n &\\int\\_0^1 du\\_3 \\, \\dot{G}\\_{13} \\dot{G}\\_{32}^2 =\\frac{1}{3}(\\dot{G}\\_{12} - \\dot{G}\\_{12}^{3})\\,, \\\\\n &\\int\\_0^1 du\\_3 \\, \\dot{G}\\_{23}^3 \\dot{G}\\_{31} = \\frac{3!}{5!}-\\frac{\\dot{G}\\_{12}^{4}}{4}\\,, \\\\\n &\\int\\_0^1 du\\_3 \\, \\dot{G}\\_{13}^2 \\dot{G}\\_{32}^2 =\\frac{4}{5!}-\\frac{\\dot{G}\\_{12}^{4}}{3!} + \\frac{\\dot{G}\\_{12}^{2}}{3!} \\,, \\\\\n &\\int\\_0^1 du\\_3 \\, \\dot{G}\\_{13}^4 \\dot{G}\\_{32} = \\frac{1}{5}(\\dot{G}\\_{12} - \\dot{G}\\_{12}^{5}) \\,,\\\\\n &\\int\\_0^1 du\\_3 \\, \\dot{G}\\_{13}^3 \\dot{G}\\_{32}^2 = \\frac{1}{10}(\\dot{G}\\_{12} - \\dot{G}\\_{12}^{5})\\,,\n \\end{split}$$ \n \n$$\\label{intGs}\n \\begin{split}\n &\\int\\_0^1 du\\_3 \\, G\\_{13} = \\frac{1}{6}\\,, \\\\\n &\\int\\_0^1 du\\_3 \\, G\\_{13} G\\_{32} = \\frac{1}{30} - \\frac{1}{6} G\\_{12}^2 \\,, \\\\\n &\\int\\_0^1 du\\_3 \\, G\\_{23} \\dot{G}\\_{31} = \\frac{1}{3}\\dot{G}\\_{12}G\\_{12}\\,.\n \\end{split}$$ \nThe functions *Y**n**l* and tensor reduction\n--------------------------------------------\nHere we study some properties of the functions *Y**n**l* defined in,\n \n$$\\label{def2Ynl}\nY\\_{nl} = \\int\\_0^{\\infty} \\frac{dT}{T}T^{n-D/2} ~\\int\\_0^1 du\\_1\\int\\_0^1 du\\_2 ~ G\\_{12}^l \\e^{-T[m^2 - G\\_{12} k\\_1 \\cdot k\\_2]} \\,.$$ \nIn terms of $\\hat{k}\\_{12} = \\frac{k\\_1 \\cdot k\\_2}{m^2} $ and using the translational invariance to fix *u*2 = 0 and *u*1 = *u*, so that *G*12 = *u*(1 − *u*), we can perform the following tensor reduction Ynl &=& 0 Tn-D/2 01 du [u(1-u)]l -Tm2[1 - u(1-u)12] \n&=& 01 du \n&=& 01 du . [Ynlcalc] This leaves us with the single integral\n \n$$\\begin{split}\n\\int\\_0^1 du ~ \\frac{1}{[1 - u(1-u)\\hat{k}\\_{12}]^{n-l-\\frac{D}{2}}} = {}\\_2F\\_1\\left(1,n-l-\\frac{D}{2};\\frac{3}{2};\\frac{\\hat{k}\\_{12}}{4} \\right) \\,.\n\\end{split}$$ \nSo, in terms of the gaussian hypergeometric function 2*F*1,\n \n$$\\label{Ynl\\_2F1}\n\\begin{split}\nY\\_{nl} = \\frac{\\Gamma\\left(n-\\frac{D}{2}-l\\right)}{m^{2n-D} } ~\\frac{d^l}{d\\hat{k}\\_{12}^l} ~ {}\\_2F\\_1 \\left(1,n-l-\\frac{D}{2}; \\frac{3}{2};\\frac{\\hat{k}\\_{12}}{4} \\right) \\,.\n\\end{split}$$ \nIn Section [section-beta] we need the elementary integral [u-integrals] (k2) Ynl & = & 01 du l-- \n& = & B (l--+1,l--+1). \nThe following identities are not used in the present paper, but let us include them here for whatever their worth might be:\n \n$$Y\\_{nl} = \\Gamma\\left(n-\\frac{D}{2}\\right) \\hat{k}\\_{12}^{-l}\\, \\sum\\_{j=0}^{l} (-1)^j \\binom{l}{j} \\left[\\Gamma\\left(n -j -\\frac{D}{2}\\right)\\right]^{-1} m^{-2j} \\, Y\\_{(n-j)0} \\,,\n \\label{YnltoYn0}$$ \nwhich can be used to recursively eliminate all *Y**n**l*’s with *l* ≠ 0. And\n \n(2*n* − *D*)(2*n* − *D* − 1) *Y**n*0 = [2 − (2*n* − *D*)(*k̂*122 − 8)] *m*2 *Y*(*n* + 1)0 + (*k̂*122 − 4)*m*4 *Y*(*n* + 2)0 , \nthat follows from the following identity of contiguous functions \n \n(*c* − *b*)2*F*1(*a*, *b* − 1; *c*; *z*) + (2*b* − *c* − *b**z* + *a**z*)2*F*1(*a*, *b*; *c*; *z*) + *b*(*z* − 1)2*F*1(*a*, *b* + 1; *c*; *z*) = 0. \nExplicit results for ${Q\\_{{\\rm scal}(34)}}$ and ${Q\\_{{\\rm spin}(34)}}$\n========================================================================\nIn this appendix, we explicitly write down the results for $Q\\_{{{\\rm scal}}}$ and $Q\\_{{{\\rm spin}}}$ after integration over *u*4 and *u*3, under the assumption that photons 3 and 4 are taken in the low-energy limit.\nScalar QED\n----------\nThe explicit form of ${Q\\_{{\\rm scal}(34)}}$ is written as\n \n$${Q\\_{{\\rm scal}(34)}}^4(1234) = \\frac{2}{3} Z\\_4(1234) \\left(G\\_{12}-4G\\_{12}^2\\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^4(2314) = \\frac{1}{9} Z\\_4(2314)\\left(1-12 G\\_{12} + 36 G\\_{12}^2 \\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^4(3124) = \\frac{2}{3} Z\\_4(3124) \\left(G\\_{12}-4G\\_{12}^2\\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^3(123;4) = -\\frac{T}{9} Z\\_3(123) k\\_2 \\cdot f\\_4 \\cdot k\\_1 \\left(G\\_{12} - 10 G\\_{12}^2 + 24 G\\_{12}^3\\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^3(234;1) = 0,$$ \n \n$${Q\\_{{\\rm scal}(34)}}^3(341;2) = 0,$$ \n \n$${Q\\_{{\\rm scal}(34)}}^3(412;3) = -\\frac{T}{9} Z\\_3(412) k\\_2 \\cdot f\\_3 \\cdot k\\_1 \\left(G\\_{12} - 10 G\\_{12}^2 + 24 G\\_{12}^3\\right),$$ \n \n$$\\begin{split}\n{Q\\_{{\\rm scal}(34)}}^2(12;34) &= -\\frac{T}{18} Z\\_2(12) \\left(1- 4G\\_{12} \\right) \\Bigg\\{\\Bigg[\\frac{1}{5} k\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot k\\_1 + \\left(\\frac{1}{5} - 6 G\\_{12}^2 \\right) k\\_1 \\cdot f\\_3 \\cdot f\\_4 \\cdot k\\_2\\\\\n&\\quad + (1\\leftrightarrow 2) \\Bigg] + 2T \\left(1- 4G\\_{12} \\right)G\\_{12}^2 \\, k\\_1 \\cdot f\\_3 \\cdot k\\_2 k\\_1 \\cdot f\\_4 \\cdot k\\_2 \\Bigg\\} \\,,\n\\end{split}$$ \n \n$${Q\\_{{\\rm scal}(34)}}^2(13;24) + {Q\\_{{\\rm scal}(34)}}^2(23;14) = -\\frac{T}{9} Z\\_2(13) k\\_1 \\cdot f\\_2 \\cdot f\\_4 \\cdot k\\_1 \\, \\left(1- 4G\\_{12} \\right)G\\_{12} + (1\\leftrightarrow 2) \\,,$$ \n \n$${Q\\_{{\\rm scal}(34)}}^2(14;23) + {Q\\_{{\\rm scal}(34)}}^2(24;13)= - \\frac{T}{9} Z\\_2(14) k\\_1 \\cdot f\\_2 \\cdot f\\_3 \\cdot k\\_1 \\, \\left(1- 4G\\_{12} \\right)G\\_{12} + (1\\leftrightarrow 2) \\,,$$ \n \n$${Q\\_{{\\rm scal}(34)}}^2(34;12) = 0,$$ \n \n$${Q\\_{{\\rm scal}(34)}}^{22}(12,34) = \\frac{1}{3}Z\\_2(12) Z\\_2(34) \\left(1-4G\\_{12}\\right),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^{22}(13,24) =\\frac{1}{9} Z\\_2(13)Z\\_2(24),$$ \n \n$${Q\\_{{\\rm scal}(34)}}^{22}(14,23) = \\frac{1}{9} Z\\_2(14) Z\\_2(23).$$ \nSpinor QED\n----------\nThe explicit form of ${Q\\_{{\\rm spin}(34)}}$ is written as\n \n$${Q\\_{{\\rm spin}(34)}}^4(1234) = - \\frac{4}{3} Z\\_4(1234) \\left(G\\_{12}+2G\\_{12}^2 \\right),$$ \n \n$${Q\\_{{\\rm spin}(34)}}^4(2314) = - \\frac{4}{9} Z\\_4(2314) \\left(2- 6G\\_{12} - 9 G\\_{12}^2 \\right),$$ \n \n$${Q\\_{{\\rm spin}(34)}}^4(3124) = - \\frac{4}{3} Z\\_4(3124) \\left(G\\_{12}+2G\\_{12}^2 \\right),$$ \n \n$${Q\\_{{\\rm spin}(34)}}^3(123;4) = \\frac{2T}{9} Z\\_3(123) k\\_2 \\cdot f\\_4 \\cdot k\\_1 \\left(G\\_{12} - G\\_{12}^2 - 12G\\_{12}^3\\right),$$ \n \n$${Q\\_{{\\rm spin}(34)}}^3(234;1) = 0,$$ \n \n$${Q\\_{{\\rm spin}(34)}}^3(341;2) = 0,$$ \n \n$${Q\\_{{\\rm spin}(34)}}^3(412;3) = \\frac{2T}{9} Z\\_3(412) k\\_2 \\cdot f\\_3 \\cdot k\\_1 \\left(G\\_{12} - G\\_{12}^2 - 12G\\_{12}^3\\right),$$ \n \n$$\\begin{split}\n{Q\\_{{\\rm spin}(34)}}^2(12;34) &= \\frac{2T}{9} Z\\_2(12) \\,G\\_{12"},"data_id":{"kind":"string","value":"arxiv_0183891"}}},{"rowIdx":45999,"cells":{"content":{"kind":"string","value":"449346390970478); 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To apply the bound on effective resistance in terms of the reciprocals of *ψ* and *φ* we couple them to *d* and *d*\\*. Lemmas [dom], [betterprob] and [sdom] will help us achieve this.\n[dom] Let G ∼ G(*n*, *p*), *I*0 : = {*u*, *v*} ⊂ *V* and *i*, *k* ≥ 0. Run MBFS(G, *I*0).\n1. [itm:gam4] Then ∣*S*1∣ ∼ *B**i**n*(*n* − 2, (1 − *p*)2) and ∣*I*1∣ ∼ *B**i**n*(*n* − 2, 2*p*(1 − *p*)).\n2. [itm:gam1] Conditioning on {*x* ∈ *I**k*} and ∣*S**k*(*x*)∣, then \n*d*\\*(*x*) ∼ *B**i**n*(∣*S**k*(*x*)∣, *p*).\n3. [itm:gam2]Conditioning on {*x* ∈ *I**k*}, ∣*S**k* + *i*∣, ∣*I**k* + *i*∣ and *d**i*\\*(*x*), then \n*d**i* + 1\\*(*x*) ∼ *B**i**n*(∣*S**k* + *i*∣, *d**i*\\*(*x*) ⋅ *p* ⋅ (1 − *p*)∣*I**k* + *i*∣ − 1).\n*Item* : a vertex in *S*0 is in *S*1 if it is not connected to either vertex in *I*0. This happens independently w.p. (1 − *p*)2 for each of the *n* − 2 vertices in *S*0 thus ∣*S*1∣ ∼ *B**i**n*(*n* − 2, (1 − *p*)2).\nA vertex in *S*0 is in *I*1 if it is connected to exactly one vertex in *I*0. This happens independently with probability 2*p*(1 − *p*) for each of the *n* − 2 vertices in *S*0 thus ∣*I*1∣ ∼ *B**i**n*(*n* − 2, 2*p*(1 − *p*)).\n*Item* : recall the definitions of Γ1\\*(*x*) and *S**k*(*x*) for *x* ∈ *I**k*, given by and respectively. Observe the following relation: \nΓ1\\*(*x*) = (Γ1(*x*) ∩ *S**k*) ∖ ⋃*y* ∈ *I**k*, *y* ≠ *x*Γ1(*y*) = Γ1(*x*) ∩ *S**k*(*x*). \n Since we completely remove the vertices if they clash, and the edges of G are independent, the order MBFS explores the neighbourhoods of each *y* ∈ *I**k* is unimportant. Assume that we have explored the neighbourhood of every *y* ∈ *I**k* with *y* ≠ *x*. We then know which vertices in the neutral set *S**k* will not clash if included in Γ1(*x*) and these are the vertices in *S**k*(*x*). Since edges occur independently with probability *p*, conditioning on ∣*S**k*(*x*)∣ yields *d*\\*(*x*) ∼ *B**i**n*(∣*S**k*(*x*)∣, *p*).\n*Item* : for a vertex *v* ∈ *S**k* + *i* we have *v* ∈ Γ*i* + 1\\*(*x*) when there is exactly one edge *y**v* ∈ *E*(G) where *y* ∈ Γ*i*\\*(*x*) and there is no edge of the form *y*ʹ*v* ∈ *E* where *y*ʹ ∈ *I**k* + *i* and *y*ʹ ≠ *y*. Conditioning on the sizes of *I**k* + *i* and Γ*i*\\*(*x*) we see that each *v* ∈ *S**k* + *i* is a member of Γ*i* + 1\\*(*x*) with probability *d**i*\\*(*x*) ⋅ *p* ⋅ (1 − *p*)∣*I**k* + *i*∣ − 1. These events are independent, thus\n \n$$\\begin{aligned}\n{d}\\_{i+1}^\\*(x) &\\sim Bin\\left(|S\\_{k+i}|,{d}\\_{i}^\\*(x)\\cdot p \\cdot (1-p)^{|I\\_{k+i}|-1}\\right), \\end{aligned}$$ \nconditional on ∣*S**k* + *i*∣, ∣*I**k* + *i*∣ and *d**i*\\*(*x*).\nLet *x* ∈ *I**k*. Choosing *i* = 0 in Lemma [dom] gives *d*\\*(*x*) ∼ *B**i**n*(∣*S**k*∣, *p*(1 − *p*)∣*I**k*∣ − 1) conditional on ∣*S**k*∣ and ∣*I**k*∣ whereas Lemma [dom] gives *d*\\*(*x*) ∼ *B**i**n*(∣*S**k*(*x*)∣, *p*) conditional on ∣*S**k*(*x*)∣. To relate to observe that conditional on ∣*S**k*∣ and ∣*I**k*∣, ∣*S**k*(*x*)∣ ∼ *B**i**n*(∣*S**k*∣, (1 − *p*)∣*I**k*∣ − 1). Item then follows as if *X* ∼ *B*(*n*, *p*) and, conditional on *X*, *Y* ∼ *B*(*X*, *q*), then *Y* is a simple binomial variable with distribution *Y* ∼ *B*(*n*, *p**q*).\nThe next two lemmas provide tail estimates on the sizes of Γ*i* and Γ*i*\\*. We prove them by induction where the inductive step comes from applying Chernoff bounds to the binomial distributions described in Lemma [dom]. For Lemma [neighup] this induction shows that w.h.p. the sequence *d*(*u*), *d*2(*u*), … is bounded above by the sequence *a*1*n**p*, *a*2(*n**p*)2, … where the *a**i* satisfy a recurrence relation. This recurrence can later be solved to give bounds on the sequence *a**i* based on the exceptional probability desired. This strategy is inspired by.\n[neighup] Let G ∼ G(*n*, *p*) where *n**p* = *ω*(1). Then for *u* ∈ *V* and *α* ∈ R, *α* ≥ 6, \nP (∣*B**i*(*u*)∣ > *α*(*n**p*)*i*) = *o*(*e*− *α**n**p*/3). \nLet ${\\ensuremath{\\mathcal E}}\\_i:=\\{ {d}\\_{i}(u) \\leq a\\_i(np)^i\\}$ and $\\mathcal{H}\\_i:=\\bigcap\\limits\\_{j=0}^i {\\ensuremath{\\mathcal E}}\\_i$. We shall show \n$${\\mathbb{P}\\!\\left(\\mathcal{H}\\_i^c\\right)} \\leq \\sum\\limits\\_{j=0}^i e^{-j}\\exp\\left(-\\frac{\\lambda^2}{2(1 + \\lambda/3\\sqrt{np})}\\right),$$ \nby induction on *i* ≥ 0, where *a**i* ≥ 0 is given by the recurrence \n$$a\\_{i+1} = a\\_{i} + \\frac{\\lambda\\_i\\sqrt{a\\_{i}}}{(np)^{(i+1)/2}}, \\; a\\_0=1,$$ \nand $\\lambda\\_i =\\sqrt{3i + \\lambda^2 }$ for some *λ* specified later. For the base case *d*0(*u*) = 1 = *a*0. Now observe\n \n$$\\begin{aligned}\n\\label{reccurance}\n a\\_i(np)^{i}np + \\lambda\\_i \\sqrt{a\\_i(np)^{i+1}}\n =(np)^{i+1} \\!\\left( \\! a\\_i + \\lambda\\_i\\frac{\\sqrt{a\\_i}}{(np)^{(i+1)/2}}\\!\\right)= a\\_{i+1} (np)^{i+1}.\n \\end{aligned}$$ \nConditional on *d**i*(*u*) we have *d**i* + 1(*u*) ≼ 1*B**i**n*(*d**i*(*u*) ⋅ *n*, *p*). Thus by above\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left(\\left({\\ensuremath{\\mathcal E}}\\_{i+1}\\right)^c\\cap {\\ensuremath{\\mathcal H}}\\_i\\right)}&:={\\mathbb{P}\\!