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Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 13
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 86124)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
File "pyarrow/error.pxi", line 154, in pyarrow.lib.pyarrow_internal_check_status
File "pyarrow/error.pxi", line 91, in pyarrow.lib.check_status
pyarrow.lib.ArrowInvalid: JSON parse error: Missing a closing quotation mark in string. in row 13
The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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text
string | meta
dict |
|---|---|
\section{Introduction}
The study of fronts interpolating from a stable solution to an unstable
solution is an important problem in mathematics, physics and biology; see
for instance \cite{AronsonWeinberger.1975, McKean.1975, DerridaSpohn.1988,
Murray.2002, vanSaarloos.2003, Munier.2015}. The archetypal model is the Fisher-KPP equation \cite{Fisher.1937,KPP.1937}
\begin{equation}
\partial_t h=\partial_x^2h +h-F(h),
\label{FKPP}
\end{equation}
where, throughout the paper, the non-linearity $F(h)$ is assumed to
satisfy the so-called ``Bramson's conditions'': \cite{Bramson.1978,Bramson.1983}
\begin{equation}
\begin{gathered}
F\in C^1[0,1],\quad F(0)=0,\quad F(1)=1,\quad F'(h)\ge0,\quad F(h)<h\text{ for
$h\in(0,1)$},
\\
F'(h)=\mathcal O(h^p)\text{ for some $p>0$ as $h\searrow0$}.
\end{gathered}
\label{BramsonsConditions}
\end{equation}
(The choice $F(h)=h^2$ is often made.)
One checks with these conditions that $h=0$ is an unstable solution and $h=1$
is a stable solution. We always assume implicitly that the initial
condition $h_0$ satisfies
$h_0\in[0,1]$; this implies, by comparison, that $0<h(x,t)<1$ for all $x$
and all $t>0$. We also always assume for simplicity that $h_0(x)\to1$ as
$x\to-\infty$, but this could be significantly relaxed.
A famous result due to Bramson \cite{Bramson.1978,Bramson.1983} (see also \cite{HamelNolenRoquejoffreRyzhik.2013,Roberts.2013}) states that,
\begin{equation}\label{Bramson's result}
h\big(2t-\tfrac32\log t + z,t\big) \xrightarrow[t\to\infty]{}\omega(z-a)
\qquad\text{iff $\int \mathrm{d} x\, h_0(x) x e^x < \infty$},
\end{equation}
where $\omega(z)$,
called the critical travelling wave, is a decreasing function
interpolating from $\omega(-\infty)=1$ to $\omega(+\infty)=0$,
and where the shift $a$ depends on the initial condition. In words, if
$h_0$ decays ``fast enough'' at infinity, then the stable solution $h=1$
on the left invades the unstable solution $h=0$ on the right, and the
position of the invasion front is $2t-\frac32\log t + a$.
The travelling wave $\omega$ is the unique solution to
\begin{equation}
\omega''+2\omega'+\omega-F(\omega)=0,\qquad
\omega(-\infty)=1,\qquad\omega(0)=\tfrac12,\qquad\omega(+\infty)=0,
\label{propomega}
\end{equation}
and there exists $\tilde\alpha>0$ and $\tilde\beta\in\mathbb{R}$
(depending on the choice of the non-linearity $F(h)$)
such that
$\omega(z)= (\tilde\alpha z + \tilde\beta ) e^{-z}+\mathcal O(e^{-(1+q)z})$ as
$z\to\infty$, where $q$ is any number in $(0,p)$. We prefer to write the equivalent statement:
\begin{equation}
\omega(z-a)= (\alpha z + \beta ) e^{-z}+\mathcal O(e^{-(1+q)z})\quad\text{as
$z\to\infty$},
\label{omegaalphabeta}
\end{equation}
where $\alpha>0$ and $\beta$ now depend also on the initial condition
$h_0$ through $a$
and are given by
$\alpha=\tilde\alpha e^a$ and
$\beta=(\tilde\beta-a\tilde\alpha)e^a$.
Let $\mu_t$ be the position where the front at time $t$ has value $1/2$ (or
the largest such position if there are more than one):
\begin{equation}
h(\mu_t,t)=\frac12.
\label{defmut}
\end{equation}
Bramson's result \eqref{Bramson's result} implies that
$\mu_t=2t-\frac32\log t + a + o(1)$ for large times if $\int \mathrm{d} x\,
h_0(x) x e^x<\infty$.
Recent results indicate that a more precise estimate of $\mu_t$ can be
given: if $h_0$ decays to zero ``fast enough'' as
$x\to\infty$, the position $\mu_t$ of the front is believed to satisfy:
\begin{equation}\label{position}
\mu_t=2t-\frac32\log t +a -3\frac{\sqrt\pi}{\sqrt t}+\frac98[5-6\log
2]\frac{\log t}t+\mathcal O\Big(\frac1t\Big),
\end{equation}
where we recall that $a$ depends on the initial condition and on the choice
of $F(h)$.
The $1/\sqrt
t$ correction is known as the Ebert-van Saarloos correction, from
a non-rigorous physics paper \cite{EbertvanSaarloos.2000}. This result was
proved \cite{NolenRoquejoffreRyzhik.2019} for $F(h)=h^2$ and $h_0$ a compact
perturbation of the step function (\textit{i.e.\@} $h_0$ differs from
the step function $\indic{x<0}$ on a compact set); see also
\cite{BBHR.2016}. The $(\log t)/t$ correction was conjectured in
\cite{BerestyckiBrunetDerrida.2017,BerestyckiBrunetDerrida.2018} using
universality argument and a implicit solution of a related model; it was
proved in \cite{Graham.2019} for $F(h)=h^2$ and $h_0$ a compact
perturbation of the step function. Arguments given in \cite{BerestyckiBrunetDerrida.2017} suggest that
the Ebert-van Saarloos term holds iff $\int \mathrm{d} x\, h_0(x) x^2 e^x
<\infty$ and that the $(\log t)/t$ terms holds iff $\int \mathrm{d} x\, h_0(x) x^3 e^x
<\infty$, for any choice of $F(h)$ satisfying~\eqref{BramsonsConditions}.
Another quantity of interest is the value of $h(ct,t)$ for $c>2$ and
large $t$. For instance, recalling \cite{McKean.1975} that $h(x,t)$, for
$F(h)=h^2$ and $h_0=\indic{x<0}$, is the probability that the rightmost
position at time $t$ in a branching Brownian motion is located on the
right of $x$, then $h(ct,t)$ would be the probability of a large deviation
where this rightmost position sustains a velocity $c>2$ some time $t$.
For a step initial condition, it is known
\cite{ChauvinRouault.1988, BovierHartung.2014, BovierHartung.2015, DerridaMeersonSasorov.2016,
BerestyckiBrunetCortinesMallein.2022} that
\begin{equation}\label{Phi}
h(ct,t) \sim \Phi(c) \frac1{\sqrt{4\pi t}} e^{(1-\frac{c^2}4)t}
\quad\text{as $t\to\infty$, for $c>2$},
\end{equation}
for some continuous function $c\mapsto \Phi(c)$, for an arbitrary
non-linearity $F(h)$ \cite{ChauvinRouault.1988,}.
(Note: The function $\Phi(c)$ in \eqref{Phi} is defined as in
\cite{DerridaMeersonSasorov.2016}. The function $\tilde C(\sigma_e)$ in \cite{BovierHartung.2015} and
$C(\rho)$ in \cite{BerestyckiBrunetCortinesMallein.2022} are identical and
related to $\Phi(c)$ by $\frac1{\sqrt{4\pi}}\Phi(c)=\frac2c C(\frac c2)$.)
We show in \autoref{prop1} below that \eqref{Phi} actually holds for any initial condition $h_0$ and any
$c>2$ such that $\int\mathrm{d} x \,h_0(x) e^{\frac c2 x}<\infty$, with
a function $\Phi(c)$ depending of course on $h_0$ and on $F(h)$.
The time dependence in
\eqref{Phi} is not surprising: the solution $h_\text{lin}$ to the
linearised Fisher-KPP equation, \textit{i.e.}\@
\eqref{FKPP} with $F(h)=0$, and with a step initial condition
$h_0(x)=\indic{x<0}$ satisfies \eqref{Phi} with a prefactor
$\Phi_\text{lin}(c)=2/c$. However, the dependence in $c$ of the prefactor
$\Phi(c)$ for the (non-linear) Fisher-KPP equation is much more
complicated. For $h_0(x)=\indic{x<0}$, it has been proved
\cite{BovierHartung.2015,BerestyckiBrunetCortinesMallein.2022} that
\begin{equation}
\Phi(2)=0,\qquad \Phi(c)\sim\frac2c \quad\text{as $c\to\infty$}.
\end{equation}
It is argued in \cite{DerridaMeersonSasorov.2016} that, for
$h_0(x)=\indic{x<0}$ and $F(h)=h^2$,
\begin{equation}
\Phi(2+\epsilon)\sim2\sqrt\pi \alpha \epsilon\quad\text{as
$\epsilon\searrow0$},\qquad\qquad
\Phi(c)\simeq\frac2c-\frac8{c^3}+\frac{6.818\ldots}{c^5}+\cdots
\quad\text{as $c\to\infty$},
\end{equation}
where $\alpha$
is the coefficient defined in \eqref{omegaalphabeta}.