\\left(\\{{d}\\_{i+1}(u)> a\\_{i+1} (np)^{i+1}\\} \\cap {\\ensuremath{\\mathcal H}}\\_i\\right)} \\\\\n &\\leq {\\mathbb{E}\\!\\left[{\\mathbb{P}\\!\\left(Bin\\left({d}\\_{i}(u)\\cdot n,p\\right)> a\\_i(np)^{i}np + \\lambda\\_i \\sqrt{a\\_i(np)^{i+1}}\\;\\bigg|{d}\\_i(u)\\right)}\\mathbf{1}\\_{\\mathcal{H}\\_i}\\right]}.\\end{aligned}$$ \nNow by the Chernoff bounds, Lemma [chb], we have\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left(\\left({\\ensuremath{\\mathcal E}}\\_{i+1}\\right)^c\\cap {\\ensuremath{\\mathcal H}}\\_i\\right)}&\\leq {\\mathbb{E}\\!\\left[\\exp\\left(-\\frac{\\lambda\\_i^2 a\\_i(np)^{i+1}}{2\\left(a\\_i(np)^{i+1} + \\lambda\\_i \\sqrt{a\\_i(np)^{i+1}}/3\\right)}\\right)\\mathbf{1}\\_{\\mathcal{H}\\_i}\\right]}\\notag \\\\\n &= {\\mathbb{E}\\!\\left[\\exp\\left(-\\frac{3i+\\lambda^2}{2\\left(1 + \\lambda\\_i /3\\sqrt{a\\_i(np)^{i+1}}\\right)}\\right) \\mathbf{1}\\_{\\mathcal{H}\\_i}\\right]}\\leq e^{-i}e^{-\\frac{\\lambda^2}{2\\left(1 + \\lambda /3\\sqrt{np}\\right)}}{\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}\\_i \\right)},\\label{dsigdsg} \\end{aligned}$$ \nfor *n* large enough since *a**i* ≥ 1 and *n**p* = *ω*(1) thus $\\lambda\\_i /\\sqrt{a\\_i(np)^{i+1}} \\leq \\lambda /\\sqrt{np} $. Now observe that ${\\ensuremath{\\mathcal H}}\\_{i+1} \\subseteq {\\ensuremath{\\mathcal H}}\\_{i}$ and ${\\ensuremath{\\mathcal H}}\\_{i}$ is the disjoint union of ${\\ensuremath{\\mathcal H}}\\_{i+1}$ and $ \\left({\\ensuremath{\\mathcal E}}\\_{i+1}\\right)^c\\cap {\\ensuremath{\\mathcal H}}\\_i$. Hence by we have\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}\\_{i+1}\\right)} &= {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}\\_i\\right)} - {\\mathbb{P}\\!\\left(\\left({\\ensuremath{\\mathcal E}}\\_{i+1}\\right)^c\\cap {\\ensuremath{\\mathcal H}}\\_i\\right)} = \\left(1- e^{-i}\\exp\\left(-\\frac{\\lambda^2}{2\\left(1 + \\lambda /3\\sqrt{np}\\right)}\\right)\\right){\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}\\_i \\right)}. \\intertext{If we continue iteratively and recall ${\\ensuremath{\\mathcal H}}\\_0=\\{{d}\\_0(u) \\leq 1 \\}$, thus ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}\\_0\\right)}=1$, then we have} \n {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}\\_{i+1}\\right)}&= \\prod\\limits\\_{j=0}^{i}\\left(1- e^{-j}\\cdot e^{-\\frac{\\lambda^2}{2\\left(1 + \\lambda /3\\sqrt{np}\\right)}}\\right){\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}\\_0 \\right)}\\geq 1- \\sum\\limits\\_{j=0}^{i}e^{-j}\\cdot e^{-\\frac{\\lambda^2}{2\\left(1 + \\lambda /3\\sqrt{np}\\right)}}.\\end{aligned}$$ \nLet $\\lambda=k\\sqrt{np}$ for any *k* ≥ 3 and observe that\n \n$$\\begin{aligned}\n\\label{phcomp}\n {\\mathbb{P}\\!\\left(\\left({\\ensuremath{\\mathcal H}}\\_i\\right)^c\\right)}&\\leq \\sum\\limits\\_{j=0}^{i} e^{-j}\\cdot e^{-\\frac{\\lambda^2}{2\\left(1 + \\lambda /3\\sqrt{np}\\right)}}={O\\!\\left(\\exp\\left(-\\frac{k^2 np }{2(1 + k/3)}\\right)\\right)}=O\\left(e^{-3knp/4}\\right),\n \\end{aligned}$$ \nwhere the last equality follows since $\\frac{k^2 }{2(1 + k/3)} \\geq \\frac{3k}{4} $ provided *k* ≥ 3, and *n**p* = *ω*(1).\nWe will show that *a**i* ≤ 2*k* for all *i*. Since *a*0 = 1 ≤ 2*k* assume *a**i* ≤ 2*k*, then by\n \n$$\\begin{aligned}\n a\\_{i+1} &=a\\_i + \\frac{\\lambda\\_i\\sqrt{a\\_i}}{(np)^{(i+1)/2}} = 1 + \\frac{\\lambda\\_0\\sqrt{a\\_0}}{\\sqrt{np}}+ \\sum\\limits\\_{j=1}^{i}\\frac{\\lambda\\_j\\sqrt{a\\_j}}{(np)^{(j+1)/2}}. \\intertext{Recall that $\\lambda\\_i = \\sqrt{3i+\\lambda^2}$ and observe that $\\lambda\\_0=\\lambda=k\\sqrt{np}$. Thus we have }\n a\\_{i+1}&= 1 + k + \\sum\\limits\\_{j=1}^{i}\\frac{ \\sqrt{3j+k^2np}\\sqrt{2k}}{(np)^{(j+1)/2}} =1+k+O\\!\\left((np)^{-1/2}\\right)\\leq 2k,\n \\end{aligned}$$ \nfor large *n*. Finally, conditional on $\\bigcap\\limits\\_{j=0}^i\\{{d}\\_j(u) \\leq 2k(np)^i \\}\\subseteq {\\ensuremath{\\mathcal H}}\\_i$ we have \n$$|B\\_i(u)| = \\sum\\limits\\_{j=0}^i{d}\\_j(u) \\leq \\sum\\limits\\_{j=0}^i 2k(np)^i \\leq \\alpha(np)^i,$$ \n where *α* = (2 + 1/100)*k*, now for this *α* we have ${\\mathbb{P}\\!\\left(\\left( {\\ensuremath{\\mathcal H}}\\_i\\right)^c\\right)} =o\\left(e^{-\\alpha np/3}\\right) $ by.\n[neighlow] Let G ∼ G(*n*, *p*), and *i* ∈ Z satisfy 1 ≤ *i* ≤ ⌊log(*n*)/log(*n**p*)⌋ − 3. Let Ψ2 be defined with respect to MBFS(G, {*u*, *v*}) for *u*, *v* ∈ *V* such that *u* ≠ *v*.\n1. [itm:bd1]Let *c* > 0, *n**p* ≥ *c*log*n*. Then ${\\mathbb{P}\\!\\left({d}\\_i^\\*(y)< 15 (np)^{i-1}\\Big| \\;y \\in \\Psi\\_2\\right)} = o\\left(e^{- 4np }\\right).$\n2. [itm:bd2] If *n**p* = *ω*(log*n*) then ${\\mathbb{P}\\!\\left({d}\\_i^\\*(y)<\\tfrac{9}{10} (np)^{i}\\Big|\\; y \\in \\Psi\\_2\\right)} = o\\left(n^{-K} \\right)$ for any fixed *K* ≥ 0.\n3. [itm:bd3] If *n**p* − log*n* → ∞ then PC (∣*B**j*(*v*)∣ < 15(*n**p*)*i* − 5) = *o*(*n*− 4) for any 5 ≤ *j* ≤ *i* − 2.\nWe will first set up the general framework for a neighbourhood growth bound and then apply this bound under different conditions to prove Items, and.\nRun MBFS(G, {*u*, *v*}) and let *y* ∈ *I**h*, *n**i* : = ∣*S**i* + *h*∣, *p**i* : = *p* ⋅ (1 − *p*)∣*I**i* + *h*∣ − 1 and *r**i* = ∏*j* = *i*0*i**n**j**p**j*. We wish to show that there exists some *i*0 ∈ Z, *i*0 ≥ 0 such that for all *i* ≥ *i*0:\n \nP (*d**i* + 1\\*(*y*) < *a**i* + 1*r**i*) ≤ (*i* + 1)exp( − *λ*2/2), \nwhere *a**i* ≥ 0 satisfies $ a\\_{i+1} = a\\_{i} - \\lambda\\sqrt{a\\_{i}}/\\sqrt{r\\_{i}}$, for some initial *a**i*0 we will find later. Observe\n \n$$\\begin{aligned}\n a\\_ir\\_{i-1} n\\_{i}p\\_{i} - \\lambda \\sqrt{a\\_ir\\_{i-1} n\\_{i}p\\_{i}}\n =\\left( \\! a\\_i - \\lambda\\frac{\\sqrt{a\\_i}}{\\sqrt{r\\_{i}}}\\!\\right)r\\_{i} = a\\_{i+1} r\\_{i}.\n \\end{aligned}$$ \nApplying Lemma [dom] and conditioning on F*i* + *h* yields *d**i* + 1\\*(*y*) ∼ *B**i**n*(*n**i*, *d**i*\\*(*y*)*p**i*).\nLet H*i* : = {*d**i*\\*(*u*) ≥ *a**i**r**i* − 1} ∈ F*i* + *h* and assume ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}\\_i^c\\right)}\\leq ie^{-\\lambda^2/2} $. Now by Lemma [chb],\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left({d}\\_{i+1}^\\*(y)< a\\_{i+1} r\\_{i}\\right)} &= {\\mathbb{E}\\!\\left[{\\mathbb{P}\\!\\left({d}\\_{i+1}^\\*(y)< a\\_ir\\_{i-1} n\\_{i}p\\_{i} - \\lambda \\sqrt{a\\_ir\\_{i-1} n\\_{i}p\\_{i}} \\big|\\mathfrak{F}\\_{i+h}\\right)} \\right]}\\\\\n &\\leq {\\mathbb{E}\\!\\left[{\\mathbb{P}\\!\\left(Bin\\left(n\\_{i},{d}\\_{i}^\\*(y) p\\_{i}\\right)< a\\_ir\\_{i} - \\lambda \\sqrt{a\\_ir\\_{i}}\\big|\\mathfrak{F}\\_{i+h}\\right)} \\mathbf{1}\\_{\\mathcal{H}\\_i}\\right]} + {\\mathbb{P}\\!\\left(\\mathcal{H}\\_i^c\\right)}\\\\\n &\\leq \\exp\\left(-\\lambda^2a\\_ir\\_{i}/(2 a\\_ir\\_{i})\\right) +i\\exp\\left(-\\lambda^2/2\\right) =(i+1)\\exp\\left(-\\lambda^2/2\\right).\n \\end{aligned}$$ \nThe above always holds, however it may be vacuous as if *i* is too large then *a**i* may be negative. This can also happen for an incorrect choice of the starting time *i*0 and initial value *a**i*0. We address this in the application making sure to condition on events where everything is well defined. In this spirit let ℓ : = ⌊log(*n*)/log(*n**p*)⌋ − *h* − 1 and define the event \nI : = ⋂*i* = 0ℓ{∣*I**i* + *h*∣ ≤ 26(*n**p*)*i* + *h*} ∩ {*d**i*\\*(*y*) ≤ 13(*n**p*)*i*} ∩ {∣*S**i* + *h*∣ ≥ *n* − 26(*n**p*)*i* + *h*}. \nConditioning on the event I and the filtration F*i* + *h* for any *i* ≤ ℓ ensures that *B**i**n*(*n**i*, *d**i*\\*(*y*)*p**i*) is a valid probability distribution and *n**i**p**i* = (1 − *o*(1))*n**p*. By Lemma [neighup] with *α* = 13,\n \n$$\\begin{aligned}\n\\label{probD}\n {\\mathbb{P}\\!\\left(\\mathcal{I}^c\\right)} &\\leq 2\\sum\\limits\\_{i=0}^{\\ell+2} {\\mathbb{P}\\!\\left(|B\\_{i}(u)| > 13(np)^{i}\\right)} =o\\left(e^{ - 4np}\\right).\n \\end{aligned}$$ \n*Item* : recall from that if *y* ∈ Ψ2(*u*) ∪ Ψ2(*v*) ⊆ *I*2 then *d*\\*(*y*) > *D*, defined at. Thus conditional on I ∩ F3, *d*2\\*(*y*) ≽ 1*B**i**n*(*n*(1 − *o* (1)), *D**p*(1 − *o* (1))). If we choose $\\lambda = 3\\sqrt{np}$, then since in this regime $D= \\max\\left\\{\\lceil\\frac{50}{c} \\rceil, 50\\right\\}$ applying Lemma [chb] yields\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left({d}\\_{2}^\\*(y)< Dn\\_1p\\_1/2\\right)}&={\\mathbb{E}\\!\\left[\\mathbb{P}\\left({d}\\_{2}^\\*(y)< Dn\\_1p\\_1/2\\big|\\mathfrak{F}\\_3\\right)\\left(\\mathbf{1}\\_{\\mathcal{I}}+\\mathbf{1}\\_{\\mathcal{I}^c} \\right)\\right]} \\leq e^{-Dnp /10}+{\\mathbb{P}\\!\\left(\\mathcal{I}^c\\right)} \\leq e^{-\\lambda^2/2}.\n \\end{aligned}$$ \nTake *i*0 = 1 and *a*2 = *D*/3 since on I we have *D*/2*n*1*p*1 ≥ *D**n**p*/3. Now *a*2 ≥ … ≥ *a**i* so on the event I we have the following for any 3 ≤ *i* ≤ ⌊log(*n*)/log(*n**p*)⌋ − 3\n \n$$\\begin{aligned}\n a\\_i = a\\_{2} - \\sum\\limits\\_{k=2 }^{i-1} \\frac{\\lambda \\sqrt{a\\_{k}}}{\\sqrt{r\\_{k}}} \\geq \\frac{D}{3} - (3 +{o\\!\\left(1\\right)})\\sqrt{\\frac{D}{3np} } \\geq 16,\n \\end{aligned}$$ \nsince conditional on I we have *r**i* = ∏*j* = *i*0*i**n**j**p**j* ≥ (1 − *o* (1))(*n**p*)*i* for any 1 ≤ *i* ≤ ⌊log(*n*)/log(*n**p*)⌋ − 3. Notice also that *d*\\*(*y*) > *D* > 15(*n**p*)0 so by\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left({d}\\_{i+1}^\\*(y)< 15 (np)^{i}\\right)} &\\leq {\\mathbb{P}\\!\\left({d}\\_{i+1}^\\*(y)< a\\_{i+1} r\\_{i}\\right)} + {\\mathbb{P}\\!\\left(\\mathcal{I}^c\\right)} \\leq (i+1)e^{-\\lambda^2/2 } +o\\left(e^{- 4np }\\right)=o\\left(e^{ - 4np}\\right).\n \\end{aligned}$$ \n*Item* : for this case on the event I we have *n**i**p**i* = (1 − *o*(1))*n**p* = *ω*(log*n*) for every 0 ≤ *i* ≤ ℓ, this is why *D* = 0 when *n**p* = *ω*(log*n*) in the definition as we do not need to rely on the fact that *d*\\*(*y*) greater than some constant to start the branching.\nFix *K* > 0 and let $\\lambda=\\sqrt{3K\\log n}$. As before conditioning on I ∩ F3 ensures that *d*\\*(*y*) ∼ *B**i**n*(*n*0, *p*0) ≽ 1*B**i**n*(*n*(1 − *o* (1)), *p*(1 − *o* (1))). By Lemma [chb],\n \n$$\\begin{aligned}\n {\\mathbb{E}\\!\\left[ {\\mathbb{P}\\!\\left({d}^\\*(y)< r\\_0 - (5/4)\\lambda\\sqrt{r\\_0} \\big|\\mathfrak{F}\\_3\\right)}\\left(\\mathbf{1}\\_{\\mathcal{I}}+\\mathbf{1}\\_{\\mathcal{I}^c} \\right)\\right]} &\\leq e^{-25\\lambda^2/32} + {\\mathbb{P}\\!\\left(\\mathcal{I}^c\\right)}\\leq \\exp\\left(-\\lambda^2/2\\right).\n \\end{aligned}$$ \nTake *i*0 = 0, *a*1 = 19/20 since on I we have $r\\_0 -(5/4)\\lambda\\sqrt{r\\_0} \\geq 19np/20$. Now *a*1 ≥ … ≥ *a**i* so on the event I we have the following for any 2 ≤ *i* ≤ ⌊log(*n*)/log(*n**p*)⌋ − 3\n \n$$\\begin{aligned}\n a\\_i = a\\_{1} - \\sum\\limits\\_{k=1 }^{i-1} \\frac{\\lambda \\sqrt{a\\_{k}}}{\\sqrt{r\\_{k}}} \\geq \\frac{19}{20} - (1 +{o\\!\\left(1\\right)})\\frac{\\sqrt{19\\cdot 3K \\log n}}{\\sqrt{20np}}= \\frac{19}{20}-o(1) \\geq \\frac{9}{10}.\n \\end{aligned}$$ \nThus for any 1 ≤ *i* ≤ ⌊log(*n*)/log(*n**p*)⌋ − 3, *K* > 0 we have\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left({d}\\_{i}^\\*(y)<9/10(np)^{i}\\right)} &\\leq {\\mathbb{P}\\!\\left({d}\\_{i}^\\*(y)< a\\_{i} r\\_{i-1}\\right)} + {\\mathbb{P}\\!\\left(\\mathcal{I}^c\\right)}\\leq (i+1)e^{-\\lambda^2/2 } +e^{- 4np }\\leq o\\left(n^{-K} \\right).\n \\end{aligned}$$ \n*Item* : We assume that *n**p* = *θ*(log*n*) for larger *p* the result follows from stochastic domination. Since G ∈ C there exists a path *u* : = *u*0, *u*1, …, *u*4 with *u**j* − 1*u**j* ∈ *E* for each 1 ≤ *j* ≤ 4. Let *f*(*u**j*) = ∣{*v* ∈ *V* ∖ {*u*0, …, *u*4} : *u**j**v* ∈ *E*}∣. Then, for *D* from, by Lemma [chb],\n \n$$\\begin{aligned}\n{\\mathbb{P}\\!\\left(f(u\\_j)< D\\right)}&={\\mathbb{P}\\!\\left(Bin(n-\\ell -1,p)<D\\right)} \\leq e^{-(n-\\ell-1)p}\\cdot \\left(\\frac{e(n-\\ell-1)p}{D}\\right)^D=e^{-(1-o(1))np}.\\end{aligned}$$ \nLet E be the event {*d*(*u**j*0) ≥ *D*for some 0 ≤ *j*0 ≤ 4}. Then as {*f*(*u**j*)}*j* = 04 are i.i.d. we have\n \n$$\\begin{aligned}\n\\label{jbd}\n {\\mathbb{P}\\_{\\mathcal{C}}\\!\\left(\\mathcal{E}^c \\right)} &\\leq {\\mathbb{P}\\!\\left(f(u\\_j)< D\\right)}^{5}/{\\mathbb{P}\\!\\left(\\mathcal{C}\\right)} \\leq e^{-5(1-o(1))\\log n} \\leq o\\left(n^{-4} \\right). \\end{aligned}$$ \nOn E there is some *u**j*0 ∈ *V* with *d*(*u*, *u**j*0) = *j*0 ≤ 4 and *d*(*u**j*0) > *D*. We use the stochastic domination *d**i*(*u**j*0) ≽ 1*d**i*\\*(*u**j*0) to bound the growth of ∣*B**i* + *j*0(*u*)∣ from below by that of *d**i*\\*(*u**j*0), where *u**j*0 ∈ *I**j*0 is defined with respect to ${\\operatorname{MBFS}}({\\ensuremath{\\mathcal G}}, \\{u,v \\} )$ for some *v* ∈ *V*. Let $\\lambda =3\\sqrt{\\log n}$ and recall $D= \\max\\left\\{\\lceil\\frac{50}{c} \\rceil, 50\\right\\} $. On I, *r**j*0 + 1 = (1 − *o*(1))*n**p*, thus by Lemma [chb] :\n \n$$\\begin{aligned}\n {\\mathbb{E}\\!\\left[\\mathbb{P}\\left({d}\\_{j\\_0+2}^\\*(u)< dn\\_{j\\_0+1}p\\_{j\\_0+1}/2\\big|\\mathfrak{F}\\_{j\\_0+1}\\right)\\mathbf{1}\\_{\\mathcal{I}\\cap\\mathcal{E}}\\right]} &\\leq {\\mathbb{E}\\!\\left[e^{-dr\\_{j\\_0+1} /8}\\mathbf{1}\\_{\\mathcal{I}\\cap\\mathcal{E}}\\right]} \\leq e^{-\\lambda^2/2}.\n \\end{aligned}$$ \nTake *i*0 = *j*0 + 1 and *a**j*0 + 2 = *d*/3 since on I ∩ E we have *d**n**j*0 + 1*p**j*0 + 1/2 ≥ *d**n**p*/3. Now *a**i*0 ≥ … ≥ *a**i* and on the event I ∩ E we have *r**i* = (1 − *o*(1))(*n**p*)*i* − *j*0. Thus we have the following for any *ɛ* > 0 and *j*0 + 3 ≤ *i* ≤ ⌊log(*n*)/log(*n**p*)⌋ − *j*0 − 1:\n \n$$\\begin{aligned}\n a\\_i = a\\_{j\\_0+2} - \\sum\\limits\\_{k=j\\_0+2 }^{i-1} \\frac{\\lambda \\sqrt{a\\_{k}}}{\\sqrt{r\\_{k}}} \\geq \\frac{d}{3} - (3 +{o\\!\\left(1\\right)})\\sqrt{\\frac{d\\log n}{3(np)^2} }\\geq 16.\n \\end{aligned}$$ \nNotice also *d**j*0 + 1\\*(*y*) > *d* > 15(*n**p*)0. Thus for any 4 ≤ *i* ≤ ⌊log(*n*)/log(*n**p*)⌋ − 5:\n \n$$\\begin{aligned}\n {\\mathbb{P}\\_{\\mathcal{C}}\\!\\left(|B\\_{i+1}(u)|< 15 (np)^{i-5}\\right)} & \\leq {\\mathbb{P}\\_{\\mathcal{C}}\\!\\left({d}\\_{i+1}(u)< 15 (np)^{i-j\\_0-1}\\big|\\mathcal{I},\\mathcal{E}\\right)} + {\\mathbb{P}\\_{\\mathcal{C}}\\!\\left(\\mathcal{I}^c\\right)}+{\\mathbb{P}\\_{\\mathcal{C}}\\!\\left(\\mathcal{E}^c\\right)} \\\\ \n &\\leq {\\mathbb{P}\\!\\left({d}\\_{i+1}^\\*(y)< a\\_{i+1} r\\_{i}\\big|\\mathcal{I},\\mathcal{E}\\right)}/{\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal C}}\\right)} + {\\mathbb{P}\\!\\left(\\mathcal{I}^c\\right)}/{\\mathbb{P}\\!\\left(\\mathcal{C}\\right)}+ {\\mathbb{P}\\_{\\mathcal{C}}\\!\\left(\\mathcal{E}^c\\right)}, \\end{aligned}$$ \nwhich is at most 2(*i* + 1)*e*− *λ*2/2 + *o*(*e*− 4*n**p*) + *o*(*n*− 4) = *o*(*n*− 4) by the bounds on ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal C}}\\right)}$, P (I) and ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal E}}^c\\right)}$ given by, and respectively.\nThe next lemma allows us to couple the complex Ψ1 and Φ neighbourhood distributions to the far more simple Γ\\* and Γ-neighbourhood distributions.\n[sdom] Let G ∼ G(*n*, *p*) where *c*log*n* ≤ *n**p* ≤ *o*(*n*1/3) for any *c* > 0. Let *I*1 and the *φ*, *ψ*, *ψ*2 and *d*\\*-distributions be defined with respect to MBFS(G, {*u*, *v*}), *u*, *v* ∈ *V*. Then\n1. [itm:d1] $\\displaystyle{{\\mathbb{P}\\!\\left(\\varphi(x)\\neq {d}^\\*(x)\\big|x \\in I\\_1\\right)}=e^{-(1-o(1))np},}$\n2. [itm:d2] P (*ψ*(*u*) ≠ *d*\\*(*u*)) = *e*− (1 − *o*(1))*n**p*,\n3. [itm:d3] P (*ψ*(*u*) ≠ *d*(*u*)or *ψ*(*v*) ≠ *d*(*v*)) ≤ 2*n**p*2 + *e*− (1 − *o*(1))*n**p*,\n4. [itm:d5] P (*ψ*2(*u*) ≠ *d*2\\*(*u*)or *ψ*2(*v*) ≠ *d*2\\*(*v*)) = *e*− (1 − *o*(1))*n**p*,\n5. [itm:d4] P (*ψ*2(*u*) ≠ *d*2(*u*)or *ψ*2(*v*) ≠ *d*2(*v*)) ≤ 4*n*3*p*4 + *e*− (1 − *o*(1))*n**p* + *O* (*n*2*p*3).\n*Item* : run MBFS(G, {*u*, *v*}) and let *x* ∈ *I*1. By the definition of *ψ*(*x*), if *d*\\*(*x̃*) > *D* for all *x̃* ∈ Γ1\\*(*x*) then *φ*(*x*) = *d*\\*(*x*), where *D* is bounded. Hence for *x* ∈ *I*1,\n \n$$\\begin{aligned}\n\\label{uniton}\n {\\mathbb{P}\\!\\left(\\varphi(x)\\neq{d}^\\*(x)\\big|\\mathfrak{F}\\_2\\right)}&= {\\mathbb{P}\\!\\left({d}^\\*(\\tilde{x})\\leq D \\text{ for some }\\tilde{x}\\in \\Gamma\\_1^\\*(x)\\big|\\mathfrak{F}\\_2\\right)} \\leq \\sum\\limits\\_{\\tilde{x} \\in \\Gamma\\_1^\\*(x)}{\\mathbb{P}\\!\\left({d}^\\*(\\tilde{x})\\leq D\\big|\\mathfrak{F}\\_2\\right)}.\n \\intertext{If $\\tilde{x}\\in \\Gamma\\_1^\\*(x), x \\in I\\_1$ then $\\tilde{x}\\in I\\_2$. Knowing the parent of $\\tilde{x}$ does not affect the ${d}^\\*$-distribution conditioned on $\\{\\tilde{x} \\in I\\_2\\}$, so by Lemma \\ref{dom} \\eqref{itm:gam2} as $|S\\_2|,|I\\_2|\\in \\mathfrak{F}\\_2$ we have }\n {\\mathbb{P}\\!\\left(\\varphi(x)\\neq{d}^\\*(x)\\big|\\mathfrak{F}\\_2\\right)}&\\leq{d}^\\*(x) {\\mathbb{P}\\!\\left(Bin\\left(|S\\_2|,p(1-p)^{|I\\_2|-1}\\right)\\leq D\\Big||S\\_2|,|I\\_2|\\right)}.\\notag\\end{aligned}$$ \nLet E*x**j* : = {∣*I**j*∣ ≤ 12(*n**p*)*j*} ∩ {*d*\\*(*x*) ≤ 6*n**p*} ∩ {∣*S**j*∣ ≥ *n* − 12(*n**p*)*j*} for *x* ∈ *I**j* − 1 and observe\n \nP ((E*x**j*)*c*) ≤ 2P (∣*B**j*(*u*)∣ > 6(*n**p*)*j*) + P (∣*B*1(*x*)∣ > 6*n**p*) = *o*(*e*− 2*n**p*), \nby Lemma [neighup] with *α* = 6. Now by Lemma [chb] and Bernoulli’s inequality\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left(\\varphi(x)={d}^\\*(x)\\big|\\mathfrak{F}\\_2\\right)}\\mathbf{1}\\_{\\mathcal{E}\\_x^1} &\\leq {d}^\\*(x) e^{-|S\\_2|p(1-p)^{|I\\_2|}}\\left(\\frac{e|S\\_2|p(1-p)^{|I\\_2|}}{D}\\right)^D\\mathbf{1}\\_{\\mathcal{E}\\_x^1} \\leq e^{-(1-o(1))np},\n \\end{aligned}$$ \nthe result follows by taking expectation of the above and using with *j* = 1.\n*Item* : for *ũ* ∈ *I*1 the distribution of *d*\\*(*ũ*) conditioned on ∣*S*1∣, ∣*I*1∣ is known by [dom]. Thus using the bound (1 − *p*)*n* ≤ exp( − *n**p*) we obtain the following for *ũ* ∈ *I*1,\n \n$$\\begin{aligned}\n\\label{nondiff}\n{\\mathbb{P}\\!\\left({d}^\\*(\\tilde{u})= 0\\big|\\mathfrak{F}\\_1\\right)}&= {\\mathbb{P}\\!\\left(Bin\\left(|S\\_1|,p(1-p)^{|I\\_1|-1}\\right)=0\\Big|\\mathfrak{F}\\_1\\right)} \\leq \\exp\\left(-|S\\_1|p(1-p)^{|I\\_1|}\\right).\\end{aligned}$$ \nRecall the definition of Ψ1(*u*). If *ũ* ∈ Γ1\\*(*u*) then *ũ* ∈ *I*1 and knowing the parent of *ũ* does not affect the *d*\\*-distribution conditioned {*ũ* ∈ *I*1}. Thus, similarly to, for *ũ* ∈ Γ1\\*(*u*),\n \n$$\\begin{aligned}\n\\label{exp6}\n{\\mathbb{P}\\!\\left(\\psi(u)\\neq {d}^\\*(u)\\big|\\mathfrak{F}\\_1\\right)} &= {\\mathbb{P}\\!\\left(\\varphi(\\tilde{u})= 0 \\text{ for some } \\tilde{u}\\in \\Gamma\\_1^\\*(u)\\big|\\mathfrak{F}\\_1\\right)}\\leq d^\\*(u) \\cdot {\\mathbb{P}\\!\\left(\\varphi(\\tilde{u})= 0\\big|\\mathfrak{F}\\_1\\right)}.\\notag \\intertext{Now using the coupling inequality \\eqref{prk}, yields the following for $\\tilde{u} \\in I\\_1 $ }\n{\\mathbb{P}\\!\\left(\\psi(u)\\neq {d}^\\*(u)\\big|\\mathfrak{F}\\_1\\right)}&\\leq{d}^\\*(u){\\mathbb{P}\\!\\left({d}^\\*(\\tilde{u})= 0\\big|\\mathfrak{F}\\_1\\right)} + {d}^\\*(u){\\mathbb{P}\\!\\left(\\varphi(\\tilde{u})\\neq {d}^\\*(\\tilde{u})\\big|\\mathfrak{F}\\_1\\right)}.\\end{aligned}$$ \nRecall E*u*1 from and observe ${\\mathbb{P}\\!\\left({d}^\\*(\\tilde{u})= 0\\big|\\mathfrak{F}\\_1\\right)}\\mathbf{1}\\_{\\mathcal{E}\\_x^1}\\leq e^{-(1-o(1))np} $ by. Thus\n \n$$\\begin{aligned}\n{\\mathbb{P}\\!\\left(\\psi(u)\\neq {d}^\\*(u)\\right)} &\\leq e^{-(1-o(1))np} + 6np{\\mathbb{P}\\!\\left(\\varphi(\\tilde{u})\\neq {d}^\\*(\\tilde{u})\\big|\\tilde{u}\\in I\\_1\\right)} + o\\left(e^{-2np}\\right)= e^{-(1-o(1))np},\\end{aligned}$$ \nby taking the expectation of conditioned on E*u*1 then using the bound on ${\\mathbb{P}\\!\\left({d}^\\*(\\tilde{u})= 0\\big|\\mathfrak{F}\\_1\\right)}$ above and bounds on ${\\mathbb{P}\\!\\left(\\varphi(\\tilde{u})\\neq {d}^\\*(\\tilde{u})\\big|\\tilde{u}\\in I\\_1\\right)}$ and ${\\mathbb{P}\\!\\left(({\\ensuremath{\\mathcal E}}\\_u^1)^c\\right)} $ from Item (i) and respectively.\n*Item* : let *I*0 = {*u*, *v*} and ${\\ensuremath{\\mathcal H}}:=\\{{d}(u)= {d}^\\*(u), \\;{d}(v)={d}^\\*(v)\\}$. By Item\n \n$$\\begin{aligned}\n\\label{lasttag}\n {\\mathbb{P}\\!\\left(\\psi(u)\\neq {d}(u) \\text{ or } \\psi(v)\\neq {d}(v) \\right)}\\notag &\\leq {\\mathbb{P}\\!\\left(\\{\\psi(u)\\neq {d}(u) \\text{ or } \\psi(v)\\neq {d}(v)\\}\\cap {\\ensuremath{\\mathcal H}}\\right)} + {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}^c\\right)}\\notag \\\\\n & \\leq {\\mathbb{P}\\!\\left(\\psi(u)\\neq {d}^\\*(u)\\text{ or }\\psi(v)\\neq{d}^\\*(u) \\right)} + {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}^c\\right)}\\notag \\\\ &\\leq 2e^{-(1-o(1))np} + {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}^c\\right)}.\n \\end{aligned}$$ \nTo calculate ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}^c\\right)}$ in the above recall the definition $\\eqref{eq:gamdef}$ of *d*\\*(*u*) and observe\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}^c\\right)}&= {\\mathbb{P}\\!\\left(\\{uv \\in E \\} \\cup \\{ xu \\in E \\text{ and } xv \\in E \\text{ for some }x \\in V\\backslash I\\_0\\}\\right)}\\notag \\\\\n &\\leq {\\mathbb{P}\\!\\left(uv \\in E\\right)} + {\\sum}\\_{x \\in V\\backslash I\\_0}{\\mathbb{P}\\!\\left(xu \\in E \\text{ and } xv \\in E \\right)} = p + (n-2)p^2. \\label{pce}\n \\end{aligned}$$ \nFinally, combining and yields the bound\n \n$$\\begin{aligned}\n{\\mathbb{P}\\!\\left(\\psi(u)\\neq {d}(u) \\text{ or } \\psi(v)\\neq {d}(v) \\right)}&\\leq 2e^{-(1-o(1))np}+ p + (n-2)p^2 \\leq 2np^2+ e^{-(1-o(1))np}.\\end{aligned}$$ \n*Item* : Define the following events \n$${\\ensuremath{\\mathcal J}}:= {\\bigcap}\\_{x \\in I\\_1} \\left\\{\\varphi(x) = {d}^\\*(x) \\right\\}, \\qquad {\\ensuremath{\\mathcal D}}= \\left\\{{d}(u),{d}(v) \\leq (1+9/\\min\\{c,1\\})np \\right\\}.$$ \n Notice $\\{\\psi\\_2(u)\\neq {d}\\_2^\\*(u)\\text{ or }\\psi\\_2(v)\\neq{d}\\_2^\\*(u) \\}= \\{\\psi(u)\\neq {d}^\\*(u)\\text{ or }\\psi(v)\\neq{d}^\\*(u) \\}\\cup {\\ensuremath{\\mathcal J}}^c $. Now\n \n$$\\label{J}{\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal J}}^c\\right)}\\leq {\\mathbb{E}\\!\\left[|I\\_1|{\\mathbb{P}\\!\\left(\\varphi(x) = {d}^\\*(x) \\big|\\mathfrak{F}\\_1\\right)}(\\mathbf{1}\\_{{\\ensuremath{\\mathcal D}}}+\\mathbf{1}\\_{{\\ensuremath{\\mathcal D}}^c})\\right]} \\leq O\\!\\left(npe^{-(1-o(1))np}\\right) + n{\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal D}}^c\\right)},$$ \nwhich is *o*(*e*− *n**p*(1 − *o*(1))) by Item and since ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal D}}^c\\right)} = o\\left(e^{-np}/n\\right)$ by Lemma [chb].\n*Item* : let *I*0 = {*u*, *v*} and ${\\ensuremath{\\mathcal L}}:=\\{{d}\\_2(u)= {d}\\_2^\\*(u), \\;{d}\\_2(v)= {d}\\_2^\\*(v)\\}$. Then\n \n$$\\label{clset}{\\ensuremath{\\mathcal L}}:= \\left(\\bigcap\\_{x \\in {d}(u), y \\in {d}(v)}\\!\\!\\!\\!\\!\\left\\{xy \\notin E \\right\\}\\!\\!\\right) \\cap \\left(\\bigcap\\_{z \\in S\\_1}\\!\\!\\left\\{ |\\{x \\in I\\_1: xz \\in E\\}|\\leq 1\\right\\}\\!\\!\\right)\\cap {\\ensuremath{\\mathcal H}},$$ \nby the definition $\\eqref{eq:gamdef}$ of *d*2\\*(*u*). Observe that by the Bernoulli inequality,\n \n$$\\begin{aligned}\n & {\\mathbb{P}\\!