The main result of this paper is an asymptotic expansion of the function
$\Phi$ for $c$ close to $2$:
\begin{thm}\label{mainthm}
For the Fisher-KPP equation \eqref{FKPP} with $F(h)=h^2$ and an initial
condition $h_0$ which is a compact perturbation of the step function, one
has
\begin{equation}
\begin{aligned}
\Phi(2+\epsilon)
=\sqrt\pi\Big(\alpha-\frac\beta2\epsilon\Big)
\Big[ 2\epsilon+3\epsilon^2\log\epsilon
-3\Big(1-\frac{\gamma_E}2\Big)\epsilon^2
+\frac94\epsilon^3\log^2\epsilon
& \\
+\frac34(3\gamma_E-6\log2-1)\epsilon^3\log\epsilon
&\Big]+\mathcal O(\epsilon^3)
\end{aligned}
\label{mainresult}
\end{equation}
where $\gamma_E$ is Euler's constant, and where $\alpha$ and $\beta$
are the coefficients defined in \eqref{omegaalphabeta}.
Actually, \eqref{mainresult} holds for any choice of $F(h)$ and of $h_0$
such that
\begin{enumerate}
\item $\int \mathrm{d} x\, h_0(x) e^{rx}<\infty$ for some $r>1$. (Otherwise,
$\Phi(c)$
would not be defined for $c>2$ and the expansion \eqref{mainresult} would be
meaningless.)
\item The position $\mu_t$ of the front satisfies the expansion~\eqref{position},
\item There exists $C>0$, $t_0\ge0$ and a neighbourhood $U$ of $1$ such that
\begin{equation}
\left |
\int \mathrm{d} z \, \Big(F[h(\mu_t+z,t)]
- F[\omega(z)]\Big)
e^{rz}
\right| \le \frac C t\qquad\text{for $t> t_0$ and $r\in U$}.
\label{technical}
\end{equation}
\end{enumerate}
\end{thm}
As will be apparent in the proofs, the expansion \eqref{mainresult} for
$\Phi(2+\epsilon)$ is closely related to the expansion~\eqref{position} for the
position $\mu_t$; in some sense, the
$\epsilon^2\log\epsilon$ and $\epsilon^3\log\epsilon^2$ terms in
\eqref{mainresult} are connected
to the $1 /\sqrt t$ term in \eqref{position}, and the
$\epsilon^3\log\epsilon$ to the $(\log t)/t$ term.
We will also see in the proof that \eqref{position} cannot hold unless $\int
\mathrm{d} x\, h_0(x) x^3e^x<\infty$. As already mentioned, we expect the converse to be true.
The technical condition \eqref{technical} should not be surprising:
The quantity $\delta(z,t):=h(\mu_t+z,t)-\omega(z)$ goes to zero
as $t\to\infty$. Moreover, it
satisfies
$\partial_t \delta = \partial_x^2\delta +\dot\mu_t \partial_x\delta +\delta
-F(\omega+\delta)+F(\omega) +(\dot\mu_t-2)\omega'$. For large times, one
can expect from \eqref{position} that $\dot\mu_t-2\sim -\frac3{2t}$ and
$\partial_t\delta\simeq
\partial_x^2\delta+2\partial_x\delta+\delta-F'(\omega)\delta
-\frac3{2t}\omega'$.
Then, it seems likely
that $\delta(z,t)\sim\frac1t\psi(z)$ with
$\psi$ a solution to $\psi''+2\psi'+\psi-F'(\omega)\psi=\frac32\omega'$.
(This is actually a result of \cite{Graham.2019} in the case $F(h)=h^2$.)
This leads to, $F[h(\mu_t+z,t)]-F[\omega(z)]\sim \delta(z,t) F'[\omega(z)]
\sim\frac1t\psi(z) F'[\omega(z)]$, of order $1/t$. Furthermore, (ignoring
polynomial prefactors), $\psi(z)$ decreases as $e^{-z}$ for large $z$ and
$F'[\omega(z)]$ should roughly decrease as $e^{-pz}$, see \eqref{BramsonsConditions}, so
that the integral in \eqref{technical} should converge quickly for $r$
around 1 for $z\to\pm\infty$, and give a result of order $1/t$.
In terms of the function $C(\rho)$ defined in
\cite{BerestyckiBrunetCortinesMallein.2022}, our result can be written as
\begin{equation}
C(1+\epsilon)
=(\alpha-\beta\epsilon)\Big[2\epsilon+ 6\epsilon^2\log\epsilon
+(3\gamma_E+6\log2-4)
\epsilon^2
+9\epsilon^3\log^2\epsilon
+3(3\gamma_E+1)\epsilon^3\log\epsilon
\Big]+\mathcal O(\epsilon^3).
\end{equation}
Note that the authors write $C(\rho)\sim\alpha(\rho-1)$ as $\rho\searrow1$
(bottom of p.\,2095),
but their $\alpha$ is twice ours.
The first part of \autoref{mainthm} is the consequence of its second part
and of the following result:
\begin{prop}[Mostly Cole Graham 2019 \cite{Graham.2019}]\label{prop1t}
For the Fisher-KPP equation \eqref{FKPP} with $F(h)=h^2$ and an initial
condition $h_0$ which is a compact perturbation of the step function,
\eqref{position} and \eqref{technical} hold.
\end{prop}
The fact that \eqref{position} holds under the hypotheses of
\autoref{prop1t} is the main result of \cite{Graham.2019}. The proofs
of \cite{Graham.2019} contain the hard parts
in showing that \eqref{technical} also holds.
The main tool used in this paper is the so-called magical relation,
which gives a relation between the initial condition, the position $\mu_t$
of the front, and the non-linear part of the equation. Introduce
\begin{equation}
\label{defgamma}
\gamma:=\sup\Big\{r>0; \int \mathrm{d} x\, h_0(x) e^{rx}<\infty\Big\},
\end{equation}
and
\begin{equation}
\varphi(\epsilon,t) :=\int\mathrm{d} z\, F[h(\mu_t+z,t)]e^{(1+\epsilon)z},\qquad
\hat\varphi(\epsilon) :=\int\mathrm{d} z\, F[\omega(z)] e^{(1+\epsilon)z}.
\label{defphi}
\end{equation}
(With these quantities, the condition \eqref{technical} can be written
$|\varphi(\epsilon,t)-\hat\varphi(\epsilon)| \le C/t$ for all $t>t_0$ and
all $\epsilon$ in some neighbourhood of 0.)
Then
\begin{prop}[Magical relation]\label{prop2}
For any $\epsilon\in(-1,\gamma-1)$
the following relation holds
\begin{equation}
\int_0^\infty\mathrm{d} t \,\varphi(\epsilon,t)e^{-\epsilon^2
t +(1+\epsilon)(\mu_t-2t)}=
\int\mathrm{d} x \,h_0(x)e^{(1+\epsilon) x}
-\indic{\epsilon>0}\Phi(2+2\epsilon).
\label{prop21}
\end{equation}
Furthermore, if $\gamma>1$ and \eqref{technical} holds,
one has
\begin{equation}
\hat\varphi(\epsilon)\int_0^\infty\mathrm{d} t \,e^{-\epsilon^2
t +(1+\epsilon)(\mu_t-2t)}=
\int\mathrm{d} x \,h_0(x)e^{(1+\epsilon) x}
-\indic{\epsilon>0}\Phi(2+2\epsilon)+\mathcal P(\epsilon)+\mathcal O(\epsilon^3).
\label{prop22}
\end{equation}
where $\mathcal P(\epsilon)$ is some polynomial in $\epsilon$.
\end{prop}
The second form \eqref{prop22} gives a relation between
$\mu_t$ and $h_0$ which does not involve the front $h(x,t)$ at any finite
time. Notice also that the non-linear term $F(h)$ only appears in
$\hat\varphi(\epsilon)$.
The magical relation was introduced in
\cite{BrunetDerrida.2015,BerestyckiBrunetDerrida.2017,BerestyckiBrunetDerrida.2018},
but only for $\epsilon<0$.
It allowed (non-rigorously) to compute the asymptotic expansion of the
position of the front for an arbitrary initial condition, and in
particular to obtain
\eqref{position}. The basic idea is the following:
for $\epsilon<0$, the whole right hand side of \eqref{prop22} can be
written as $\mathcal P(\epsilon)+\mathcal O(\epsilon^3)$ for some polynomial
$\mathcal P(\epsilon)$ if $h_0$ goes to zero fast enough. (Specifically, it can be shown that
the necessary and sufficient condition is $\int\mathrm{d} x\,h_0(x) x^3 e^x<\infty$.)
However, the left
hand side produces very easily some singular terms of $\epsilon$ in a small
$\epsilon$ expansion; it
turns out that $\mu_t$ should satisfy \eqref{position} in order to avoid
all the singular terms up to order $\epsilon^3$.
In this paper, by considering both sides $\epsilon<0$ and
$\epsilon>0$, we can eliminate the unknown polynomial $\mathcal P(\epsilon)$
in \eqref{prop22} and obtain \eqref{mainresult}.
The rest of the paper is organized as follow; in \autoref{sec:Phi}, we show that the function $\Phi(c)$ is well
defined, and we give a useful representation. In \autoref{sec:magic}, we
prove the first part of \autoref{prop2}, \textit{i.e.}\@ \eqref{prop21}. We
state and prove some technical lemmas in \autoref{sec:lem}, which allow us
to finish the proof of \autoref{prop2} and to prove \autoref{mainthm} in
\autoref{sec:expansion}.
In \autoref{proofprop1t}, we prove \autoref{prop1t}.
Finally, a technical lemma is proved in \autoref{appendix}.
\section{The function \texorpdfstring{$\Phi(c)$}{Φ(c)}}\label{sec:Phi}
\begin{prop}\label{prop1}
For a given initial condition $h_0$ such that $\int \mathrm{d} x\, h_0(x)
e^x<\infty$,
let $h(x,t)$ be the solution to \eqref{FKPP}.