\\left( |\\{x \\in I\\_1: xz \\in E\\}|>1\\big|\\mathfrak{F}\\_1\\right)} = 1- {\\sum}\\_{a =0,1}{\\mathbb{P}\\!\\left( |\\{x \\in I\\_1: xz \\in E\\}|=a\\big|\\mathfrak{F}\\_1\\right)}\\\\ \n &\\qquad= 1 - (1-p)^{|I\\_1|} - |I\\_1|p(1-p)^{|I\\_1|-1}\\leq 1 - (1-|I\\_1|p) -|I\\_1|p(1-|I\\_1|p)= \\left(|I\\_1|p\\right)^2.\\end{aligned}$$ \nBy $\\eqref{clset}$, the above estimate on ${\\mathbb{P}\\!\\left( |\\{x \\in I\\_1: xz \\in E\\}|>1\\big|\\mathfrak{F}\\_1\\right)}$ and ${\\ensuremath{\\mathcal H}}\\in \\mathfrak{F}\\_1$, we have\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal L}}^c\\big|\\mathfrak{F}\\_1\\right)}&\\leq \\sum\\limits\\_{ x \\in {d}(u), y \\in {d}(v)}{\\mathbb{P}\\!\\left(xy \\in E\\big|\\mathfrak{F}\\_1\\right)}+\\sum\\limits\\_{z \\in S\\_1}{\\mathbb{P}\\!\\left( |\\{x \\in I\\_1: xz \\in E\\}|>1\\big|\\mathfrak{F}\\_1\\right)} +\n {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}^c\\big|\\mathfrak{F}\\_1\\right)}\\\\\n &\\leq {d}(u){d}(v)p + |S\\_1|\\left(|I\\_1|p\\right)^2+ {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}^c\\big|\\mathfrak{F}\\_1\\right)}. \n \\end{aligned}$$ \nThen by the bound on ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal L}}^c\\big|\\mathfrak{F}\\_1\\right)}$ above, the tower property and Cauchy Schwartz inequality,\n \n$$\\begin{aligned}\n\\label{ell}\n {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal L}}^c\\right)} &\\leq p\\sqrt{{\\mathbb{E}\\!\\left[{d}(u)^2\\right]}{\\mathbb{E}\\!\\left[{d}(v)^2\\right]}} + p^2\\sqrt{{\\mathbb{E}\\!\\left[|S\\_1|^2\\right]}{\\mathbb{E}\\!\\left[|I\\_1|^4\\right]}} + {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}^c\\right)}=4n^3p^4 + O\\!\\left(n^2p^3 \\right), \\end{aligned}$$ \nthe last equality holds by and as ∣*S*1∣ ∼ *B**i**n*(*n* − 2, (1 − *p*)2), ∣*I*1∣ ∼ *B**i**n*(*n* − 2, 2*p*(1 − *p*)) by Lemma [dom], then applying the bound on moments of binomial r.v.’s from.\nFinally let ${\\ensuremath{\\mathcal F}}:= \\left\\{\\psi\\_2(u)= {d}\\_2(u),\\; \\psi\\_2(v)= {d}\\_2(v) \\right\\}$. Then by the definitions and of the vertex sets Ψ1(*u*) and Ψ2(*u*) we have ${\\ensuremath{\\mathcal F}}:= \\left\\{\\psi(u)= {d}(u),\\; \\psi(v)= {d}(v) \\right\\} \\cap {\\ensuremath{\\mathcal J}}\\cap {\\ensuremath{\\mathcal L}}$. Thus\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal F}}^c\\right)}& \\leq {\\mathbb{P}\\!\\left(\\psi(u)\\neq {d}(u)\\text{ or } \\psi(v)\\neq {d}(u) \\right)}+ {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal J}}^c\\right)} + {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal L}}^c\\right)}\\leq 4n^3p^4 +e^{-(1-o(1))np}+ O\\!\\left(n^2p^3 \\right),\n \\end{aligned}$$ \nby Item, and.\nRecall from Definition that A*u*, *v* is the event that ${\\ensuremath{\\mathcal G}}(n,p) $ satisfies the strong path property for *u*, *v* ∈ *V* and some *k* ≤ log(*n**p*)/(2log*n*) + 2. Recall also Definition of B*u*, *v* which is the event the pruned first neighbourhoods Ψ1(*u*), Ψ1(*v*) are both non-empty.\n[setlem] Let G ∼ G(*n*, *p*) where *c*log*n* ≤ *n**p* < *n*1/10, *c* > 0. Then for *u*, *v* ∈ *V*, *u* ≠ *v*, \nP ((A*u*, *v*)*c*) = *o*(*e*− 7min{*n**p*, log*n*}/2) and P ((B*u*, *v*)*c*) = *e*− (1 − *o*(1))*n**p*. \nRun MBFS(G, {*u*, *v*}), *u*, *v* ∈ *V*. For *k* ≥ 0 let T : = T*u*, *v*, *k* = T1 ∩ T2 where \n$$\\mathcal{T}\\_1:=\\left\\{ \\left|S\\_{k+2}\\right|\\geq n - n^{5/6} \\right\\},\\quad \\mathcal{T}\\_2:=\\left\\{|\\Gamma\\_{k}^\\*(x)\\times \\Gamma\\_{k}^\\*(y)| \\geq 4n, \\text{for all } (x,y) \\in \\\\ \\Psi\\_2(u)\\times\\Psi\\_2(v)\\right\\}.$$ \n On the event T1 when MBFS(G, {*u*, *v*}) has run for *k* + 2 iterations there is still a lot of the graph yet to explore and the algorithm will run for at least one more iteration. The *k* in the definition of T will be the one occurring in the description of A*u*, *v*. Set the value of *k* to be\n \n$$\\label{kchoice}\n k:=k(n,p)=\\begin{cases}\n \\left\\lceil\\log \\left( \\frac{4n}{(15)^2} \\right) /\\left(2\\log(np) \\right) \\right\\rceil +1 & \\text{if } np=c\\log n \\text{ where }c>0\\\\\n \\left\\lceil\\log \\left( \\frac{400n}{81} \\right) /\\left(2\\log(np)\\right) \\right\\rceil &\\text{if }np=\\omega\\left(\\log n \\right).\n \\end{cases}$$ \nThus *k* ≤ log(*n**p*)/(2log*n*) + 2 for large *n*. It remains to show that for *k* given by : \n$${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal G}}\\notin {A}\\_{u,v}^{n,k}\\right)} = o\\left(e^{-7\\min \\left\\{np,\\log n \\right\\}/2}\\right).$$ \nLet R : = R*u*, *v* be the event {∣Ψ2(*u*) × Ψ2(*v*)∣ ≤ (12(*n**p*)2)2}. Since *ψ*2(*u*) ≤ *d*2(*u*) for any *u* ∈ *V*, we have P (R*c*) = *o*(*e*− 4*n**p*) by Lemma [neighup] with *α* = 12. Thus by the tower property\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left(\\mathcal{T}^c\\right)}&\\leq {\\mathbb{P}\\!\\left(\\mathcal{T}\\_1^c \\right)}+{\\mathbb{P}\\!\\left(\\mathcal{R}^c\\right)}+ {\\mathbb{E}\\!\\left[{\\mathbb{P}\\!\\left(\\mathcal{T}\\_2^c \\big| \\mathfrak{F}\\_3\\right)}\\mathbf{1}\\_{\\mathcal{R}}\\right]} \\leq 2{\\mathbb{P}\\!\\left(|B\\_{k+2}(u)|> n^{5/6}/2\\right)}+2{\\mathbb{P}\\!\\left(|B\\_{2}(u)|> 12(np)^2\\right)}\\notag\\\\\n &\\qquad+ 2{\\mathbb{E}\\!\\left[\\psi\\_2(u)\\psi\\_2(v)\\mathbf{1}\\_{\\mathcal{R}}{\\mathbb{P}\\!\\left({d}\\_{k}^\\*(w) < 2n^{1/2} \\big| \\{w \\in \\Psi\\_2\\},\\mathfrak{F}\\_3\\right)}\\right]}\\notag,\n \\intertext{as $\\{{d}^\\*(x)\\cdot {d}^\\*(y) < k\\}\\subseteq \\{{d}^\\*(x) \\text{ or }{d}^\\*(y)< \\sqrt{ k}\\}$. Provided $np\\leq n^{1/10}$ the choice of $k$ given by \\eqref{kchoice} satisfies the conditions of Lemma \\ref{neighlow}. Thus by Lemmas \\ref{neighup}, \\ref{neighlow} \\eqref{itm:bd1} and \\ref{neighlow} \\eqref{itm:bd2} we have} {\\mathbb{P}\\!\\left(\\mathcal{T}^c\\right)}&\\leq o\\left(e^{-9np/2}\\right)+ {O\\!\\left(\n np\\right)}^4\\cdot {\\mathbb{P}\\!\\left({d}\\_{k}^\\*(w) < 2n^{1/2} \\big| w \\in \\Psi\\_2\\right)} =o\\left(e^{-7\\min \\left\\{np,\\log n \\right\\}/2}\\right),\\label{prrnt}\\end{aligned}$$ \nwhere the bound ${\\mathbb{P}\\!\\left({d}\\_{k}^\\*(w) < 2n^{1/2} \\big| w \\in \\Psi\\_2\\right)}\\leq e^{-4\\min\\{np,\\log n\\}}$ above covers the different values of *n**p* and comes from amalgamating Lemmas [neighlow] and, with *K* = 4 in the latter.\nLet L*x*, *y* be the following event indexed by (*x*, *y*) ∈ Ψ2(*u*) × Ψ2(*v*), \n$$\\mathcal{L}\\_{x,y}:=\\{x'y'\\notin E, \\text{for every pair } (x',y') \\in \\Gamma\\_k^\\*(x)\\times\\Gamma\\_k^\\*(y)\\}.$$ \n This is independent of F*k* + 2 as each *x*ʹ*y*ʹ has not been checked up to iteration *k* + 2, thus\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left(\\mathcal{L}\\_{x,y}\\big|\\mathfrak{F}\\_{k+2}\\right)}\\mathbf{1}\\_{\\mathcal{T}}&= {\\mathbb{P}\\!\\left(x'y'\\notin E \\right)}^{{d}\\_k^\\*(x){d}\\_k^\\*(y)}\\mathbf{1}\\_{\\mathcal{T}} \\leq (1-p)^{4n} \\leq e^{-4np }. \\label{conbd}\n \\end{aligned}$$ \nRecall Definition [A] of the strong path property *A**u*, *v**n*, *k* which we can express as\n \n$$\\left\\{{\\ensuremath{\\mathcal G}}\\notin A\\_{u,v}^{n,k}\\right\\} = \\bigcup\\limits\\_{(x,y) \\in \\Psi\\_2(u)\\times\\Psi\\_2(v) }\\left\\{\\Gamma\\_k^\\*(x)= \\emptyset\\right\\}\\cup\n \\left\\{\\Gamma\\_k^\\*(y)= \\emptyset\\right\\}\\cup\\mathcal{L}\\_{x,y}.$$ \nObserve that for each *i*, *j* ≥ 0 the random variables {*d**j*\\*(*w*)}*w* ∈ *I**i* are identically distributed. Recall also that Ψ1(*u*), Ψ1(*v*), R ∈ F3. Let P denote ${\\mathbb{P}\\!\\left(\\left\\{{\\ensuremath{\\mathcal G}}\\notin A\\_{u,v}^{n,k}\\right\\}\\cap \\mathcal{R}\\cap \\mathcal{T}\\right)}$. By the union bound, tower property and since *ψ*1(*u*)*ψ*1(*v*) ≤ 122(*n**p*)4 on R, we have\n \n$$\\begin{aligned}\n \\mathfrak{P}&\\leq {\\mathbb{E}\\!\\left[ \\sum\\limits\\_{(x,y) \\in \\Psi\\_2(u)\\times\\Psi\\_2(v) }{\\mathbb{E}\\!\\left[\\left(\\mathbf{1}\\_{\\mathcal{L}\\_{x,y}\\cup\\{{d}\\_{k}^\\*(x)=0\\}\\cup\\{{d}\\_{k}^\\*(y)=0\\}}\\right) \\mathbf{1}\\_{\\mathcal{R}}\\mathbf{1}\\_{\\mathcal{T}}\\Big|\\mathfrak{F}\\_3 \\right]} \\right]}\n \\\\& \\leq{\\mathbb{E}\\!\\left[ 12^2(np)^4\\mathbf{1}\\_{\\mathcal{R}}{\\mathbb{E}\\!\\left[\\left(\\mathbf{1}\\_{\\mathcal{L}\\_{x,y}}+\\mathbf{1}\\_{\\{{d}\\_{k}^\\*(x)=0\\}}+\\mathbf{1}\\_{\\{{d}\\_{k}^\\*(y)=0\\}}\\right) \\mathbf{1}\\_{\\mathcal{T}}\\Big|\\mathfrak{F}\\_3 \\right]} \\right]}.\\intertext{Now since $x,y \\in \\Psi\\_2$ and ${d}\\_j^\\*(x),{d}\\_j^\\*(y)$ are identically distributed for any $j\\geq 0$:}\n \\mathfrak{P} &\\leq {O\\!\\left(np\\right)}^4\\cdot\\left({\\mathbb{E}\\!\\left[{\\mathbb{E}\\!\\left[\\mathbf{1}\\_{\\mathcal{L}\\_{x,y}}\\mathbf{1}\\_{\\mathcal{T}}\\Big|\\mathfrak{F}\\_{k+2} \\right]} \\right]}+ 2{\\mathbb{P}\\!\\left({d}\\_{k}^\\*(w)=0\\big| \\{w \\in \\Psi\\_2\\}\\right)}\\right).\\intertext{By Lemma \\eqref{neighlow} \\eqref{itm:bd1}, \\eqref{conbd} and since $\\mathcal{T} \\in \\mathfrak{F}\\_{k+2}$ we have } \n \\mathfrak{P}&\\leq {O\\!\\left(np\\right)}^4\\cdot \\left({\\mathbb{E}\\!\\left[{\\mathbb{P}\\!\\left(\\mathcal{L}\\_{x,y}\\Big|\\mathfrak{F}\\_{k+2} \\right)}\\mathbf{1}\\_{\\mathcal{T}} \\right]}+ 2e^{-4\\min\\{np,\\log n\\}}\\right)= o\\left(e^{-7\\min \\left\\{np,\\log n \\right\\}/2}\\right).\\end{aligned}$$ \nRecall P (R*c*) = *o*(*e*− 4*n**p*), so by and the bound on ${\\mathbb{P}\\!\\left(\\left\\{{\\ensuremath{\\mathcal G}}\\notin A\\_{u,v}^{n,k}\\right\\}\\cap \\mathcal{R}\\cap \\mathcal{T}\\right)} $ above,\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal G}}\\notin A\\_{u,v}^{n,k}\\right)}&\\leq {\\mathbb{P}\\!\\left(\\left\\{{\\ensuremath{\\mathcal G}}\\notin A\\_{u,v}^{n,k}\\right\\}\\cap \\mathcal{R}\\cap \\mathcal{T}\\right)} + {\\mathbb{P}\\!\\left(\\left(\\mathcal{R}\\cap \\mathcal{T}\\right)^c\\right)} \\leq o\\left(e^{-7\\min \\left\\{np,\\log n \\right\\}/2}\\right). \n \\end{aligned}$$ \nFor P ((B*u*, *v*)*c*) we apply the coupling inequality to the *ψ* and *d*\\*-distributions:\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left((\\mathcal{B}\\_{u,v})^c\\right)} &\\leq {\\mathbb{P}\\!\\left(\\psi\\_1(u)=0 \\right)} +{\\mathbb{P}\\!\\left(\\psi\\_1(v)=0 \\right)}\\leq 2{\\mathbb{P}\\!\\left({d}^\\*(u)=0 \\right)} + 2{\\mathbb{P}\\!\\left(\\psi\\_1(u)\\neq {d}^\\*(u)\\right)}. \\notag \\intertext{Then since ${\\mathbb{P}\\!\\left(\\psi\\_1(u)\\neq {d}^\\*(u)\\right)}$ is known by Lemma \\ref{sdom} we have}\n {\\mathbb{P}\\!\\left((\\mathcal{B}\\_{u,v})^c\\right)}&\\leq 2{\\mathbb{P}\\!\\left({d}^\\*(u)=0 \\left. \\right|{d}^\\*(v)\\leq 6np\\right)} + 2{\\mathbb{P}\\!\\left({d}^{\\phantom{\\*}}(v) >6np\\right)} +2e^{-(1-o(1))np}.\\notag \\\\&\\leq 2(1-p)^{n-6np-1} + o\\left(e^{-2np}\\right) +e^{-(1-o(1))np} = e^{-(1-o(1))np} \\notag,\n \\end{aligned}$$ \nby applying Lemma [dom] to the first term and Lemma [neighup] with *α* = 6 to the second.\nThe following crude but resilient bound on *R*(*i*, *j*) is useful when conditioning on $ {\\ensuremath{\\mathcal A}}\\_{i,j}^c$.\n[basicres] Let G ∼ G(*n*, *p*) be such that *n**p* − log*n* → ∞. Then for *i*, *j* ∈ *V*, \nPC (*R*(*i*, *j*) > 3log*n*/log(*n**p*)) = *o*(*n*− 4). \nSince *G* ∈ C the effective resistance between two points is bounded from above by the graph distance. Let J*i*, *j* : = {∣*B**k*(*i*)∣ ⋅ ∣*B**k*(*j*)∣ ≥ 4*n*} where $k:= \\left\\lceil \\log\\left(\\frac{4n}{15^2}\\right)/(2 \\log np)\\right\\rceil +5.$ Using Lemma [neighlow] to bound PC (J*i*, *j**c*), since 5 ≤ *k* ≤ ⌊log(*n*)/log(*n**p*)⌋ − 5 when *n* large:\n \n$$\\begin{aligned}\n {\\mathbb{P}\\_{\\mathcal{C}}\\!