For $c\ge2$, the following (finite or infinite) limits exist and are equal:
\begin{equation}\label{prop11}
\Phi(c):=\lim_{t\to\infty} \sqrt{4\pi t}\,h(ct,t)e^{\big(\frac{c^2}4-1\big)t}
= \lim_{t\to\infty} e^{-t\big(1+\frac {c^2} 4\big)}\int\mathrm{d} x\, h(x,t)
e^{\frac c 2 x}\in[0,\infty].
\end{equation}
Furthermore, $\Phi(2)=0$, $\Phi(c)>0$ for $c>2$ and
\begin{equation}
\Phi(c)
<\infty
\quad\iff\quad
\int\mathrm{d} x\, h_0(x)e^{\frac c2 x}<\infty.
\label{Phih0}
\end{equation}
The function $c\mapsto \Phi(c)$ is continuous in the domain where it is
finite.
\end{prop}
\noindent\textbf{Remark:} the condition
$\int \mathrm{d} x\,h_0(x)e^{ x}<\infty$ implies, in
particular, that the front has a velocity 2.
Before doing a rigorous proof, here is a quick and dirty argument to show
that the second limit in \eqref{prop11} is equal to the first:
starting from the integral in that limit, make the change of variable
$x=vt$ (with $v$ being the new variable) and boldly replace $h(vt,t)$ under the integral sign using
the equivalent implied by the first limit to obtain
\begin{equation}\begin{aligned}
\int \mathrm{d} x\,h(x,t)e^{\frac c2x}&=t\int\mathrm{d}
v\, h(vt,t)e^{\frac c2vt}\\
&\simeq \frac t{\sqrt{4\pi t}}\int\mathrm{d}
v\,\Phi(v) e^{(1-\frac{v^2}4+\frac c2v)t}=
e^{(1+\frac{c^2}4)t}\frac{\sqrt t}{\sqrt{4\pi}}\int\mathrm{d}
v\,\Phi(v) e^{-\frac14(v-c)^2t}.
\end{aligned}
\end{equation}
(The fact that the substitution only makes sense for $v\ge2$ is not a problem
since, clearly, the part of the integral for $v<2$ does not contribute
significantly.)
The remaining integral is dominated by $v$ close to $c$ in the large time
limit. Replacing $\Phi(v)$ by $\Phi(c)$ and computing the remaining
Gaussian integral gives the second limit.
We will need in the proof a bound on how $h(x,t)$ decreases for large $x$:
For $r>0$, introduce
\begin{equation}
g(r,t):=\int \mathrm{d} x\, h(x,t)e^{rx}.
\label{defg}
\end{equation}
\begin{lem}
\label{lem1}
For all $x$, all $t>0$, and all $r>0$ such that $g(r,0)=\int \mathrm{d} x\,
h_0(x)e^{rx}<\infty$,
\begin{equation}
h(x,t)\le \frac{e^{(1+r^2)t}}{\sqrt{4\pi t}}g(r,0)e^{-rx},\qquad
g(r,t) \le e^{(1+r^2)t} g(r,0).
\label{lem11}
\end{equation}
\end{lem}
\begin{proof}
Using the comparison principle, one obtains that $h(x,t)\le h_\text{lin}(x,t)$, where
$h_\text{lin}(x,t)$ is the solution to $\partial_t
h_\text{lin}=\partial_x^2 h_\text{lin}+h_\text{lin}$ with initial condition
$h_0$. Solving for $h_\text{lin}$, we get
\begin{equation}
h(x,t)\le \frac{e^t}{\sqrt{4\pi t}} \int \mathrm{d} y \, h_0(y)
e^{-\frac{(x-y)^2}{4t}},
\label{29}
\end{equation}
and then,
\begin{equation}
h(x,t)e^{rx}\le \frac{e^t}{\sqrt{4\pi t}} \int \mathrm{d} y \, h_0(y) e^{ry}\times e^{r(x-y)-\frac{(x-y)^2}{4t}}
=\frac{e^{(1+r^2)t}}{\sqrt{4\pi t}} \int \mathrm{d} y \, h_0(y) e^{ry}\times
e^{-\frac{(x-y-2rt)^2}{4t}}.
\end{equation}
Both inequalities in \eqref{lem11} are obtained from that last relation,
respectively by writing that the Gaussian term is smaller than 1, or by
integrating over $x$.
\end{proof}
\begin{proof}[Proof of \autoref{prop1}]
We write the non-linearity in \eqref{FKPP} as $F(h)=h\times
G(h)$. From \eqref{BramsonsConditions}, the function $G$, defined on $[0,1]$, is continuous, satisfies $0\le G(h)\le 1$, $G(h)=\mathcal
O(h^p)$ for some $p>0$ as $h\to0$ and $G(1)=1$. To avoid parentheses, we will write
$G(h(x,t))$ as $G\circ h(x,t)$ using the composition operator $\circ $.
We write the solution $h(x,t)$ of \eqref{FKPP} using the Feynman-Kac
representation (see \cite[Theorem 5.3 p. 148]{Friedman.1975} or, for
a short proof, \cite[Proposition 3.1]{BerestyckiBrunetPenington.2019}):
\begin{equation}\label{FK1}
h(x,t)=e^t \mathbb E_x\Big[h_0(B_t) e^{-\int_0^t\mathrm{d}
s\,G\circ h(B_s,t-s)}\Big],
\end{equation}
where under $\mathbb E_x$, $B$ is a Brownian with diffusivity $\sqrt2$
started from $x$.
(so that $\mathbb E_x(B_t^2)=x^2+2t$.)
In \eqref{FK1}, we condition the Brownian to end at $B_t=y$ and we
integrate over $y$:
\begin{equation}
h(x,t)=e^t\int\mathrm{d} y \frac{e^{-\frac{(x-y)^2}{4t}}}
{\sqrt{4\pi t}}h_0(y) \mathbb E_{t:x\to y}\Big[e^{-\int_0^t\mathrm{d} s\,G\circ h(B_s,t-s)}\Big]
\end{equation}
where, under $\mathbb E_{t:x\to y}$, $B$ is a Brownian bridge going from $x$ to $y$
in a time $t$, with a diffusivity $\sqrt2$. We reverse time and remove the
linear part from the bridge:
\begin{align}
h(x,t)&=e^t\int\mathrm{d} y \frac{e^{-\frac{(x-y)^2}{4t}}}
{\sqrt{4\pi t}}h_0(y) \mathbb E_{t:y\to x}\Big[e^{-\int_0^t\mathrm{d} s\,G\circ h(B_s,s)}\Big]
\label{FK2}
\\
&=\int\mathrm{d} y \frac{e^{t-\frac{(x-y)^2}{4t}}}{\sqrt{4\pi t}}h_0(y)
\mathbb E_{t:0\to 0}\Big[e^{-\int_0^t\mathrm{d} s\,G\circ h\big(B_s+(x-y)\tfrac
st+y,s\big)}\Big].
\end{align}
Then, at $x=ct$,
\begin{equation}
h(ct,t)=\frac{e^{(1-\frac{c^2}4)t}}{\sqrt{4\pi t}}
\int\mathrm{d} y\,e^{\frac c 2y-\frac{y^2}{4t}}
h_0(y)\mathbb E_{t:0\to 0}\Big[e^{ -\int_0^t\mathrm{d} s\,G\circ h\big(B_s+c s-y\tfrac
st+y,s\big)}\Big]
\end{equation}
We move the prefactors to the left hand side and write
the Brownian bridge as
a time-changed Brownian path
\begin{equation}
\sqrt{4\pi t}\, e^{(\frac{c^2}4-1)t}
h(ct,t)=
\int\mathrm{d} y\,e^{\frac c 2y-\frac{y^2}{4t}}
h_0(y)\mathbb E_{0}\Big[e^{ -\int_0^t\mathrm{d} s\,G\circ
h\big(\frac{t-s}tB_{\frac{ts}{t-s}}+c s-y\tfrac
st+y,s\big)}\Big].
\label{hctt}
\end{equation}
For any $y$, any $c\ge2$, and almost all Brownian path $B$, one has
\begin{equation}
\int_0^t\mathrm{d} s\,G\circ
h\Big(\frac{t-s}tB_{\frac{ts}{t-s}}+c s-y\tfrac
st+y,s\Big) \to \int_0^\infty\mathrm{d} s\,G\circ h\big(B_{s}+c s+y,s\big)
\qquad\text{as $t\to\infty$}.
\label{domconv}
\end{equation}
Indeed, first consider the case $c>2$, pick $\tilde c \in (2,c)$ and $t_0$
such that $c-y/t_0>\tilde c$.
Recall that, for almost all path $B$, there exists a constant $A$
(depending on $B$) such that $|B_u|\le
A(1+u^{0.51})$ for all $u$; this implies that
$\frac{t-s}t\big|B_{\frac{ts}{t-s}}\big|\le
A(1+s^{0.51})$ for all $t$ and all $s<t$. Then
\begin{equation}
\frac{t-s}tB_{\frac{ts}{t-s}}+c s-y\tfrac
st+y \ge \tilde c s + C+y\quad\text{for all $t>t_0$ and all $s\in(0,t)$,}
\end{equation}
where $C$ is some constant depending on
$B$. (Indeed, the function $s\mapsto cs-y s/t-\tilde
c s -As^{0.51}$ is uniformly bounded from below for $t>t_0$.)
Using
\eqref{lem11} for $r=1$, we obtain that
\begin{equation}
h(\frac{t-s}tB_{\frac{ts}{t-s}}+c s-y\tfrac
st+y,s)\le \frac{C}{\sqrt s} e^{-(\tilde c-2)s-y}\quad\text{for all $t>t_0$ and
all $s\in(0,t)$},
\end{equation}
with $C$ another
constant depending on $B$.