\\left(R(i,j) > 2k+1\\right)}&\\leq {\\mathbb{P}\\_{\\mathcal{C}}\\!\\left(d(i,j)>2k+1\\Big|\\mathcal{J}\\_{i,j} \\right)} + {\\mathbb{P}\\_{\\mathcal{C}}\\!\\left(\\mathcal{J}\\_{i,j}^c\\right)}\\\\\n &\\leq {\\mathbb{P}\\!\\left(xy \\notin E,\\; \\forall (x,y) \\in B\\_k(i)\\times B\\_k(j), B\\_k(i)\\cap B\\_k(j)= \\emptyset \\Big| \\mathcal{J}\\_{i,j} \\right)}/{\\mathbb{P}\\!\\left(\\mathcal{C}\\right)} \\\\ &\\qquad\\qquad\\quad + 2{\\mathbb{P}\\_{\\mathcal{C}}\\!\\left(|B\\_k(j)|< 2 \\sqrt{n}\\right)} \\leq 2(1-p)^{4n } + 2\\,o\\left(n^{-4} \\right)=o\\left(n^{-4} \\right).\n \\end{aligned}$$ \nThe result follows since $2k+1 = 2\\left(\\left\\lceil \\log\\left(\\frac{4n}{15^2}\\right)/(2 \\log np)\\right\\rceil +5\\right) +1 \\leq \\frac{3 \\log n}{\\log(np)}$ for large *n*.\nProof of Theorems [resconc], [resconc2] & [bolthom]\n===================================================\nMost of our main theorems result from Corollary [Gnpbdd], Lemma [supbddlem] below simplifies this application.\n[supbddlem] Let G ∼ G(*n*, *p*) where log*n* + logloglog*n* ≤ *n**p* < *o*(*n*1/3). Let *α* ≥ 1 and Ψ1(*u*), Ψ1(*v*) be defined with respect to MBFS(G, {*u*, *v*}), *u*, *v* ∈ *V*. Then\n1. [itm:superr] $\\displaystyle \\mathbb{E}\\_{\\mathcal{C}}\\!\\left[\\left(\\sup\\_{x\\in \\Psi\\_1(u)}\\frac{\\mathbf{1}\\_{\\mathcal{B}\\_{u}^{u,v}}}{\\varphi(x)} \\right)^{\\alpha}\\right]^{1/\\alpha}= O\\!\\left(\\frac{1}{np}\\right)$.\n2. [itm:simres2] If *c*log*n* ≤ *n**p* ≤ *n*1/10, for any fixed *c* > 0, then \n$${\\mathbb{P}\\!\\left(R\\left(u,v\\right) >\\left(\\frac{1}{\\psi(u)} + \\frac{1}{\\psi(v)} \\right)\\!\\!\\left(1 + 9{\\varepsilon\\_n}\\right) \\right)}= o\\left(e^{-np/4} \\right)+o\\left(n^{-7/2} \\right).$$\n*Item* : Let P*u* : = P (inf*x* ∈ Ψ1(*u*)*φ*(*x*) < *K**p*) and $ K\\_p:= \\left(1 - \\sqrt{3}/2\\right) np(1-66np^2)$.\nRecall Ψ1(*u*) ⊂ *I*1 for *u* ∈ *I*0 and observe that,\n \n$$\\begin{aligned}\n \\mathfrak{P}\\_u&\\leq {\\mathbb{E}\\!\\left[{\\mathbb{P}\\!\\left(\\inf\\_{x\\in I\\_1}{d}^\\*(x) < K\\_p\\big|\\mathfrak{F}\\_1\\right)}\\right]} + {\\mathbb{E}\\!\\left[{\\mathbb{P}\\!\\left(\\varphi(x)\\neq {d}^\\*(x) \\text{ for some }x \\in I\\_1 \\big|\\mathfrak{F}\\_1\\right)}\\right]}, \\notag \\intertext{by the tower property. Applying the union bound since $I\\_1 \\in \\mathfrak{F}\\_1$ yields } \n \\mathfrak{P}\\_u &\\leq {\\mathbb{E}\\!\\left[|I\\_1|{\\mathbb{P}\\!\\left( {d}^\\*(x) < K\\_p \\big|x\\in I\\_1,\\mathfrak{F}\\_1\\right)}\\right]} + {\\mathbb{E}\\!\\left[|I\\_1|{\\mathbb{P}\\!\\left(\\varphi(x)\\neq {d}^\\*(x) \\big|x \\in I\\_1,\\mathfrak{F}\\_1\\right)}\\right]}.\\notag\n \\intertext{Let $a:= 13/\\min\\{c,1\\}$ where $c>0$ is the largest real number such that $np \\geq c \\log n$. Separate the expectations into parts $\\left\\{ |I\\_1| \\leq anp\\right\\}$ and $\\left\\{ |I\\_1| >anp\\right\\}$ to give}\n \\mathfrak{P}\\_u &\\leq anp {\\mathbb{E}\\!\\left[{\\mathbb{P}\\!\\left( {d}^\\*(x) < K\\_p \\big|x\\in I\\_1,\\mathfrak{F}\\_1\\right)}\\right]} + anp{\\mathbb{P}\\!\\left(\\varphi(x)\\neq{d}^\\*(x) \\big|x \\in I\\_1\\right)} + 2n{\\mathbb{P}\\!\\left(|I\\_1| >anp\\right)}.\\notag\\intertext{Since ${d}^\\*(x)\\sim Bin(|S\\_1(x)|,p)$ by Lemma \\ref{dom}, $S\\_1(x) \\in \\mathfrak{F}\\_2$, and by Lemma \\ref{sdom} \\eqref{itm:d1} we have }\n \\mathfrak{P}\\_u &\\leq anp{\\mathbb{E}\\!\\left[{\\mathbb{P}\\!\\left(Bin(|S\\_1(x)|,p) < K\\_p\\big|\\mathfrak{F}\\_2 \\right)}\\right]} + a(np)e^{-(1-o(1))np}+ 4 n{\\mathbb{P}\\!\\left({d}(u) >a np/2\\right)} \\notag.\n \\notag\\intertext{Applying Lemma \\ref{chb} to the first term and Lemma \\ref{neighup} with $\\alpha=a$ to the last yields}\n \\mathfrak{P}\\_u &\\leq anp{\\mathbb{E}\\!\\left[e^{-\\left(|S\\_1(x)|p-K\\_p\\right)^2/\\left(2|S\\_1(X)|p\\right) }\\right]} + anp\\cdot e^{-(1-o(1))np}+ 4 n\\cdot o\\left(e^{-anp/6 }\\right).\\notag \\intertext{Once again by separating the expectation into the two disjoint parts $\\left\\{ |S\\_1(x)| \\leq n-12(np)^2\\right\\}$ and $\\left\\{ |S\\_1(x)| >n-12(np)^2\\right\\}$ the applying Lemma \\ref{neighup} with $\\alpha=6$ we have the following}\n \\mathfrak{P}\\_u &\\leq anp\\cdot e^{- np/3 } + 2{\\mathbb{P}\\!\\left(|B\\_2(u)| > 6(np)^2\\right)}+ o\\left(e^{-np }\\right) = o\\left(e^{-np/4}\\right). \\label{Ptag}\n \\end{aligned}$$ \nRecall sup*x* ∈ Ψ1(*u*)**1**B*u**u*, *v*/*φ*(*x*) < 1/*D*, see &. Bernoulli’s inequality provides\n \n$$\\begin{aligned}\n\\label{expectationbdd}\n \\mathbb{E}\\_{\\mathcal{C}}\\!\\left[\\left(\\sup\\_{x\\in \\Psi\\_1(u)}\\frac{\\mathbf{1}\\_{\\mathcal{B}\\_{u}^{u,v}}}{\\varphi(x)} \\right)^{\\alpha}\\right]^{1/\\alpha}&\\leq \\left(\\frac{1}{(K\\_p)^\\alpha} + \\frac{1}{D^\\alpha{\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal C}}\\right)}}{\\mathbb{P}\\!\\left(\\inf\\_{x\\in \\Psi\\_1(u)}\\varphi(x) < K\\_p\\right)} \\right)^{1/\\alpha}\\notag \\\\&\\leq \\frac{1}{(K\\_p)}\\left(1 + (K\\_p)^\\alpha e^{-np/4}/D^\\alpha{\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal C}}\\right)} \\right)^{1/\\alpha}= O\\!\\left(\\frac{1}{np}\\right).\n \\end{aligned}$$ \nNote that the bound on P\\_u holds for any *n**p* ≥ *c*log*n*, *c* > 0 fixed. The restriction on *n**p* to *n**p* ≥ log*n* comes from, where we need ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal C}}\\right)}$ bounded below by a constant.\n*Item* : Observe that $K\\_p=\\left(1 - \\sqrt{3}/2\\right) np(1-66np^2) \\geq np/9 $ for large *n* and *k* ≤ *n**p**ɛ**n* in the definition of event A*u*, *v*. Thus conditional on {*φ*(*x*) ≥ *K**p*for all *x* ∈ Ψ1} ∩ A*u*, *v*, \n*R*(*u*, *v*) ≤ (1/*ψ*(*u*) + 1/*ψ*(*v*))(1 + (*k* + 2)/*K**p*) ≤ (1/*ψ*(*u*) + 1/*ψ*(*v*))(1 + 9*ɛ**n*), \nby Corollary [Gnpbdd]. The result follows since we have P (inf*x* ∈ Ψ1(*u*)*φ*(*x*) < *K**p*) = *o*(*e*− *n**p*/4) by and P ((A*u*, *v*)*c*) = *o*(*n*− 7/2) by Lemma [setlem].\nProof of Theorems [resconc] & [resconc2]\n----------------------------------------\nTo begin let *i*, *j* ∈ *V* and define the following three functions for ease of notation \n*r**i*, *j* : = 1/*d*(*i*) + 1/*d*(*j*), *f**i*, *j* : = 1/*d*(*i*)2 + 1/*d*(*j*)2, *g**i*, *j* : = *ɛ**n* ⋅ *r**i*, *j*. \nDefine the four events ${\\ensuremath{\\mathcal I}}, {\\ensuremath{\\mathcal H}}$ ${\\ensuremath{\\mathcal F}}$ and ${\\ensuremath{\\mathcal L}}$ as follows, where event ${\\ensuremath{\\mathcal L}}$ is w.r.t. ${\\operatorname{MBFS}}({\\ensuremath{\\mathcal G}}, \\{i,j \\} )$,\n \n$$\\label{heeq}\n\\begin{alignedat}{2}\n{\\ensuremath{\\mathcal I}}&:= \\bigcap\\_{\\{i,j \\}\\subseteq V}\\left\\{\\left|R(i,j) - \\frac{2}{np}\\right| \\leq \\frac{7\\sqrt{\\log n}}{(np)^{3/2}} \\right\\}, \\qquad {\\ensuremath{\\mathcal H}}&:= \\left\\{\\left|R(i,j) - r\\_{i,j}\\right| \\leq 9\\left(2f\\_{i,j}+ g\\_{i,j}\\right)\\right\\}, \\\\\n {\\ensuremath{\\mathcal F}}&:= \\left\\{ R(i,j)\\leq \\left(\\frac{1}{\\psi(i)} + \\frac{1}{\\psi(j)} \\right)\\!\\! \\left(1+ 9{\\varepsilon\\_n}\\right)\\right\\},\\qquad {\\ensuremath{\\mathcal L}}&:= \\bigcap\\_{z\\in \\{i,j \\}}\\left\\{ \\frac{1}{d^\\*(z) } \\leq \\frac{1}{d(z) } + \\frac{16}{d(z)^2 } \\right\\}.\n\\end{alignedat}$$ \n[betterprob] Let *c*log*n* ≤ *n**p* ≤ *n*1/10 where *c* > 0. Then ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal L}}^c\\right)}= o(1/n^3 ) + o(e^{-np/3})$\nLet *I*0 = {*i*, *j*} and recall that Γ\\*(*i*) = Γ(*i*) ∖ ({*j*} ∩ Γ(*j*)), see and. Let ${\\ensuremath{\\mathcal E}}\\_a = \\{ d^\\*(v)\\leq d(v) -a\\}$. For each *z* ∈ {*j*} ∩ Γ(*y*), provided *z* ≠ *i*, we have *i**z* ∈ *E* independently with probability *p*. Thus ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal E}}\\_a \\mid ij\\notin E, d(j)=k\\right)} = {\\mathbb{P}\\!\\left(Bin(k,p) \\geq a \\right)}$ and similarly $ {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal E}}\\_a \\mid ij\\in E, d(j)=k\\right)} = {\\mathbb{P}\\!\\left(Bin(k-1,p) \\geq a -1\\right)}$. Thus we have \n$${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal E}}\\_a\\right)} \\leq {\\mathbb{P}\\!\\left(Bin(k,p) \\geq a -1\\right)} + {\\mathbb{P}\\!\\left(d(x)\\geq k\\right)}.$$ \n Let *k* = 3*a*max{log*n*, *n**p*} and apply Lemma [chb] and respectively to give\n \n$$\\begin{aligned}\n\\label{split}{\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal E}}\\_a\\right)} &\\leq e^{-kp}\\left(\\frac{ekp}{a-1}\\right)^{a-1} +e^{-\\frac{k^2}{2(np+k/3)}}\\leq \\left(\\frac{2e(np)^2}{n} \\right)^{a-1} + n^{-3a^2/2} = n^{-a/2}. \\end{aligned}$$ \nConditional on the event ${\\ensuremath{\\mathcal E}}\\_j^c\\cap \\{d(v) \\geq 2a \\} $ we have \n$$\\frac{1}{d^\\*(x) } \\leq \\frac{1}{d(x) - j }\\leq \\frac{1}{d(x) } + \\frac{j}{d(x)^2 -jd(x) } \\leq \\frac{1}{d(x) } + \\frac{2j}{d(x)^2 }.$$ \n If we let *a* = 8 then P (*d*(*v*) < 16) ≤ *o*(*e*− *n**p*/3) by Lemma [chb] and ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal E}}\\_8\\right)} = o(1/n^3) $ by.\n[Proof of Theorem [resconc]] To begin, by Lemma [lowjoh] we have\n \n$$\\begin{aligned}\n R(i,j) - r\\_{i,j}&\\geq -\\left(\\frac{1}{{d}(i)^2+{d}(i)} +\\frac{1}{{d}(j)^2+{d}(j)}\\right)> -\\left(\\frac{1}{{d}(i)^2} +\\frac{1}{{d}(j)^2}\\right)=-f\\_{i,j}. \n \\end{aligned}$$ \nLet ${\\ensuremath{\\mathcal B}}$ be the event {*ψ*(*i*) = *d*\\*(*i*), *ψ*(*j*) = *d*\\*(*j*)}. Conditional on ${\\ensuremath{\\mathcal B}}\\cap{\\ensuremath{\\mathcal L}}\\cap {\\ensuremath{\\mathcal F}}$ we have \n$$R(i,j) - r\\_{i,j} \\leq \\frac{16}{d(i)^2} + \\frac{16}{d(j)^2}+ \\left(\\frac{1}{d(i)} + \\frac{1}{d(j)}+ \\frac{16}{d(i)^2} + \\frac{16}{d(j)^2} \\right)9{\\ensuremath{\\epsilon}}\\_n \\leq 9(2 f\\_{i,j}+g\\_{i,j}).$$ \n Bounding ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal B}}^c\\right)}$, ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal L}}^c\\right)}$ and ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal F}}^c\\right)}$ using Lemmas [sdom], [betterprob], and [ubredux] respectively:\n \n$$\\begin{aligned}\n{\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}^c\\right)} &\\leq 2 e^{-(1-o(1))np} +o(1/n^3 ) + o(e^{-np/3})+ o\\!\\left(e^{-np/4} \\right)+o\\!\\left(n^{-7/2} \\right), \\end{aligned}$$ \nwhich is *o* (*e*− *n**p*/4) + *o* (*n*− 3) as required.\nFor *S* ⊆ *V* and *λ* : = *λ*(*n*) = *o*(*n**p*) let $ {\\ensuremath{\\mathcal E}}(S,\\lambda)$ be the event\n \n$$\\begin{aligned}\n\\label{eventelam}\n{\\ensuremath{\\mathcal E}}(S,\\lambda):=\\bigcap\\_{u\\in S}\\left\\{\\left|{d}(u)- np\\right| \\leq \\sqrt{\\lambda np }\\right\\}, \\end{aligned}$$ \nfor which we have ${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal E}}(S,\\lambda)^c\\right)} \\leq 2|S|e^{-\\lambda(n)/3} $ by the union bound and Lemma [chb].\n[Proof of Theorem [resconc2]] *Item* : conditional on the event ${\\ensuremath{\\mathcal E}}\\left(\\{i,j\\},{\\varepsilon\\_n}^2 np/19\\right) \\cap {\\ensuremath{\\mathcal H}}$ we have\n \n$$\\begin{aligned}\n\\label{triandge}\n \\Big|R(i,j)-\\frac{2}{np}\\Big|&\\leq \\Big|R(i,j)-r\\_{i,j}\\Big| +\\Big|r\\_{i,j}- \\frac{2}{np}\\Big|\\leq \\frac{19{\\varepsilon\\_n}}{2np}+\\frac{2\\sqrt{\\lambda(n)}}{(np)^{3/2}}\\leq \\frac{10{\\varepsilon\\_n}}{np},\n \\end{aligned}$$ \nsince 9(2*f**i*, *j* + *g**i*, *j*) ≤ 19*ɛ**n*/(2*n**p*) on ${\\ensuremath{\\mathcal E}}(\\{i,j\\},\\lambda) \\cap {\\ensuremath{\\mathcal H}}$. Thus by Theorem [resconc]:\n \n$$\\begin{aligned}\n {\\mathbb{P}\\!\\left(\\left|R(i,j) - 2/(np)\\right| > 10{\\varepsilon\\_n}/(np)\\right)} &\\leq {\\mathbb{P}\\!\\left(\\left({\\ensuremath{\\mathcal H}}\\cap{\\ensuremath{\\mathcal E}}\\right)^c\\right)} \\leq {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal H}}^c\\right)}+ 4e^{-\\lambda(n)/3}\\leq e^{-{\\varepsilon\\_n}^2 np/60}. \n \\end{aligned}$$ \n*Item* : Recall the definition of ${\\ensuremath{\\mathcal H}}$ from and notice we supressed dependence on *i*, *j* that is ${\\ensuremath{\\mathcal H}}:={\\ensuremath{\\mathcal H}}\\_{i,j}$. Similarly to, conditional on $\\left(\\bigcap\\_{\\{i,j\\} \\subseteq v }{\\ensuremath{\\mathcal H}}\\_{i,j}\\right)\\cap {\\ensuremath{\\mathcal E}}\\left(V,9\\log n\\right) $ we have \n$$\\left|R(i,j)- \\frac{2}{np}\\right|\\leq 2\\cdot \\frac{3\\sqrt{\\log n}}{(np)^{3/2}} + \\frac{19{\\varepsilon\\_n}}{2np} \\leq \\frac{7\\sqrt{\\log n}}{(np)^{3/2}}.$$ \n Recall event ${\\ensuremath{\\mathcal I}}$ from. The result now follows since by Theorem [resconc] and we have \n$${\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal I}}^c\\right)}\\leq n^2\\left( o\\!\\left(e^{-np/4} \\right)+ o\\!\\left(n^{-3}\\right)\\right) + 2ne^{-3\\log n} = o\\left(\\frac{1}{n}\\right).$$ \n *Item* : Recall that *m* : = ∣*E*∣ and let M be the event $\\left\\{\\left|m- \\binom{n}{2}p\\right| \\leq 3\\sqrt{\\log(n)\\binom{n}{2}p } \\right\\}.$\nConditional on ${\\ensuremath{\\mathcal E}}\\left(V,9\\log n \\right) \\cap \\mathcal{M} \\cap \\mathcal{I}$ we have the following for any {*i*, *j*} ⊆ *V*,\n \n$$\\begin{aligned}\n |mR(i, j) -n| \\leq 4n\\sqrt{\\frac{\\log n }{np}}, \\quad \\text{and} \\quad \\left|\\sum\\limits\\_{u\\in V} \\frac{{d}(u)}{2}\\left[R(j,u)-R(u, i)\\right]\\right|\\leq 8n\\sqrt{\\frac{\\log n }{np}},\n \\end{aligned}$$ \nthus $ \\left| h(i, j) -n \\right|\\leq 12 \\sqrt{\\log(n) /np } $ by Tetali’s formula and the Triangle inequality. Now \n$${\\mathbb{P}\\!\\left(\\left({\\ensuremath{\\mathcal E}}\\left(V,9\\log n \\right) \\cap \\mathcal{M} \\cap \\mathcal{I}\\right)^c\\right)}= o\\left(1/n^3\\right) + o\\left(1/n^3\\right) +o\\left(1/n\\right) = o\\left(1/n\\right)$$ \n by, Lemma [chb], since $m\\sim Bin\\left(\\binom{n}{2},p \\right)$ and Theorem [resconc2] respectively.\nProof of Theorem [bolthom]\n--------------------------\nRecall that *p**a**t**h**s*2(*i*, *j*, *l*) is the maximum number of paths of length at most *l* between vertices *i* and *j* that are vertex disjoint on *V* ∖ (*B*1(*i*) ∪ *B*1(*j*)) of a graph *G*.\n[Proof of Theorem [bolthom]]*Item* : For *i*, *j* ∈ *V* let ${\\ensuremath{\\mathcal E}}\\_{i,j}$ be the event that there is no path from *i* to *j* of length less than 4. Then by over-counting the number of paths we have\n \n$$\\label{E}\n {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal E}}\\_{i,j}^c \\right)}\\leq \\sum\\_{l=1}^3 {\\mathbb{P}\\!\\left(\\text{path from }i\\text{ to }j\\text{ of length }l \\right)}\\leq p + (n-2)p^2 +\\binom{n-2}{2}p^3 \\leq n^2p^3.$$ \nConditional on ${\\ensuremath{\\mathcal E}}\\_{i,j}$ every path between *i* and *j* must pass through at least one vertex from each of *d*2(*i*) and *d*2(*j*), though these vertices may not be distinct. So there cannot be more than min{*d*2(*i*), *d*2(*j*)} paths between *i*, *j* ∈ *V* which are vertex disjoint on *V*\\* : = *V* ∖ (*B*1(*i*) ∪ *B*1(*j*)) since Γ2(*i*) ∪ Γ2(*j*) ⊆ *V*\\*. Thus conditional on ${\\ensuremath{\\mathcal E}}\\_{i,j}$ for any *l* ≥ 0 we have\n \n*p**a**t**h**s*2(*i*, *j*, *l*) ≤ min{*d*2(*i*), *d*2(*j*)}. \nTo bound *p**a**t**h**s*2(*i*, *j*, *l*) from below we construct min{*ψ*2(*i*), *ψ*2(*j*)} vertex disjoint paths between *i* and *j* conditional on ${\\ensuremath{\\mathcal A}}\\_{i,j}$, Definition [A], then couple *ψ*2(*i*) to *d*2(*i*) and *ψ*2(*j*) to *d*2(*j*).\nFor the path construction condition on A*i*, *j* and w.l.o.g. assume *ψ*2(*i*) ≤ *ψ*2(*j*). Take any subset Ψ2(*j*)\\* ⊆ Ψ2(*j*) with *ψ*2(*i*) elements and any bijection *M* between Ψ2(*i*) and Ψ2(*j*)\\*. Given any pair (*x*, *y*) in *M*, conditional on A*i*, *j*, there is some *k* and some pair (*x**k*, *y**k*) ∈ Γ*k*\\*(*x*) × Γ*k*\\*(*y*) such that *x**k**y**k* ∈ *E*. We define the path *P**x*, *y* : = *i*, *i**x*, *x*, *x*1, …, *x**k*, *y**k*, *y**k* − 1, …, *y*, *j**y*, *j*, where *x*, *x*1, …, *x**k* is the unique path from *x* to *x**k* in the tree *T**k*(*x*) : = ∪ *i* = 0*k*Γ*i*\\*(*x*) and *i**x* is the unique vertex in Γ1\\*(*i*) connected to *x*. The equivalent descriptions hold for *y*, *y*1, …, *y**k* ∈ *T**k*(*y*) and *j**y* ∈ Γ1(*j*) with respect to *y* and *j*. The paths {*P**x*, *y*}(*x*, *y*) ∈ *M* are all vertex disjoint on *V*\\* since the trees {*T**K*(*u*)}*u* ∈ Ψ2 are all vertex disjoint. Each path in *P**i*, *j* has length *l* : = 2*k* + 5 where the *k* is given by the event A*i*, *j*. Thus conditional on the event A*i*, *j* we have\n \n*p**a**t**h**s*2(*i*, *j*, *l*) ≥ ∣{*P**x*, *y*}(*x*, *y*) ∈ *M*∣ = min{*ψ*2(*i*), *ψ*2(*j*)}. \nExchanging the *ψ*2 and *d*2 distributions on the event {*ψ*2(*i*) ≠ *d*2(*i*)or *ψ*2(*j*) ≠ *d*2(*j*)} yields\n \n$$\\begin{aligned}\n \\mathfrak{P}&:={\\mathbb{P}\\!\\left(paths\\_2(i,j,l)\\neq \\min\\{{d}\\_2(i),{d}\\_2(j)\\}\\right)} \\leq {\\mathbb{P}\\!\\left(\\psi\\_2(i) \\neq {d}\\_2(i) \\text{ or } \\psi\\_2(j) \\neq {d}\\_2(j) \\right)} \\\\ &\\qquad+{\\mathbb{P}\\!\\left(paths\\_2(i,j,l)< \\min\\{\\psi\\_2(i),\\psi\\_2(j)\\}\\right)} +{\\mathbb{P}\\!\\left(\\left\\{paths\\_2(i,j,l)> \\min\\{{d}\\_2(i),{d}\\_2(j)\\}\\right\\}\\right)}.\\intertext{Now by \\eqref{pathsA} and \\eqref{pathsE} we have the following} \\mathfrak{P}&\\leq {\\mathbb{P}\\!\\left(\\psi\\_2(i) \\neq {d}\\_2(i) \\text{ or } \\psi\\_2(j) \\neq {d}\\_2(j) \\right)} + {\\mathbb{P}\\!\\left(( \\mathcal{A}\\_{i,j})^c\\right)} + {\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal E}}\\_{i,j}^c\\right)} \\leq 5n^3p^4 + o\\left(e^{-7\\min \\left\\{np,\\log n \\right\\}/2}\\right), \n \\end{aligned}$$ \nby Lemma [sdom], Lemma [setlem] and respectively. On the event A*i*, *j* the strong path property is satisfied for some $k\\leq \\lfloor\\tfrac{\\log n}{2\\log (np)}\\rfloor +2$, thus $l=2k+5 \\leq \\tfrac{\\log n}{\\log (np)} +9$.\n*Item* : observe that *d*2(*u*) ∼ *B**i**n*(*n* − 1 − *d*(*u*), 1 − (1 − *p*)*d*(*u*)), conditional on *d*(*u*) for any *u* ∈ *V*. Notice that (1 − *p*)*k* ≤ 1 − *k**p* + (*k**p*)2 when (*k**p*)*i* ≥ (*k**p*)*i* + 1 for all *i* by the Bernoulli inequality. Thus conditional on ${\\ensuremath{\\mathcal E}}(\\{i,j\\},3\\log(np))$, see, we have the following\n \n$$\\begin{aligned}\n Bin\\left(n-2np, np^2 - 2p\\sqrt{\\log (np)np}\\right)\\preceq\\_1 {d}\\_2(i),{d}\\_2(j) \\preceq\\_1 Bin\\left(n, np^2 + p\\sqrt{3\\log (np)np}\\right). \n \\end{aligned}$$ \nLet R*i*, *j* be the event $\\left\\{ \\left|\\min\\{{d}\\_2(i),{d}\\_2(j)\\}-(np)^2\\right|\\leq 3(np)^{3/2}\\sqrt{\\log np}\\right\\}$. Observe that we have\n \n$$\\begin{aligned}\n\\label{rrrtag}\n {\\mathbb{P}\\!\\left(\\mathcal{R}\\_{i,j}^c\\right)}\\leq {\\mathbb{P}\\!\\left(\\mathcal{R}\\_{i,j}^c\\big|{\\ensuremath{\\mathcal E}}(\\{i,j\\},3\\log(np))\\right)} +{\\mathbb{P}\\!\\left({\\ensuremath{\\mathcal E}}(\\{i,j\\},3\\log(np))^c \\right)} = o\\left(1/np\\right),\n \\end{aligned}$$ \nby and applying Chernoff bounds to *d*2(*i*) conditional on ${\\ensuremath{\\mathcal E}}(\\{i,j\\},3\\log(np))$. We now have\n \n$$\\begin{aligned}\n& {\\mathbb{P}\\!\\left(\\left|paths\\_2(i,j,l) -(np)^2\\right|> 3(np)^{3/2}\\sqrt{\\log np}\\right)} \\leq {\\mathbb{P}\\!\\left(paths\\_2(i,j,l) \\neq \\min\\{{d}\\_2(i),{d}\\_2(j) \\}\\right)}+{\\mathbb{P}\\!\\left(\\mathcal{R}\\_{i,j}^c\\right)} \\\\ &\\qquad \\leq 5n^3p^4 + o\\left(e^{-7\\min \\left\\{np,\\log n \\right\\}/2}\\right) + o\\left(1/np\\right)= o\\left(1/np\\right), \\end{aligned}$$ \nby Item and the bound on P (R*i*, *j**c*) from.\nProof of Theorems [exthm], [concentrationnew], [exthm2] & [concentration]\n=========================================================================\nRecall $\\varepsilon\\_n : = \\frac{\\log n}{ np\\log (np)}$ from, that *m* = ∣*E*∣ and Tetali’s formula, which is given by\n \n$$\\begin{aligned}\n h(i, j) &= m R(i, j) + \\sum\\limits\\_{u\\in V} \\frac{{d}(u)}{2}\\left[R(j,u)-R(u, i)\\right].\n \\end{aligned}$$ \nOur results on hitting times and other random walk indices come from applying our bounds on resistance to Tetali’s formula to obtain moments hitting times. The following two Lemmas help us calculate the terms arising during these computations.\n[ubredux]Let G ∼ G(*n*, *p*) where log*n* + logloglog*n* ≤ *n**p* < *o*(*n*1/3). Let *α* ≥ 1 and Ψ1(*u*), Ψ1(*v*) be defined with respect to MBFS(G, {*u*, *v*}), *u*, *v* ∈ *V*. Then \n$$\\mathbb{E}\\_\\mathcal{C}\\left[\\frac{\\mathbf{1}\\_{\\mathcal{B}\\_{u}^{u,v}}}{\\psi(u)^\\alpha} \\right]^{1/\\alpha}= \\frac{ 1 + {O\\!\\left({\\ensuremath{\\epsilon}}\\_n\\right)}}{np}.$$ \nWe restrict to the event ${\\ensuremath{\\mathcal B}}\\_{u}^{u,v}$ to ensure the expectation is bounded,\n \n$$\\begin{aligned}\n \\mathfrak{E}&:= \\mathbb{E}\\_\\mathcal{C}\\left[ \\frac{\\mathbf{1}\\_{\\mathcal{B}\\_{u}^{u,v}}}{\\psi(u)^\\alpha}\\right]= \\sum\\limits\\_{k=1}^{n} \\frac{1}{k^\\alpha}{\\mathbb{P}\\_{\\mathcal{C}}\\!\\left(\\psi(u) =k \\right)} = \\sum\\limits\\_{k=1}^{n} \\frac{1}{k^\\alpha}\\frac{{\\mathbb{P}\\!\\left(\\{\\psi(u) =k \\}\\cap \\mathcal{C}\\right)}}{{\\mathbb{P}\\!\\left(\\mathcal{C}\\right)}}. \\notag \\intertext{Applying the coupling inequality \\eqref{prk}, and then Lemma \\ref{sdom} to bound ${\\mathbb{P}\\!\\left({d}^\\*(u) \\neq \\psi(u)\\right)}$ gives} \\mathfrak{E} &\\leq \\sum\\limits\\_{k=1}^{n} \\frac{1}{k^\\alpha}\\frac{{\\mathbb{P}\\!\\left({d}^\\*(u) =k\\right)} +{\\mathbb{P}\\!\\left({d}^\\*(u) \\neq \\psi(u)\\right)} }{{\\mathbb{P}\\!\\left(\\mathcal{C}\\right)}} =\\sum\\limits\\_{k=1}^{n} \\frac{1}{k^\\alpha}\\frac{{\\mathbb{P}\\!\\left({d}^\\*(u) =k\\right)} +e^{-(1-o(1))np} }{{\\mathbb{P}\\!\\left(\\mathcal{C}\\right)}}.\\notag \\intertext{ Let $\\tilde{{d}}\\_1(v):= \\left|\\Gamma\\_1(v) \\cap S\\_0 \\right| \\sim Bin(n-2,p)$. By Lemma \\ref{dom} we have ${d}^\\*(u) \\sim Bin(n-2-h,p)$ conditional on $\\{\\tilde{{d}}\\_1(v)=h \\}$. By the law of total expectation and the generalised harmonic series, }\n \\mathfrak{E}&\\leq \\sum\\limits\\_{k=1}^{"},"data_id":{"kind":"string","value":"arxiv_0183892"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":459,"numItemsPerPage":100,"numTotalItems":46503,"offset":45900,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NjE3ODc4NCwianRpIjoiZTVmZTdkOWEtMThlYy00Y2Y0LThmOTYtOTM3ZDhkNzM2M2U1Iiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3ODYxODIzODQsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.iHvQbfVQGrq-ZrYWn_fxk7f5SyO-SD_VOLtUFkQc5kp1zsy5dqOLXalh-SX8m_R1TBCxPylVfoWrLO9mO85MCQ","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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Exploratory X-ray Monitoring of Luminous Radio-Quiet Quasars at High Redshift:Initial Results
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Department of Physics, University of North Texas, Denton, TX 76203, USA; ohad@unt.edu Department of Astronomy & Astrophysics, The Pe... | arxiv_0183486 |
Feedback through graph motifs relates structure and function in complex networks
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[1](#fn1)
In physics, biology and engineering, network systems abound. How does the connectivity of a network system combine with the behavior of its individu... | arxiv_0183495 |
Eq. ([fJ]) for the fraction of time spent by the watcher in different vacua can no longer be used here. The reason is that the lifetimes of dS and AdS vacua are influenced by their decays into *M*-vacua, which are ignored in this case.