As $G(h)=\mathcal O(h^p)$ for some $p>0$ as $h\to0$ and $G(1)=1$, there exists
a constant $C$ such that $G(h)\le C h^{\min(1,p)}$. Then, we see by
dominated convergence that \eqref{domconv} holds for $c>2$, and furthermore
we see that the right hand side is smaller than $C e^{-\min(1,p)y}$ for
some constant $C$ depending on $B$.
For $c=2$, the right hand side of \eqref{domconv} is $+\infty$. Indeed,
$B_s+cs+y$ is infinitely often smaller than $2s-\sqrt s$, where the front
$h$ is close to 1. Then, noticing that \eqref{domconv} with the upper limits
of both integrals replaced by some $T>0$ clearly holds by dominated
convergence, and that, by choosing $T$ large enough, the right hand side is
arbitrarily large, we see that the left hand side of \eqref{domconv} must
diverge as $t\to\infty$.
From \eqref{domconv}, we immediately obtain by dominated convergence
\begin{equation}
\mathbb E_{0}\Big[e^{ -\int_0^t\mathrm{d} s\,G\circ
h\big(\frac{t-s}tB_{\frac{ts}{t-s}}+c s-y\tfrac
st+y,s\big) }\Big] \to
\mathbb E_{0}\Big[e^{ - \int_0^\infty\mathrm{d} s\,G\circ h(B_{s}+c s+y,s)
}\Big] \qquad\text{as $t\to\infty$},
\end{equation}
where the right hand side is 0 if $c=2$ and positive if $c>2$.
(As we have shown, the integral in the exponential is almost surely
infinite if $c=2$, and almost surely finite if $c>2$.) Furthermore, for
$c>2$, the right hand side converges to 1 as $y\to\infty$. (Recall that,
for $c>2$, the integral in the exponential is smaller than
$Ce^{-\min(1,p)y}$.)
If $\int h_0(y)e^{cy/2}\mathrm{d} y<\infty$, then a last application of
dominated convergence in \eqref{hctt} shows that the first limit defining
$\Phi(c)$ in \eqref{prop11} does exist and is
given by:
\begin{equation}
\Phi(c):=\lim_{t\to\infty} {\sqrt{4\pi t}}\, e^{(\frac{c^2}4-1)t}h(ct,t) =\int\mathrm{d} y\,e^{\frac c 2y}
h_0(y)\mathbb E_0\Big[e^{ -\int_0^\infty\mathrm{d} s\,G\circ h(B_s+c s
+y,s)}\Big]<\infty,
\label{1stexpressionphi}
\end{equation}
and furthermore $\Phi(2)=0$ and $\Phi(c)>0$ for $c>2$. Note that \cite{BerestyckiBrunetCortinesMallein.2022} gives a similar
expression.
We now assume that $\int h_0(y)e^{cy/2}\mathrm{d} y=\infty$ and show that the
limit of \eqref{hctt}
diverges. Notice that we must be in the $c>2$ case since we also assumed
that $\int h_0(y)e^y\,\mathrm{d} y<\infty$.
Cutting the integral in \eqref{hctt} at some arbitrary value $A$ and then
sending $t\to\infty$ gives
\begin{equation}
\liminf_{t\to\infty}
{\sqrt{4\pi t}}\, e^{(\frac{c^2}4-1)t}h(ct, t)
=\int_{-\infty}^A\mathrm{d} y\,e^{\frac c 2y}
h_0(y)\mathbb E_0\Big[e^{ -\int_0^\infty\mathrm{d} s\,G\circ h(B_s+c s
+y,s)}\Big].
\label{33}
\end{equation}
As the expectation appearing in the integral goes to
1 as $y\to\infty$, the hypothesis $\int h_0(y)e^{cy/2}\mathrm{d} y=\infty$
implies that the right hand side diverges as $A$ to $\infty$, and then that
$\Phi(c)$ exists and is infinite.
Using the same methods, one can show from \eqref{1stexpressionphi} that $\Phi(c)$ is a continuous
function (in the range of $c$ where $\Phi$ is finite)
by first showing that $\int_0^\infty\mathrm{d}
s\,G\circ h(B_s+c_n s +y,s)\to\int_0^\infty\mathrm{d} s\,G\circ
h(B_s+c s +y,s)$ if $c_n\to c$, by dominated convergence, using the
same bounds as above (specifically that the integrands are uniformly bounded by an
exponentially decreasing function of $s$ if $c>2$ and that the result is
infinity if $c=2$.)
To show that the second expression in \eqref{prop11} is equal to the first, start again from \eqref{FK2}:
$$\begin{aligned}
h(x,t)=e^t\int\mathrm{d} y \frac{e^{-\frac{(x-y)^2}{4t}}}
{\sqrt{4\pi t}}h_0(y) \mathbb E_{t:y\to x}\Big[e^{-\int_0^t\mathrm{d} s\,G\circ h(B_s,s)}\Big]
=e^t\int\mathrm{d} y \,
h_0(y) \mathbb E_{y}\Big[e^{-\int_0^t\mathrm{d} s\,G\circ h(B_s,s)}\delta(B_t-x)\Big].
\end{aligned}$$
Then
\begin{equation}
\begin{aligned}
e^{-t\big(1+\frac {c^2} 4\big)}\int\mathrm{d} x\, h(x,t) e^{\frac c 2 x}
&=e^{-\frac{c^2}4t}\int\mathrm{d} y \,
h_0(y) \mathbb E_{y}\Big[e^{-\int_0^t\mathrm{d} s\,G\circ h(B_s,s)}e^{\frac c 2 B_t}\Big]
\\&=\int\mathrm{d} y\,e^{\frac
c2 y}h_0(y)\mathbb E_y\Big[\text
e^{-\int_0^t\mathrm{d} s\,G\circ h(B_s+cs,s)}\Big].
\end{aligned}\end{equation}
where the last transform is through Girsanov's Theorem (or a change of
probability of the Brownian).
Taking the limit $t\to\infty$ is immediate and gives back the
expression of $\Phi(c)$ written in \eqref{1stexpressionphi}.
\end{proof}
\section{Magical relation}\label{sec:magic}
In this section, we prove the first part of \autoref{prop2}. Many of the
arguments presented here were already written in
\cite{BerestyckiBrunetDerrida.2018}
for the case $\epsilon<0$.
Recall the definitions \eqref{defgamma} of $\gamma$ and \eqref{defg} of
$g(r,t)$:
\begin{equation}
\gamma:=\sup\Big\{r; \int \mathrm{d} x\, h_0(x) e^{rx}<\infty\Big\},
\qquad
g(r,t):=\int \mathrm{d} x\, h(x,t) e^{rx}.
\end{equation}
According to \autoref{lem1},
\begin{equation}
g(r,t)\le e^{(1+r^2)t}g(r,0)<\infty\qquad\text{for $r\in(0,\gamma)$ and $t\ge0$}.
\end{equation}
We wish to write an expression for $\partial_t g(r,t)$, and the first step
is to justify that we can differentiate under the integral sign:
\begin{equation}
\partial_t g(r,t) =\int\mathrm{d} x\,\partial_t h(x,t) e^{rx}=\int\mathrm{d} x\,\big[\partial_x^2 h(x,t) + h(x,t)-F[h(x,t)]\big] e^{rx}
\qquad\text{for $0<r<\gamma$},
\label{36}
\end{equation}
and then (still assuming $0<r<\gamma$) that we can integrate twice by parts
the $\partial_x^2h$ term:
\begin{equation}
\partial_t g(r,t) =\int\mathrm{d} x\, \big[(r^2 h(x,t) + h(x,t)-F[h(x,t)]\big] e^{rx}
=(1+r^2)g(r,t) -\int\mathrm{d} x\,F[h(x,t)]e^{rx}.
\label{37}
\end{equation}
Both steps \eqref{36} and \eqref{37} are justified by using bounding
functions provided by the following
lemma with $\beta$ chosen in $(r,\gamma)$:
\begin{lem}\label{lem2}
Let $\beta\in(0,\gamma)$. For $t>0$, the quantities $h(x,t)$, $|\partial_x
h(x,t)|$, $|\partial_x^2h(x,t)|$ and $|\partial_t h(x,t)|$ are bounded by
$A(t)\max(1,e^{-\beta x})$ for some locally bounded function $A$.
\end{lem}
The proof of \autoref{lem2} is given in \autoref{appendix}.
Recall the definition \eqref{defphi} of $\varphi$:
\begin{equation}
\varphi(\epsilon,t) :=\int\mathrm{d} z\, F[h(\mu_t+z,t)]e^{(1+\epsilon)z};
\end{equation}
we have
\begin{equation}
\int\mathrm{d} x\,F[h(x,t)]e^{rx}=e^{r\mu_t}\int\mathrm{d}
z\,F[h(\mu_t+z,t)]e^{rz}=e^{r\mu_t}\varphi(r-1,t),
\end{equation}
and so, in \eqref{37},
\begin{equation}
\partial_t g(r,t) =(1+r^2)g(r,t) - e^{r\mu_t}\varphi(r-1,t).
\end{equation}
Integrating, we obtain
\begin{equation}
g(r,t)e^{-(1+r^2)t}
=g(r,0) - \int_0^t \mathrm{d} s\, \varphi(r-1,s) e^{r\mu_s-(1+r^2)s}.
\label{54}
\end{equation}
We now send $t\to\infty$ in \eqref{54}, distinguishing two cases
\begin{itemize}
\item If $\gamma>1$ and $1\le r<\gamma$;\quad notice that the left hand
side is the expression appearing in the second limit in
\eqref{prop11} with $c=2r$. This implies that
\begin{equation}
\Phi(2r) = g(r,0) -\int_0^\infty \mathrm{d} s\, \varphi(r-1,s) e^{r\mu_s-(1+r^2)s}
\qquad\text{if $1\le r<\gamma$}.
\label{r>1}
\end{equation}
\item If $0<r<\min(1,\gamma)$;\quad we claim that the left hand
side of \eqref{54} goes to 0 as $t\to\infty$, and so:
\begin{equation}
0 = g(r,0) -\int_0^\infty \mathrm{d} s\, \varphi(r-1,s) e^{r\mu_s-(1+r^2)s}
\qquad\text{if $0<r<\min(1,\gamma)$}.
\label{r<1}
\end{equation}
Indeed,
take $\beta\in\big(r,\min(1,\gamma)\big)$.
Applying \eqref{lem11} with $\beta$ instead of $r$, we have
\begin{equation}
h(x,t)\le\min\left[1, \frac{e^{(1+\beta^2)t}}{\sqrt{4\pi
t}}g(\beta,0)e^{-\beta x}\right].
\label{hmin}
\end{equation}
For $t$ given, let $X$ be the point where both expressions inside the min are equal:
\begin{equation}
e^{\beta X}=\frac{e^{(1+\beta^2)t}}{\sqrt{4\pi
t}}g(\beta,0).
\end{equation}
We obtain from \eqref{hmin}
\begin{equation}
g(r,t)=\int\mathrm{d} x\,h(x,t)e^{rx}\le
\frac{e^{rX}}r+ \frac{e^{(1+\beta^2)t}}{\sqrt{4\pi
t}}g(\beta,0)\frac{e^{-(\beta-r) X}}{\beta-r}
=\left(\frac1r+\frac1{\beta-r}\right)e^{rX}
=C\frac{e^{r(\beta^{-1}+\beta)t}}{t^{\frac r{2\beta}}}
,
\end{equation}
where $C$ is some quantity depending on $r$ and $\beta$, but independent of
time. As $\beta>r$ and as
the function $\beta\to \beta^{-1}+\beta$ is decreasing for $\beta<1$, we
obtain
$r(\beta^{-1}+\beta)<r(r^{-1}+r)=1+r^2$. We conclude that, indeed, the left hand side of
\eqref{54} goes to zero as $r\to\infty$.
\end{itemize}
Combining \eqref{r>1} and \eqref{r<1}, we have shown that
\begin{equation}
\int_0^\infty \mathrm{d} s\, \varphi(r-1,s) e^{r\mu_s-(1+r^2)s}
= g(r,0) -\indic{r>1}\Phi(2r)\qquad\text{for $r\in(0,\gamma)$}.
\label{bothr}
\end{equation}
(Recall that $\Phi(2)=0$, hence the right hand side is continuous at
$r=1$.)
Writing now that $r\mu_s-(1+r^2)s= r(\mu_s-2s)-(1-r)^2s$, and taking
$r=1+\epsilon$ and $s=t$ in \eqref{bothr}, we obtain
\begin{equation}
\int_0^\infty\mathrm{d} t \,\varphi(\epsilon,t)e^{-\epsilon^2
t +(1+\epsilon)(\mu_t-2t)}=
g(1+\epsilon,0)
-\indic{\epsilon>0}\Phi(2+2\epsilon)\quad\text{for
$\epsilon\in(-1,\gamma-1)$},
\end{equation}
which is the first part \eqref{prop21} of \autoref{prop2}.
To prove the second part of the proposition, we need to show that, for some
polynomial $\mathcal P(\epsilon)$,
\begin{equation}
\int_0^\infty\mathrm{d} t \,\big[\varphi(\epsilon,t)-\hat\varphi(\epsilon)\big]e^{-\epsilon^2
t +(1+\epsilon)(\mu_t-2t)}=\mathcal P(\epsilon)+\mathcal O(\epsilon^3),
\label{42}
\end{equation}
if $ \big|\varphi(\epsilon,t)-\hat\varphi(\epsilon)\big|\le\frac C t $ for
$t$ large enough and
$\epsilon$ in some real neighbourhood of 0 and if $\gamma>1$,
which implies that
$\mu_t = 2t-\frac32\log t +a +o(1)$ (by Bramson's result).
To do so, we need several technical lemmas. We
thus take a pause in the proof of \autoref{prop2} to state and prove these
lemmas, and we resume in \autoref{sec:expansion}.
\section{Some technical lemmas}\label{sec:lem}
We begin by recalling a classical result on the analyticity of functions
defined by an integral:
\begin{lem}\label{lem:a}
Let $f(\epsilon,t)$ be a family of functions such that
\begin{itemize}
\item $\epsilon\mapsto f(\epsilon,t)$ is analytic on some simply
connected open domain $U$ of $\mathbb{C}$ (independent of $t$) for almost all $t\in \mathbb{R}$,
\item $|f(\epsilon,t)|\le g(t)$ for all $\epsilon\in U$, where $g$ is
some integrable function: $\int g(t)\,\mathrm{d} t <\infty$.
\end{itemize}
Then, $\epsilon\mapsto F(\epsilon):=\int\mathrm{d} t\, f(\epsilon,t)$ is
analytic on $U$.
\end{lem}
\begin{proof}
On any closed path $\gamma$ in $U$,
one has with Fubini
\begin{equation}
\oint_\gamma F(\epsilon) \,\mathrm{d}\epsilon=\int\mathrm{d} t \oint_\gamma
\mathrm{d}\epsilon\,f(\epsilon,t).
\end{equation}
This last integral is 0 since $\epsilon\mapsto f(\epsilon,t)$ is analytic
and $U$ is simply connected. Then, by Morera's theorem, $F$ is analytic.
\end{proof}
The next lemma states that some functions of $\epsilon$ which are variations
on the incomplete gamma functions have small $\epsilon$ expansions with
only one or two singular terms
\begin{lem}\label{lem:1s}
Let $\alpha,\beta$ be real numbers such that either
$\alpha\not\in\{1,2,3,\ldots\}$, or $\beta\ne0$. There exist
functions $\epsilon\mapsto {\mathcal A}_{\alpha,\beta}(\epsilon)$ and
$\epsilon\mapsto \tilde {\mathcal A}_{\alpha,\beta}(\epsilon)$ which are analytic around
$\epsilon=0$, such that for $\epsilon$ real, non-zero and $|\epsilon|$
small enough,
\begin{align}\label{lemeq1}
\int_1^\infty \mathrm{d} t\, e^{-\epsilon^2 t}
\frac1{t^{\alpha+\beta\epsilon}}&=
\lvert\epsilon\rvert^{2\alpha-2+2\beta\epsilon}\Gamma(1-\alpha-\beta\epsilon)
+\frac{\indic{\alpha=1}}{\beta\epsilon}
+{\mathcal A}_{\alpha,\beta}(\epsilon),
\\
\label{lemeq2}
\int_1^\infty \mathrm{d} t\, e^{-\epsilon^2 t}
\frac{\log t}{t^{\alpha+\beta\epsilon}}&=
\lvert\epsilon\rvert^{2\alpha-2+2\beta\epsilon}\Big[-2\log\lvert\epsilon\rvert
\Gamma(1-\alpha-\beta\epsilon)+\Gamma'(1-\alpha-\beta\epsilon)\Big]
+\frac{\indic{\alpha=1}}{(\beta\epsilon)^2}+\tilde {\mathcal A}_{\alpha,\beta}(\epsilon).
\end{align}
\end{lem}
\noindent\textbf{Remarks:} the condition $\alpha\not\in\{1,2,3,\ldots\}$ or
$\beta\ne0$ ensures that the gamma functions appearing in the result are
well defined for $\epsilon$ small enough. For $\alpha=n\in\{1,2,3,\ldots\}$ and $\beta=0$
one would have
$$\int_1^\infty \mathrm{d} t\, e^{-\epsilon^2 t} \frac1{t^{n}}
=\frac{2(-1)^n}{(n-1)!} \epsilon^{2n-2}\log|\epsilon|+\mathcal A_{n,0}(\epsilon),$$
but we don't need this result in the present paper, and we skip the proof.
For convenience, we give the
results we actually use, writing simply ${\mathcal A}(\epsilon)$ for
the analytic functions:
\begin{align}
\int_1^\infty \mathrm{d} t\, e^{-\epsilon^2 t} \frac1{t^{\frac32+\frac32\epsilon}}
&=|\epsilon|^{1+3\epsilon}\Gamma(-\tfrac12-\tfrac32\epsilon)
+{\mathcal A}(\epsilon),
\qquad\int_1^\infty \mathrm{d} t\, e^{-\epsilon^2 t} \frac1{t^{2+\frac32\epsilon}}
=|\epsilon|^{2+3\epsilon}\Gamma(-1-\tfrac32\epsilon) +{\mathcal A}(\epsilon),
\notag\\
\int_1^\infty \mathrm{d} t\, e^{-\epsilon^2 t} \frac{\log t}{t^{\frac52+\frac32\epsilon}}
&=|\epsilon|^{3+3\epsilon}\Big[-2\log|\epsilon|\Gamma(-\tfrac32-\tfrac32\epsilon)+\Gamma'(-\tfrac32-\tfrac32\epsilon)\Big]
+{\mathcal A}(\epsilon).
\label{lem1seq}
\end{align}
\begin{proof}[Proof of \autoref{lem:1s}]
Fix $\alpha$ and $\beta$ such that either $\alpha\not\in\{1,2,3,\ldots\}$
or $\beta\ne0$. We restrict $\epsilon$ to be real, non-zero and
$|\epsilon|$ to be small enough so that
$\alpha+\beta\epsilon\not\in\{1,2,3,\ldots\}$.