With this prescription, vast Minkowski regions of spacetime are excluded from consi... | arxiv_0183498 |
The LLV decomposition of hyper-Kähler cohomology(The known cases and the general conjectural behavior)
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[1](#fn1)
Looijenga–Lunts and Verbitsky showed that the cohomology of a compact hyper-Kähler manifold *X* admits a... | arxiv_0183500 |
*T*1 = 0.3 and *n* = *n*1 = 1 in our rescaled variables. This approach differs significantly from some previous investigations that focussed exclusively on the energetics of the dissipation region — for example on Joule heating. Our approach is, however, entirely consistent with the Petschek scenario in which most of ... | arxiv_0183501 |
S. [1998], The Euler-Poincaré equations and semidirect products with applications to continuum theories, *Adv. Math.*, **137** (1), 1–81.
Holm, D. D., Schmah, T., and Stoica, C. [2011], *Geometric mechanics and symmetry, From finite to infinite dimensions*, Oxford Texts in Applied and Engineering Mathematics, 12. Oxfo... | arxiv_0183505 |
Grand orbit relations in wandering domains
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One of the fundamental distinctions in McMullen and Sullivan’s description of the Teichmüller space of a complex dynamical system is between discrete and indiscrete grand orbit relations. We investigate these on the Fatou set of trans... | arxiv_0183511 |
Bayes EMbedding (BEM): Refining Representation by Integrating Knowledge Graphs and Behavior-specific Networks
=============================================================================================================
Low-dimensional embeddings of knowledge graphs and behavior graphs have proved remarkably powerful i... | arxiv_0183515 |
A covariant approach to parameterised cosmological perturbations
================================================================
### Received 30 June 2017; published 8 September 2017
We present a covariant formulation for constructing general quadratic actions for cosmological perturbations, invariant under a given se... | arxiv_0183518 |
The geometry of the ball quotient model of the moduli space of genus four curves
================================================================================
[1](#fn1)
[2](#fn2)
S. Kondo has constructed a ball quotient compactification for the moduli space of non-hyperelliptic genus four curves. In this paper, we s... | arxiv_0183523 |
Two loop soft function for secondary massive quarks
===================================================
We present the calculation of the O(*α**s*2*C**F**T**F*) massive quark corrections to the soft function for the double hemisphere jet mass distribution in *e*+*e*− collisions, a necessary ingredient for the calculati... | arxiv_0183524 |
as well as the more quiescent galaxies have significantly steeper curves, up to (*δ* ∼ − 0.9), which is steeper than the slope of SMC extinction curve (roughly *δ* = − 0.45). There are no regions with average slopes shallower than the Calzetti curve.
For star-forming galaxies (lower left panel) the primary trend in ... | arxiv_0183529 |
& 0.949 & 0.939 & 0.954 & 0.889 & 0.926 & 0.936 & 0.935
& 0.95 & 0.972 & 0.978 & 0.979 & 0.905 & 0.961 & 0.966 & 0.968
& 0.99 & 0.983 & 0.997 & 0.997 & 0.916 & 0.995 & 0.996 & 0.992 & 0.90 & 0.921 & 0.919 & 0.942 & 0.907 & 0.929 & 0.934 & 0.942
& 0.95 & 0.966 & 0.964 & 0.976 & 0.946 & 0.966 & 0.971 & 0.974
& 0... | arxiv_0183530 |
A comparative study on machine learning models combining with outlier detection and balanced sampling methods for credit scoring
================================================================================================================================
**A comparative study on machine learning models combining wi... | arxiv_0183536 |
Calabi-Yau properties of ribbon graph orders
============================================
We pursue the order-theoretic approach to ribbon graphs initiated by Kauer and Roggenkamp. We show that any ribbon graph order is twisted 1-Calabi-Yau in general and 1-Calabi-Yau if the ribbon graph is bipartite. We derive analogo... | arxiv_0183546 |
Posterior Concentration for Bayesian Regression Trees and Forests
=================================================================
,
Since their inception in the 1980’s, regression trees have been one of the more widely used non-parametric prediction methods. Tree-structured methods yield a histogram reconstruction... | arxiv_0183547 |
*c**n*). Letting *n* go to infinity implies that *ϕ*1[0 ↔ ∞] = 0. It is classical that the absence of infinite connected component implies *ϕ*1 = *ϕ*0 (one can see it as a simple application of and that we leave as an interesting exercise; or see ).
[cor:3] Assume **(SupCrit)**. There exists *c* > 0 such that for every... | arxiv_0183549 |
Deep ReLU Networks Have Surprisingly Simple Polytopes
=====================================================
A ReLU network is a piecewise linear function over polytopes. Figuring out the properties of such polytopes is of fundamental importance for the research and development of neural networks. So far, either theoret... | arxiv_0183554 |
the centroid positions determined by AE from its “CHECK\_POSITIONS” stage were adopted as the improved positions of these sources. We registered the absolute astrometry of each observation to a common frame by matching the X-ray source positions to the NIR source positions in the Taiwan ECDFS Near-Infrared Survey (TEN... | arxiv_0183557 |
unknown subset of nodes are intervened. We consider a general latent causal model (parametric or nonparametric) and focus on the *linear* transformations as an important class of parametric transformation models. We establish identifiability results and design algorithms to achieve them under both soft and hard interv... | arxiv_0183563 |
Approximate group context tree
==============================
We study a variable length Markov chain model associated with a group of stationary processes that share the same context tree but each process has potentially different conditional probabilities. We propose a new model selection and estimation method which ... | arxiv_0183570 |
, a few degrees, depend mostly on the assumption taken for the dust mass opacity.
Clump catalog
-------------
[]
[tab:MDCs]
Compact sources were extracted from the column density map at 18.2$\arcsec$ resolution (see ) using *getsf* (v200124, Men’shchikov in prep.), a new multi-scale, multi-wavelength source and fi... | arxiv_0183573 |
Counting solutions to random CNF formulas
=========================================
### 24th May 2021
We give the first efficient algorithm to approximately count the number of solutions in the random *k*-SAT model when the density of the formula scales exponentially with *k*. The best previous counting algorithm for t... | arxiv_0183575 |
On almost Blow-analytic equivalence
===================================
We study the analytic equivalence of real analytic function germs after desingularization and state the cardinality of the classes under this equivalence relation. We consider also the Nash case, and compare these equivalences with the blow-analyti... | arxiv_0183591 |
iterations. The composite pulse sequence used in achieves a fidelity of 0.994. In both cases, the deviation from unity is caused by off-resonant excitation on the carrier transition. It is quite remarkable that the Krotov algorithm finds a time-dependent phase *ϕ*(*t*) with similar amplitude and period as the composi... | arxiv_0183596 |
ambiguous] gives an algorithm producing an ambiguous system of combinatorial equations describing ${\ensuremath{\mathcal{C}}\xspace}$. Next, Section [sec:disambiguation] describes how to adapt this algorithm to obtain a combinatorial specification for ${\ensuremath{\mathcal{C}}\xspace}$. Finally, Section [sec:ex] illus... | arxiv_0183614 |
point to an overall lower energy budget, and potentially lower circumburst densities. Taken in conjunction, the emission and environmental properties point to a distinct origin for short and long GRBs, and indicate that the short GRB progenitors are not dominated by a young population (e.g., massive stars or young mag... | arxiv_0183616 |
)$ is called the volume of V1, while the boundary of V1, defined as ∂V1 = {*e* ∈ E : *e* ∩ V1 ≠ *ϕ*, *e* ∩ V1*c* ≠ *ϕ*}, denotes the set of edges that are cut when the nodes are divided into V1 and V ∖ V1. The volume of ∂V1 is defined as
$$\begin{aligned}
{\textup{vol}}(\partial{\mathcal{V}}\_1) = \sum\_{e\in\partia... | arxiv_0183623 |
The Red Supergiant Binary Fraction of theLarge Magellanic Cloud
===============================================================
The binary fraction of unevolved massive stars is thought to be 70-100% but there are few observational constraints on the binary fraction of the evolved version of a subset of these stars, th... | arxiv_0183632 |
Hunting Majorana Fermions in Kitaev Magnets
===========================================
Introduction
============
Majorana fermions are charge-neutral spin-1/2 particles that are their own antiparticles. They were theoretically discovered by Ettore Majorana in 1937 in a real solution for the Dirac equation . The Majora... | arxiv_0183633 |
Time-to-Event Prediction with Neural Networksand Cox Regression
===============================================================
New methods for time-to-event prediction are proposed by extending the Cox proportional hazards model with neural networks. Building on methodology from nested case-control studies, we propose... | arxiv_0183637 |
the most popular MDS codes and together with the very similar information dispersal algorithm (IDA) , have been investigated in distributed storage applications (e.g.). Fountain codes and LDPC codes are recent code designs that offer approximate MDS properties and fast encoding-and-decoding complexity. Finally there... | arxiv_0183638 |
#1(see Fig. #1.)
1.4cm
***E*11-extended spacetime and gauged supergravities**
Fabio Riccioni and Peter West
*Department of Mathematics*
King’s College London
Strand London WC2R 2LS
UK
1.5cm
addtoresetequationsection
Introduction
============
One of the most surprising discoveries in the development of superg... | arxiv_0183642 |
for *E* as defined in , n,ij & = & 0, (i, j) D
n,ij & = & n, ij, (i, j) E {(i,i), i = 1, …, p}. Hence the entries in the covariance matrix $\hat{\Sigma}\_n$ for the chosen set of edges in *E* and the diagonal entries are set to their corresponding values in *Ŝ**n*. Indeed, we can derive the above relationships via ... | arxiv_0183644 |
Long-Term Profile Variability in Active Galactic Nuclei with Double-Peaked Balmer Emission Lines
================================================================================================
An increasing number of Active Galactic Nuclei (AGNs) exhibit broad, double-peaked Balmer emission lines,which represent some ... | arxiv_0183649 |
(\alpha\_0 y) = \alpha\_0 g\_\gamma(y) = {\hat{\mathcal{F}}}^0 (g\_\gamma(y)),
\label{eq:filt\_diff\_rot\_zero\_deg\_shift}$$
is identical to the proof of Theorem [t:imrot], which validates the base case for the *trivial* filter of degree 0. Then we need to show that Eq. ([eq:transtrivdiff]) is valid for any *trivia... | arxiv_0183652 |
*β* ↑ 1, so $\calR\_1^{\rm K}{\stackrel{d}{=}}\{\calV\_1\}$ and $\calM\_{\alpha,1}^{\rm K}{\stackrel{d}{=}}\calM\_\alpha^{\rm is}$.
Now to put this random sup-measure into our aggregation framework, it suffices to find discrete random variables *V**n* ∈ {0, …, *n*}/*n* such that $V\_n\weakto \calV\_1$, and consider it... | arxiv_0183656 |
Photospheric Emission in Gamma-Ray Bursts
=========================================
A major breakthrough in our understanding of gamma-ray bursts (GRB) prompt emission physics occurred in the last few years, with the realization that a thermal component accompanies the over-all non-thermal prompt spectra. This thermal ... | arxiv_0183660 |
$0.7043 & +*118%*
& *σ* = 20% & $0.6362 & $1.4534 & +*128%*
& *σ* = 25% & $1.0060 & $2.4051 & +*139%*
The above table illustrates the impact of investment (RPI) volatility (*σ*) on both the RCLA and S-RCLA value, assuming the same Gompertz mortality with parameters *λ* = 0, *m* = 86.3 and *b* = 9.5. Note that bo... | arxiv_0183663 |
be closely related to the above three classes of games.
[D.FR] [d1] A game *G* is said to be
1. *forced* if each move from a (0, 1)-position results in a (1, 0)-position and vice versa;
2. *returnable* if the following, weaker, implications hold: let *x* be a (0, 1)-position (resp., a (1, 0)-position) movable to a non... | arxiv_0183667 |
Deconfinement in $\mathbf{{\cal N}=1}$ super Yang-Mills theory on **\R^3 \times \S^1** via dual-Coulomb gas and ``affine" XY-model
==================================================================================================================================
Introduction
============
The dynamics of ${\cal{N}}=1$ su... | arxiv_0183672 |
when fitting SFR instead of *i*. Namely, $\log M\_{\mathrm{{H\_2}}} = d + e \,(\log \mathrm{H\alpha/H\beta}-0.62) + f \log \mathrm{SFR}$, where *d* = 8.95 ± 0.14, *e* = 1.92 ± 0.30 and *f* = 0.80 ± 0.05. This gas scaling of star-forming galaxies with SFR and H\alpha/H\beta is similar to what we found for the total sam... | arxiv_0183691 |
High Dimensional Causal Discovery Under Non-Gaussianity
=======================================================
We consider graphical models based on a recursive system of linear structural equations. This implies that there is an ordering, *σ*, of the variables such that each observed variable *Y**v* is a linear funct... | arxiv_0183693 |
}^T $. In this section the linearized coupled advection scheme is investigated, while the full non-linear coupled advection scheme is studied in section [ss:Channel].