This ensures that the $\Gamma$ function and its derivative in
\eqref{lemeq1} and \eqref{lemeq2} are defined,
and we define $A_{\alpha,\beta}(\epsilon)$ and $\tilde
A_{\alpha,\beta}(\epsilon)$ by (respectively) \eqref{lemeq1} and
\eqref{lemeq2}. We now show
that the functions thus defined can be extended into
analytic functions around $\epsilon=0$ for $\alpha<1$.
We first consider $\alpha<1$. Note that by our restriction on the range of
allowed $\epsilon$, one also has $\alpha+\beta\epsilon<1$, and one can
write
\begin{equation}
\int_1^\infty \mathrm{d} t\,
\frac{ e^{-\epsilon^2 t}}{t^{\alpha+\beta\epsilon}}=
\int_0^\infty \mathrm{d} t\,
\frac{ e^{-\epsilon^2 t}}{t^{\alpha+\beta\epsilon}}-
\int_0^1\mathrm{d} t\,
\frac{ e^{-\epsilon^2 t}}{t^{\alpha+\beta\epsilon}}
=\lvert\epsilon\rvert^{2\alpha-2+2\beta\epsilon}\Gamma(1-\alpha-\beta\epsilon)-
\int_0^1\mathrm{d} t\,
\frac{ e^{-\epsilon^2 t}}{t^{\alpha+\beta\epsilon}}
.
\end{equation}
By identification with \eqref{lemeq1}, one obtains
\begin{equation}
A_{\alpha,\beta}(\epsilon)=-\int_0^1\mathrm{d} t\, e^{-\epsilon^2 t}
\frac1{t^{\alpha+\beta\epsilon}}\qquad\text{for $\alpha<1$}.
\end{equation}
Similarly,
\begin{equation}
\tilde A_{\alpha,\beta}(\epsilon)=-\int_0^1\mathrm{d} t\, e^{-\epsilon^2 t}
\frac{\log t}{t^{\alpha+\beta\epsilon}}\qquad\text{for $\alpha<1$}.
\end{equation}
Let $\tilde\alpha\in(\alpha,1)$, and let $U$ a simply connected
neighbourhood of $0$ in $\mathbb{C}$ such that
$\alpha+\beta\Re(\epsilon)<\tilde{\alpha}$ and $|e^{-\epsilon^2t}|<2$ for
all $\epsilon\in U$ and $t\in[0,1]$. One
can apply Lemma~\ref{lem:a} with the bounding function
$g(t)=2(1+|\log t|)/t^{\tilde\alpha}\indic{t\in(0,1)}$ to show that
$A_{\alpha,\beta}(\epsilon)$ and $\tilde
A_{\alpha,\beta}(\epsilon)$ are analytic around 0.
To extend the result to $\alpha\ge1$, we integrate by parts the left hand
side of \eqref{lemeq1}
\begin{equation}
\int_1^\infty \mathrm{d} t\, e^{-\epsilon^2 t}
\frac1{t^{\alpha+\beta\epsilon}}=
\frac1{1-\alpha-\beta\epsilon}\Big[-e^{-\epsilon^2}+\epsilon^2\int_1^\infty\mathrm{d}
t\,e^{-\epsilon^2 t}\frac1{t^{\alpha+\beta\epsilon-1}}\Big].
\end{equation}
Then, rewriting the integrals in terms of the functions ${\mathcal A}_{\alpha,\beta}$
as in \eqref{lemeq1},
\begin{multline}
\lvert\epsilon\rvert^{2\alpha-2+2\beta\epsilon}\Gamma(1-\alpha-\beta\epsilon)
+\frac{\indic{\alpha=1}}{\beta\epsilon}
+{\mathcal A}_{\alpha,\beta}(\epsilon)
\\
=\frac1{1-\alpha-\beta\epsilon}\Big[-e^{-\epsilon^2}+
\lvert\epsilon\rvert^{2\alpha-2+2\beta\epsilon}\Gamma(2-\alpha-\beta\epsilon)
+\frac{\indic{\alpha=2}}{\beta}\epsilon
+\epsilon^2{\mathcal A}_{\alpha-1,\beta}(\epsilon)\Big].
\end{multline}
With the property $x\Gamma(x)=\Gamma(x+1)$, the terms with the $\Gamma$
functions cancel and one is left with
\begin{equation}
{\mathcal A}_{\alpha,\beta}(\epsilon)
=
-\frac{\indic{\alpha=1}}{\beta\epsilon}
+
\frac1{1-\alpha-\beta\epsilon}\Big[-e^{-\epsilon^2}+
\frac{\indic{\alpha=2}}{\beta}\epsilon
+\epsilon^2{\mathcal A}_{\alpha-1,\beta}(\epsilon)\Big].
\end{equation}
For convenience let us also write the special case $\alpha=1$:
\begin{equation}
{\mathcal A}_{1,\beta}(\epsilon)
=\frac{e^{-\epsilon^2}-1}{\beta\epsilon}
-\frac\epsilon\beta {\mathcal A}_{0,\beta}(\epsilon).
\label{lema1}
\end{equation}
It is then clear from these equations that, except for
$(\alpha=1,\beta=0)$ or $(\alpha=2,\beta=0)$,
one has
\begin{equation}\label{emimplies}
\{ \epsilon\mapsto {\mathcal A}_{\alpha-1,\beta}(\epsilon)\text{ analytic around $0$}\} \implies
\{ \epsilon\mapsto {\mathcal A}_{\alpha,\beta}(\epsilon)\text{ analytic around $0$}\}.
\end{equation}
As ${\mathcal A}_{\alpha,\beta}$ is analytic around 0 for $\alpha<1$,
this implies by induction that ${\mathcal A}_{\alpha,\beta}$ is analytic around
0 for all $\alpha$ if $\beta\ne0$, and for all
$\alpha\not\in\{1,2,3,\ldots\}$ if $\beta=0$.
We proceed in the same way for $\tilde {\mathcal A}_{\alpha,\beta}$. Integrating by
parts the integral in \eqref{lemeq2},
\begin{equation}
\int_1^\infty \mathrm{d} t\, e^{-\epsilon^2 t}
\frac{\log t}{t^{\alpha+\beta\epsilon}}=
\frac1{1-\alpha-\beta\epsilon}\Big[\epsilon^2\int_1^\infty\mathrm{d}
t\,e^{-\epsilon^2 t}\frac{\log t}{t^{\alpha+\beta\epsilon-1}}-
\int_1^\infty \mathrm{d} t\, e^{-\epsilon^2 t}
\frac1{t^{\alpha+\beta\epsilon}}\Big].
\end{equation}
We replace all the integrals using \eqref{lemeq1} and \eqref{lemeq2} and
notice, using $x\Gamma(x)=\Gamma(x+1)$ and
$\Gamma(x)+x\Gamma'(x)=\Gamma'(x+1)$, that all the terms involving $\Gamma$
functions cancel, \textit{i.e.}\@:
\begin{multline}
\Big[-2\log\lvert\epsilon\rvert
\Gamma(1-\alpha-\beta\epsilon)+\Gamma'(1-\alpha-\beta\epsilon)\Big]
\\=
\frac1{1-\alpha-\beta\epsilon}\Big[-2\log\lvert\epsilon\rvert
\Gamma(2-\alpha-\beta\epsilon)+\Gamma'(2-\alpha-\beta\epsilon)
-\Gamma(1-\alpha-\beta\epsilon)\Big].
\end{multline}
Then, the remaining terms are
\begin{equation}
\frac{\indic{\alpha=1}}{(\beta\epsilon)^2}+\tilde{\mathcal A}_{\alpha,\beta}(\epsilon)=\frac1{1-\alpha-\beta\epsilon}\Big[\frac{\indic{\alpha=2}}{\beta^2}+\epsilon^2\tilde
{\mathcal A}_{\alpha-1,\beta}(\epsilon)-\frac{\indic{\alpha=1}}{\beta\epsilon}-{\mathcal A}_{\alpha,\beta}(\epsilon)\Big]
\end{equation}
In particular, for $\alpha=1$,
\begin{equation}
\tilde
{\mathcal A}_{1,\beta}(\epsilon)=-\frac\epsilon{\beta}\tilde
{\mathcal A}_{0,\beta}(\epsilon)+\frac{{\mathcal A}_{1,\beta}(\epsilon)}{\beta\epsilon}.
\end{equation}
Notice from \eqref{lema1} that ${\mathcal A}_{1,\beta}(0)=0$. Hence we have again,
except if $\alpha\in\{1,2,3,\ldots\}$ and $\beta=0$
\begin{equation}
\{ \epsilon\mapsto \tilde {\mathcal A}_{\alpha-1,\beta}(\epsilon)\text{ analytic around $0$}\} \implies
\{ \epsilon\mapsto \tilde {\mathcal A}_{\alpha,\beta}(\epsilon)\text{ analytic
around $0$}\},
\end{equation}
and the proof is finished in the same way as for ${\mathcal A}_{\alpha,\beta}$.
\end{proof}
\autoref{lem:1s} gives asymptotic expansions of $e^{-\epsilon^2 t}$ times
exact power laws of $t$. The next lemma deals with the case of approximate
power laws.
\begin{lem}\label{lem:O}
Let $f(\epsilon,t)$ be a family of functions such that, for
a certain neighbourhood $U$ of 0 in $\mathbb{C}$,
\begin{itemize}
\item $\epsilon\mapsto f(\epsilon,t)$ is analytic in $U$ for all $t>0$,
\item $\displaystyle|f(\epsilon,t)| \le
C\min\left(1,\frac1{t^{\alpha+\beta\epsilon}}\right)$ for all $t>0$ and
$\epsilon\in U\cap\mathbb{R}$, where $C>0$, $\alpha$ and $\beta$ are some real constants.
\end{itemize}
Then there exists a polynomial $\mathcal P$ such that, for $\epsilon$ real,
non-zero, and $|\epsilon|$ small enough,
\begin{equation}
\int_0^\infty\mathrm{d} t\, e^{-\epsilon^2t}f(\epsilon,t)
= \mathcal P(\epsilon)+\begin{cases}
\mathcal O\big(|\epsilon|^{2\alpha-2}\big)&\text{if
$\alpha\not\in\{1,2,3,\ldots\}$},\\[1ex]
\mathcal O\big(|\epsilon|^{2\alpha-2}\log|\epsilon|\big)&\text{if
$\alpha\in\{1,2,3,\ldots\}$},
\end{cases}
\qquad\text{as $\epsilon\to0$}.
\end{equation}
\end{lem}
\begin{proof}
For the case $\alpha\le 1$, the polynomial $\mathcal P(\epsilon)$ plays no role as
it is asymptotically smaller than the $\mathcal O$ term. Thus, we simply need to
bound the integral for $\epsilon\in U\cap\mathbb{R}$:
\begin{equation}
\bigg|\int_0^\infty\mathrm{d} t\, e^{-\epsilon^2t}f(\epsilon,t)
\bigg|\le C +C\int_1^\infty\mathrm{d} t\,
e^{-\epsilon^2t}\frac1{t^{\alpha+\beta\epsilon}}.
\end{equation}
The remaining integral is given by \eqref{lemeq1} except for the case
$\alpha=1$ and $\beta=0$:
\begin{equation}
\int_1^\infty\mathrm{d} t\,
e^{-\epsilon^2t}\frac1{t^{\alpha+\beta\epsilon}}=
\begin{cases}
\lvert\epsilon\rvert^{2\alpha-2+2\beta\epsilon}\Gamma(1-\alpha-\beta\epsilon)
+{\mathcal A}_{\alpha,\beta}(\epsilon)
=\mathcal O(|\epsilon|^{2\alpha-2})&\text{if $\alpha<1$},\\[1ex]
\lvert\epsilon\rvert^{2\beta\epsilon}\Gamma(-\beta\epsilon)
+\frac{1}{\beta\epsilon}
+{\mathcal A}_{\alpha,\beta}(\epsilon)
=\mathcal O(\log|\epsilon|)&\text{if $\alpha=1$},
\end{cases}
\end{equation}
where we used $|\epsilon|^{2\beta\epsilon}=1+\mathcal O(\epsilon\log|\epsilon|)$ and
$\Gamma(-\beta\epsilon)=\frac{\Gamma(1-\beta\epsilon)}{-\beta\epsilon}=-\frac{1}{\beta\epsilon}+\mathcal O(1)$.
One checks independently that the case $\alpha=1$ and $\beta=0$ gives also
$\mathcal O(\log|\epsilon|)$.
For $\alpha>1$, we proceed by induction. Pick $\tilde\alpha\in(1,\alpha)$,
and make the neighbourhood $U$ of 0 small enough that
$\alpha+\beta\epsilon>\tilde\alpha$ for $\epsilon\in U\cap \mathbb{R}$. Then
\begin{equation}
|f(\epsilon,t)|\le
C\min\left(1,\frac1{t^{\alpha+\beta\epsilon}}\right)
\le
C\min\left(1,\frac1{t^{\tilde\alpha}}\right)\qquad\text{for all $t>0$ and
$\epsilon\in U\cap \mathbb{R}$}.
\end{equation}
Integrating by parts,
\begin{equation}
\int_0^\infty\mathrm{d} t\, e^{-\epsilon^2t}f(\epsilon,t)
=F(\epsilon,0)-\epsilon^2\int_0^\infty\mathrm{d}
t\,e^{-\epsilon^2t}F(\epsilon,t)\qquad\text{with }F(\epsilon,t)=\int_{t}^\infty\mathrm{d}
t'\,f(\epsilon,t').
\label{68}
\end{equation}
Using \autoref{lem:a}, the function $\epsilon\mapsto F(\epsilon,t)$ is
analytic for all $t\ge0$ and $\epsilon\in U$. Furthermore, for some~$\tilde C$,
\begin{equation}
|F(\epsilon,t)|\le \tilde C \min
\left(1,\frac1{t^{\alpha-1+\beta\epsilon}}\right)\qquad\text{for all $t>0$ and
$\epsilon\in U\cap \mathbb{R}$}.
\end{equation}
Then, assuming that the lemma holds for $\alpha-1$, we can apply it to
the integral with $F$ in \eqref{68}; after Taylor-expanding
the analytic function $F(\epsilon,0)$, we see that the result holds for $f$.
\end{proof}
\section{Expansions in \texorpdfstring{$\epsilon$}{ε}}\label{sec:expansion}
We now finish the proof of \autoref{prop2}. It remains to show \eqref{42}:
\begin{equation}
\int_0^\infty\mathrm{d} t \,\big[\varphi(\epsilon,t)-\hat\varphi(\epsilon)\big]e^{-\epsilon^2
t +(1+\epsilon)(\mu_t-2t)}=\mathcal P(\epsilon)+\mathcal O(\epsilon^3),
\label{42bis}
\end{equation}
for some polynomial $\mathcal P(\epsilon)$,
under the hypotheses that $\gamma>1$ and there exists $C>0$, $t_0\ge0$
and a (real) neighbourhood $U$ of 0 such that
\begin{equation}
\big|\varphi(\epsilon,t)-\hat\varphi(\epsilon)\big|\le\frac C t
\qquad\text{for $t>t_0$ and $\epsilon\in U$}.
\label{67}
\end{equation}
Recall from \eqref{BramsonsConditions} that $F(h)\le
C h^{1+p}$ for some $p>0$ and some constant $C$.
Recall the definitions \eqref{defphi} of $\varphi$ and $\hat\varphi$:
\begin{equation}
\varphi(\epsilon,t) :=\int\mathrm{d} z\, F[h(\mu_t+z,t)]e^{(1+\epsilon)z},\qquad
\hat\varphi(\epsilon) :=\int\mathrm{d} z\, F[\omega(z)] e^{(1+\epsilon)z}.
\end{equation}
For each $t>0$, these functions of $\epsilon$ are analytic in the region
$V=\{\epsilon\in \mathbb{C}\,;\,-1<\Re\epsilon <p\}$.
Indeed,
\autoref{lem1} with $r=1$
and \eqref{BramsonsConditions}
give that $0\le F[h(\mu_t+z,t)]\le C_{t}
e^{-(1+p)z}$ for some function of time $C_{t}$. Using that bound for $z>0$
and the bound $0\le F[h(\mu_t+z,t)]\le1$ for $z<0$, we get from
\autoref{lem:a} that $\varphi(\epsilon)$ is analytic in the domain
$\{\epsilon\in\mathbb{C}\,;\,-1+a<\Re\epsilon<p-a\}$ for any $a>0$, and is
therefore
analytic on the domain $V$ defined above. The same argument
works in the same way for~$\hat\varphi(\epsilon)$.
Then, as $\gamma>1$, Bramson's result implies that $\mu_t = 2t-\tfrac32\log t +a +o(1)$. With \eqref{67}, we see that there is a constant $C'>0$ such that
\begin{equation}
\big|\varphi(\epsilon,t)-\hat\varphi(\epsilon)\big|e^{
(1+\epsilon)(\mu_t-2t)}
\le \frac {C'}
{t^{\frac52+\frac32\epsilon}}
\qquad\text{for $t>t_0$ and $\epsilon\in U$}.
\label{77}
\end{equation}
Pick $\beta\in(1,\gamma)$, and make the neighbourhood $U$ smaller if needed so that
$\epsilon\in U \implies 0.5 <1+ \epsilon<\beta$.
Then, recalling the definition \eqref{defg} of $g$,
since $F(h)<h$, and since $e^{(1+\epsilon)z}\le e^{0.5z}+e^{\beta z}$ for
all $z$,
\begin{equation}
\varphi(\epsilon,t)\le g(0.5,t)+g(\beta,t)\qquad\text{for $t>0$ and $\epsilon\in U$},
\end{equation}
which remains bounded for $t\in[0,t_0]$ according to \autoref{lem1}.
Similarly, $\hat\phi(\epsilon)$ is bounded for $\epsilon\in\ U$ and we see
that
the left hand side of \eqref{77} is uniformly bounded by some constant for
$t\le t_0$ and $\epsilon\in U$.
Then, \eqref{42bis} is a direct application of \autoref{lem:O}.
Combined with \eqref{prop21}, this implies that \eqref{prop22} holds:
\begin{equation}\label{prop22bis}
\hat\varphi(\epsilon)\int_0^\infty\mathrm{d} t \,e^{-\epsilon^2
t +(1+\epsilon)(\mu_t-2t)}=
\int\mathrm{d} x \,h_0(x)e^{(1+\epsilon) x}
-\indic{\epsilon>0}\Phi(2+2\epsilon)+\mathcal P(\epsilon)+\mathcal O(\epsilon^3).
\end{equation}
This concludes the proof of \autoref{prop2}.