W.l.o.g. the advected velocity ${{\underline{u}}}\_{1}$ is represented by the two staggered XLES-grids containing ${\hat{{u}}\_{{2,1}}}$ and ${\hat{{u}}\... | arxiv_0183695 |
Binary progenitor models of type IIb supernovae
===============================================
### Received 16 July 2010 / Accepted 30 January 2011
Introduction
============
Core collapse supernovae are the bright explosions marking the end of the lives of massive stars. Their light curves and spectral signatures come... | arxiv_0183697 |
Secure GDoF of the *Z*-channel with Finite Precision CSIT:How Robust are Structured Codes?
==========================================================================================
Under the assumption of perfect channel state information at the transmitters (CSIT), it is known that structured codes offer significant ... | arxiv_0183700 |
overset{T}{\longrightarrow} \quad \sum\_{m,n} (-1)^m \, \Lambda\_{m,n+\frac{1}{2}}$$
hold both for odd and even *s*.
99
E. Dudas and J. Mourad, *Brane solutions in strings with broken supersymmetry and dilaton tadpoles,* Phys. Lett. B **486** (2000), 172-178 [arXiv:hep-th/0004165 [hep-th]].
O. DeWolfe, D. Z. Freedman... | arxiv_0183702 |
Expected Complexity of Routing in Θ6 and Half-Θ6 Graphs[4](#fn4)
=================================================================
We study online routing algorithms on the Θ6-graphand the half-Θ6-graph(which is equivalent to a variant of the Delaunay triangulation). Given a source vertex *s* and a target vertex *t* i... | arxiv_0183703 |
&
Qatar & & Singapore &
United Arab Emirates & & Canada &
& & USA &
The contribution of this real-world dataset cluster analysis is two-fold. First, we explored and determined the principal force that brings structural difference in global capital markets, which potentially predicts the “comovement” pattern of... | arxiv_0183705 |
Effective estimates for the degrees of maximal special subvarieties
===================================================================
Let *Z* be an algebraic subvariety of a Shimura variety. We extend results of the first author to prove an effective upper bound for the degree of a non-facteur maximal special subvari... | arxiv_0183706 |
Integral Fluctuation Relations for Entropy Production at Stopping Times
=======================================================================
A stopping time *T* is the first time when a trajectory of a stochastic process satisfies a specific criterion. In this paper, we use martingale theory to derive the integral f... | arxiv_0183711 |
35 | 3.50 ± 1.33 |
Table [table:detection:gal] shows that only a fraction of the initial sample is detected in the far-IR. To overcome this low detection rate, we can use a stacking method to improve our detection threshold. We start from a catalog of 24 *μ*m sources divided into bins of redshift and flux and then sta... | arxiv_0183712 |
Strong, weak or no balance?Testing structural hypotheses in real heterogeneous networks
=======================================================================================
The abundance of data about social, economic and political relationships allows social theories to be tested against empirical evidence and huma... | arxiv_0183717 |
target receive power at the FAP, the handed off cellular user in open access generally transmits with higher power to the FAP (thereby creating increased femto-to-macro interference) as compared to the in home user. Thus open access potentially deteriorates QoS provided to cellular users remaining in the macrocell (ar... | arxiv_0183719 |
charging each agent the expected externality that her report causes to other agents. This expected externality vanishes to zero in the size of the group, so asymptotically, the AGV mechanisms do not require large transfers. However, to appropriately charge any given report, the AGV mechanism needs to know the exact ex... | arxiv_0183720 |
*b*-monotone Hurwitz numbers:Virasoro constraints, BKP hierarchy, and *O*(*N*)-BGW integral
===========================================================================================
We study a *b*-deformation of monotone Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We give an ... | arxiv_0183721 |
és trop restrictifs, et des modèles non-paramétriques classiques, dont l’estimation peut être difficile en présence d’un nombre élevé de variables. L’*apprentissage statistique*, ou *apprentissage machine*, propose de nouvelles méthodes d’estimation non-paramétriques, performantes dans un cadre général et en présence d... | arxiv_0183724 |
Constrained path-finding and structure from acyclicity
======================================================
[1](#fn1)
This note presents several results in graph theory inspired by the author’s work in the proof theory of linear logic; these results are purely combinatorial and do not involve logic.
We show that trai... | arxiv_0183727 |
[-2 D (-1 + p\_G) p\_G+ A (1 - 2 p\_G + 2 p\_G^2)]\}\right),\nonumber\\
D'&=\frac{1}{{P\_D}}\left\{-2 (C (-1 + p\_G) p\_G (C + D + 2 A p\_G - 2 C p\_G - 2 D p\_G - 2 A p\_G^2 + 2 C p\_G^2 +
2 D p\_G^2) +B^2 p\_G (-1 + 3 p\_G- 4 p\_G^2 + 2 p\_G^3)\right. \nonumber\\
&\left.-B \{C (1 - 2 p\_G + 2 p\_G^2)^2- (-1 + p\_G)... | arxiv_0183728 |
presence of spotted cooler regions, thus inducing a radius inflation and a change in the expected colours [spectral energy distribution (SED)]. Depending on the assumptions made on the spots (i.e. their depth, temperature and coverage fraction) a radius variation up to about 10 percent can be obtained. showed that mo... | arxiv_0183730 |
Quantifying the statistics of CMB-lensing-derived galaxy cluster mass measurements with simulations
===================================================================================================
### Accepted XXX. Received YYY; in original form ZZZ
[firstpage]
CMB lensing is a promising, novel way to measure galaxy... | arxiv_0183731 |
Algebraic canonical quantization of lumped superconducting networks
===================================================================
We present a systematic canonical quantization procedure for lumped-element superconducting networks by using a redundant configuration-space description. The algorithm is based on an ... | arxiv_0183732 |
A Simple Adjustment for Bandwidth Snooping[3](#fn3)
===================================================
Kernel-based estimators such as local polynomial estimators in regression discontinuity designs are often evaluated at multiple bandwidths as a form of sensitivity analysis. However, if in the reported results, a res... | arxiv_0183744 |
Smooth and polyhedral norms viafundamental biorthogonal systems
===============================================================
[1](#fn1)
Let X be a Banach space with a fundamental biorthogonal system and let Y be the dense subspace spanned by the vectors of the system. We prove that Y admits a *C*∞-smooth norm that lo... | arxiv_0183748 |
Homogenous and heterogeneous magnetism in (Zn,Co)O
==================================================
A series of (ZnO)*m*(CoO)*n* digital alloys (*m* = 2, 8, *n* = 1) and superlattices (*m* = 80, *n* = 5, 10) grown by atomic layer deposition has been investigated by a range of experimental methods. The data provide e... | arxiv_0183758 |
{m}} \frac{1}{\sqrt{\frac{V\_{1,p}}{n}+\frac{V\_{2,q}}{m}}}\right)\\
&+& \ \
\frac{1}{\sqrt{\frac{V\_{1,p}}{n}+\frac{V\_{2,q}}{m}}}R\_{n,m}.\end{aligned}$$
That leads to
$$\begin{aligned}
\sqrt{\frac{nm}{mV\_{1,p}+nV\_{2,q}}}\left( J(\widehat{\textbf{p}}\_n,\widehat{\textbf{q}}\_m)-J(\textbf{p},\textbf{q})\right)... | arxiv_0183761 |
To
*My Parents*
*and*
*Teachers*
Declaration
===========
I hereby declare that the work reported in this thesis is entirely original and has been carried out by me under the supervision of Prof. Manjunath Krishnapur at the Department of Mathematics, Indian Institute of S... | arxiv_0183764 |
Thermal transport in isotopically disordered carbon nanotubes
=============================================================
We present a study of the phononic thermal conductivity of isotopically disordered carbon nanotubes. In particular, the behavior of the thermal conductivity as a function of the system length is i... | arxiv_0183768 |
in our analytic discussion.
Milestone temperatures
----------------------
It is useful to define several temperatures corresponding to milestones in the phase transition. The nucleation, percolation and completion temperatures, among others, are often introduced to capture the moment of their respective milestones. He... | arxiv_0183769 |
51 × 51 arcsec2 field of view of the camera. The dispersion for the first order spectrum is about 8.0 nm per pixel, which results in spectra about 105 pixels long.
In March 2002, the NICMOS Cryocooler System (NCS) was installed. With the NCS, the NICMOS detectors operate at a higher temperature and this changed the d... | arxiv_0183770 |
**Random Orderings and Unique Ergodicity of Automorphism Groups**
=================================================================
### 9 August 2012
We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this impli... | arxiv_0183775 |
Comparison of Domain Wall Fermion Multigrid Methods
===================================================
We present a detailed comparison of several recent and new approaches to multigrid solver algorithms suitable for the solution of 5d chiral fermion actions such as Domain Wall fermions in the Shamir formulation, and ... | arxiv_0183777 |
Shock Dissipation in Magnetically Dominated Impulsive Flows
===========================================================
### Received/Accepted
The idea that cosmic relativistic jets are magnetically driven Poynting-dominated flows has many attractive features but also some problems. One of them is the low efficiency of ... | arxiv_0183787 |
X-ray Properties of Radio-Selected Dual Active Galactic Nuclei
===============================================================
Merger simulations predict that tidally induced gas inflows can trigger kpc-scale dual active galactic nuclei (dAGN) in heavily obscured environments. Previously with the Very Large Array, we ... | arxiv_0183791 |
Deformations and moduli of structures on manifolds: general existence theorem and application to the sasakian case
==================================================================================================================
In this paper, we deal with the moduli problem for geometric structures on manifolds. We p... | arxiv_0183798 |
� at *z* = 0.4 and *z* = 1, respectively.
We analysed the properties of the sample and the evolution of pure-disc systems with redshift. As expected, the bulk of bulgeless galaxies occupies the low-mass end region of the colour-mass diagram (log(*M*⋆/*M*⊙) < 10) and is predominantly blue, indicating recent star-formati... | arxiv_0183799 |
The Wilkinson Microwave Anisotropy Probe (WMAP) Source Catalog
==============================================================
We present the list of point sources found in the *WMAP* 5-year maps. The technique used in the first-year and three-year analysis now finds 390 point sources, and the five-year source catalog i... | arxiv_0183800 |
Magnetar Driven Bubbles and the Origin of Collimated Outflows in Gamma-ray Bursts
=================================================================================
### Accepted. Received ; in original form
[firstpage]
We model the interaction between the wind from a newly formed rapidly rotating magnetar and the surrou... | arxiv_0183801 |
Nonuniform Negative Sampling and Log Odds Correction with Rare Events Data
==========================================================================
We investigate the issue of parameter estimation with nonuniform negative sampling for imbalanced data. We first prove that, with imbalanced data, the available informati... | arxiv_0183806 |
Engineering Topological Modelswith a General-Purpose Symmetry-to-Hamiltonian Approach
=====================================================================================
Symmetry is at the heart of modern physics. Phases of matter are classified by symmetry breaking, topological phases are characterized by non-local ... | arxiv_0183811 |
Categoricity and covering spaces
================================
Categoricity and covering spaces
Adam Harris
Mathematical Institute
Merton College
A thesis submitted for the degree of
DPhil in Mathematics
University of Oxford, 2014
Funded by EPSRC
First and foremost I w... | arxiv_0183818 |
we actually see is that when the red continuum gets faint (presumably because clouds are rotating into view), the flux in the alkali lines *brightens*. In retrospect, it seems clear that also observed the same phenomenon in broadband photometry on the nights of UT 2013 April 16 and 22.
saw further deviations from th... | arxiv_0183824 |
Local magneto-shear instability in Newtonian gravity
====================================================
The magneto-rotational instability (MRI)—which is due to an interplay between a sheared background and the magnetic field—is commonly considered a key ingredient for developing and sustaining turbulence in the oute... | arxiv_0183830 |
Phonon Scattering at Kinks in Suspended Graphene
================================================
Recent experiments have shown surprisingly large thermal time constants in suspended graphene ranging from 10 to 100 ns in drums with a diameter ranging from 2 to 7 microns. The large time constants and their scaling with ... | arxiv_0183831 |
Constructing lattice surfaces with prescribed Veech groups: an algorithm
========================================================================
The Veech group of a translation surface is the group of Jacobians of orientation-preserving affine automorphisms of the surface. We present an algorithm which constructs all... | arxiv_0183832 |
FL]) as follows: ()=t() - (t())<, [eq:omegastarprop] where finiteness of Ω(*γ*) follows by the fact that *t*(*γ*) ∈ R, and by finiteness of *ω*(*t*). By further noting that *ω*′(*t*) is differentiable and *ω*′′(*t*) > 0, the theorem about differentiation of the inverse function allows concluding that the derivative of... | arxiv_0183838 |
Existence of stationary weak solutions for the heat conducting flows
====================================================================
1. University of Warsaw
Institute of Applied Mathematics and Mechanics
Banacha 2, Warsaw, Poland
e-mail: p.mucha@mimuw.edu.pl
2. Charles University in Prague
Faculty of Mat... | arxiv_0183840 |
, Toronto Book Corpus crawled 11,038 e-books from Smashwords, forming a large-scale corpus of books. This corpus was once publicly available but is no longer accessible. In 2019, PG-19 collected books published before 1919 from Project Gutenberg and removed short-text books, resulting in a final count of 28,752 books... | arxiv_0183846 |
ac *d*log*Y**i* ∧ *d*log*Y**j* = ⟨*α**i*, *α**j*⟩Dirac *d*log*R**i* ∧ *d*log*R**j*.
The rational map *Y**i* → *R**i* is simply the classical limit of the monodromy action, from which we may recover the full quantum action by taking the normal–order prescription for the operators (this is true for all simply–laced, *... | arxiv_0183848 |
Principal components in linear mixed models with general bulk
=============================================================
We study the principal components of covariance estimators in multivariate mixed-effects linear models. We show that, in high dimensions, the principal eigenvalues and eigenvectors may exhibit bia... | arxiv_0183849 |
{pmatrix},$$
with *γ*(*t*) a smooth function with support in the interval indicated in Fig. [Fig:LandaFlow3](b). This choice makes $\tilde s(t)$ smooth in *t*. If
$$\hat{\bm f}\_0 = \begin{pmatrix} \hat{\bm h} & 0 \\ 0 & \hat{\bm h}' \end{pmatrix}$$
is the adiabatic family of Hamiltonians for the uncoupled syste... | arxiv_0183853 |
), Wilman et al. (2008), and reference therein. All the simulated sky maps are plotted in a circular field of view (FOV) with a radius of 5∘ centered at the north celestial pole, as currently being observed with the 21CMA array, and the obtained images are pixelated with grids of 1024 × 1024 pixels, each image pixel co... | arxiv_0183855 |
) – (2.943013355541281,0.5748759344428223) – cycle; (3.2438189126237993,0.6280566799665583) – (3.1418629537083134,0.6280566799665585) – (3.1418629537083134,0.5261007210510723) – (3.2438189126237993,0.5261007210510723) – cycle; (3.1418629537083134,0.6280566799665585) – (3.0460861438180094,0.6280566799665586) – (3.046086... | arxiv_0183859 |
{aligned}$$
So we have
$$\begin{aligned}
\frac{1}{N}R(\hat{\kappa}\_1,\ldots,\hat{\kappa}\_K)=\frac{1}{N}R(\kappa^\*\_1,\ldots,\kappa\_K^\*)+O(N^{-\frac{q-3-2q\alpha}{2q}})-\frac{1}{N}R(\kappa\_1^0,\ldots,\kappa\_K^0)<0\end{aligned}$$
as *N* → ∞. This is a contradiction because *κ̂**k* are the maximizers of *R*(... | arxiv_0183860 |
Tri-hypercharge: a separate gauged weak hypercharge for each fermion family as the origin of flavour
====================================================================================================
fpheader
Introduction
============
The existence of the three families of quarks and leptons is one of the fundamental... | arxiv_0183864 |
Wave propagation and imaging in moving random media
===================================================
We present a study of sound wave propagation in a time dependent random medium and an application to imaging. The medium is modeled by small temporal and spatial random fluctuations in the wave speed and density, and... | arxiv_0183866 |
*P**A* and gives the updating scheme
$$\begin{aligned}
h\_{ij}^{t+1} \leftarrow h\_{ij}^t \frac{(W^TXQ+2\lambda H(PA))\_{ij}}{(W^TWHQ + 2\lambda H(PD))\_{ij}}. \end{aligned}$$
*k*-NN induced Persistent Laplacian
-----------------------------------
One major issue with top-GNMF and top-rGNMF is that the parameters... | arxiv_0183869 |
Early-type Eclipsing Binaries with Intermediate Orbital Periods
===============================================================
We analyze 221 eclipsing binaries (EBs) in the Large Magellanic Cloud with B-type main-sequence (MS) primaries (*M*1 ≈ 4-14 ) and orbital periods *P* = 20-50 days that were photometrically m... | arxiv_0183889 |
The QED four-photon amplitudes off-shell: part 2
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This is the second one of a series of four papers devoted to a first calculation of the scalar and spinor QED four-photon amplitudes completely off-shell. We use the worldline formalism which provides a gauge-invariant dec... | arxiv_0183891 |
449346390970478); (5.574252828322784,5.505499646566473)– (7.165261736875962,4.569612053299891); (5.574252828322784,4.513458797703897)– (7.240132744337288,3.3529581820533356); (5.798865850706763,3.165780663400019)– (7.09887593910803,4.441876131589434); (5.720429241945538,4.804251294245557)– (7.09887593910803,5.280312782... | arxiv_0183892 |
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