We now turn to the proof of \autoref{mainthm}. The first statement (for the
$F(h)=h^2$ case with $h_0$ a compact perturbation of the step function) is
a consequence, with \autoref{prop1}, of the second statement (for a general
$F(h)$). We assume
that the hypotheses of that second statement hold; they imply in particular that the second form
of the magical relation holds, \eqref{prop22} or \eqref{prop22bis}. Furthermore, the hypothesis that $\int
\mathrm{d} x\,h_0(x)e^{rx}<\infty$ for some $r>1$ (\textit{i.e.}\@
$\gamma>1$) implies (see \autoref{lem:a}) that $\int \mathrm{d} x\,h_0(x)
e^{(1+\epsilon)x}$ is an analytic function of $\epsilon$ around 0,
so that it can be absorbed into the
$\mathcal P(\epsilon)+\mathcal O(\epsilon^3)$ term. Finally, we write \eqref{prop22bis}
as
\begin{equation}
\label{prop22ter}
\hat\varphi(\epsilon)I(\epsilon)=
-\indic{\epsilon>0}\Phi(2+2\epsilon)+\mathcal P(\epsilon)+\mathcal O(\epsilon^3)
\qquad\text{with }I(\epsilon)=\int_1^\infty\mathrm{d} t \,e^{-\epsilon^2
t +(1+\epsilon)(\mu_t-2t)}.
\end{equation}
Notice that we defined $I(\epsilon)$ as an integral from 1 to $\infty$, not
0 to $\infty$. We are allowed to do this because the remaining
integral from 0 to~1 is an analytic function of $\epsilon$ around~0;
multiplied by $\hat\varphi(\epsilon)$ (another analytic function), it can be
absorbed into $\mathcal P(\epsilon)+\mathcal O(\epsilon^3)$ term.
Using the lemmas proved in \autoref{sec:lem}, we now compute a small
$\epsilon$ expansion of $I(\epsilon)$. We have assumed that the position $\mu_t$ has
a large $t$ expansion given by
\eqref{position}; actually, let us simply write
\begin{equation}\label{position2}
\mu_t=2t-\frac32\log t +a +\frac{b}{\sqrt t}+c
\frac{\log t}t+r(t)\qquad\text{with }r(t)=\mathcal O\Big(\frac1t\Big),
\end{equation}
and we will recover the values of $b$ and $c$ as given in \eqref{position}.
We have
\begin{equation}
\label{prop22terx}
\begin{aligned}
e^{(1+\epsilon)(\mu_t-2t)}
&=\frac{e^{(1+\epsilon)a}}{t^{\frac32+\frac32\epsilon}}e^{\frac{b(1+ \epsilon)}{\sqrt t}+\frac{c(1+ \epsilon)\log t}{t}+(1+\epsilon) r(t)}
\\
&=e^{(1+\epsilon)a}\Big[\frac1{t^{\frac32+\frac32\epsilon}}+\frac{b(1+\epsilon)}{t^{2
+\frac{3\epsilon}2}}
+\frac{c(1+\epsilon)\log
t}{t^{\frac52+\frac32\epsilon}}
\Big]
+R(t,\epsilon)
\end{aligned}
\end{equation}
With
\begin{equation}
R(t,\epsilon)=\frac{e^{(1+\epsilon)a}}{t^{\frac32+\frac32\epsilon}}\Big[e^{\frac{b(1+ \epsilon)}{\sqrt t}+\frac{c(1+ \epsilon)\log
t}{t}+(1+\epsilon) r(t)} - \Big(1+ {\frac{b(1+ \epsilon)}{\sqrt t}+\frac{c(1+ \epsilon)\log
t}{t}}\Big)\Big].
\end{equation}
For $|u|<1$, we have the bound $|e^{u}-(1+u-v)|\le |e^{u}-(1+u)|+|v|\le
u^2+|v|$. Applying this to
$u=\frac{b(1+ \epsilon)}{\sqrt t}+\frac{c(1+ \epsilon)\log
t}{t}+(1+\epsilon)r(t)$ and $v=(1+\epsilon) r(t)$, we see easily that there exists a
$C>0$ and a real neighbourhood $U$ of 0 such that
\begin{equation}
\label{prop22tery}
|R(t,\epsilon)|\le\frac C {t^{\frac52+\frac32\epsilon}}\qquad\text{for all $t>1$ and all $\epsilon\in
U$.}
\end{equation}
As $\epsilon\mapsto R(t,\epsilon)$ is analytic around 0 for all $t>0$, a direct application of \autoref{lem:1s}, and in particular of
\eqref{lem1seq}, and of \autoref{lem:O} gives from \eqref{prop22ter},
\eqref{prop22terx} and \eqref{prop22tery}:
\begin{equation}
\begin{aligned}
I(\epsilon)&=
e^{(1+\epsilon)a}
\Big[|\epsilon|^{1+3\epsilon}\Gamma(-\tfrac12-\tfrac32\epsilon)
+b(1+\epsilon)
|\epsilon|^{2+3\epsilon}\Gamma(-1-\tfrac32\epsilon)
\\&\qquad
+c(1+\epsilon)|\epsilon|^{3+3\epsilon}\Big(
-2\log|\epsilon|\,\Gamma(-\tfrac32-\tfrac32\epsilon)
+\Gamma'(-\tfrac32-\tfrac32\epsilon)
\Big)
\Big]
+\mathcal P(\epsilon)+\mathcal O(\epsilon^3).
\end{aligned}
\end{equation}
The three analytic functions from \autoref{lem:1s} have been absorbed
into the $\mathcal P(\epsilon)+\mathcal O(\epsilon)$ term of
\autoref{lem:O}. We expand all the Gamma functions; the expansion of
the second one is irregular:
\begin{equation}
\Gamma(-1-\tfrac32\epsilon)=
\frac{\Gamma(-\tfrac32\epsilon)}{-1-\tfrac32\epsilon}
=\frac{\Gamma(1-\tfrac32\epsilon)}{-\tfrac32\epsilon(-1-\tfrac32\epsilon)}
=
\frac2{3\epsilon}\frac{1+\gamma_E\tfrac32\epsilon+\mathcal O(\epsilon^2)}{1+\tfrac32\epsilon}
=
\frac{2}{3 \epsilon }+\gamma_E-1
+\mathcal O(\epsilon),
\end{equation}
where $\gamma_E=-\Gamma'(1)\simeq0.577$ is Euler's gamma constant.
We obtain
\begin{align}
\label{expI}
I(\epsilon)&=
e^{(1+\epsilon)a}|\epsilon|^{3\epsilon}\Big[\Gamma(-\tfrac12)|\epsilon|-\tfrac32\Gamma'(-\tfrac12)\epsilon|\epsilon|+b\big(\tfrac23\epsilon+(\gamma_E-\tfrac13)\epsilon^2\big)-2c\Gamma(-\tfrac32)|\epsilon|^3\log|\epsilon|\Big]+\mathcal O(\epsilon^3)
\\&=e^{(1+\epsilon)a}|\epsilon|^{3\epsilon}\Big[
\Big(\Gamma(-\tfrac12)|\epsilon|
+\tfrac23b \epsilon\Big)
+
\Big( b(\gamma_E-\tfrac13)\epsilon^2
-\tfrac32 \Gamma'(-\tfrac12)\epsilon |\epsilon|\Big)
-2c\Gamma(-\tfrac32)|\epsilon|^3\log|\epsilon|
\Big]+\mathcal O(\epsilon^3).\notag
\end{align}
Notice how the expansion mixes terms such as $\epsilon$ and $|\epsilon|$.
For reference, we recall that
\begin{equation}\label{Gamma}
\Gamma(-\tfrac12)=-2\sqrt\pi,\qquad
\Gamma'(-\tfrac12)=-2\sqrt\pi(2-\gamma_E-2\log2),\qquad
\Gamma(-\tfrac32)=\tfrac43\sqrt\pi.
\end{equation}
Before going further, we show how to recover the values of $b$ and $c$.
Notice in \eqref{prop22ter} that, for $\epsilon<0$ (and since
$\hat\varphi(\epsilon)$ is analytic around 0), we must have $I(\epsilon)=\mathcal
P(\epsilon)+\mathcal O(\epsilon)$. In particular, there must remain no
$\log|\epsilon|$ term in the expansion~\eqref{expI} for $\epsilon<0$.
There is a $\log|\epsilon|$ term explicitly written in \eqref{expI}, and
others in the expansion of the prefactor
$|\epsilon|^{3\epsilon}=1+3\epsilon\log|\epsilon|+\tfrac92\epsilon^2\log^2|\epsilon|+\cdots$.
Developing, we obtain a term $ \big(\Gamma(-\tfrac12)|\epsilon|
+\tfrac23b \epsilon\big)3\epsilon\log|\epsilon|$; that term must
cancel for $\epsilon<0$, hence, with \eqref{Gamma}
\begin{equation}
b=\frac32\Gamma(-\tfrac12)= -3\sqrt\pi.
\label{valb}
\end{equation}
Then, $c$ must be chosen in order to prevent a term
$\epsilon^3\log|\epsilon|$ from appearing when $\epsilon<0$.
This
leads to
\begin{equation}
3\Big(b(\gamma_E-\tfrac13)+\tfrac32\Gamma'(-\tfrac12)\Big)+2c\Gamma(-\tfrac32)=0
\end{equation}
With \eqref{Gamma} and \eqref{valb}, this leads to
\begin{equation}\label{valc}
c=\tfrac98(5-6\log2).
\end{equation}
Using the values \eqref{valb} and \eqref{valc} of $b$ and $c$ in
\eqref{position2} gives back the expression \eqref{position} of the
position $\mu_t$ of the front.
Let us make two remarks:
\begin{itemize}
\item
If we try to add in \eqref{position2} extra terms of the
form $C (\log t)^n / t^\alpha$, we would obtain non-cancellable
singularities (terms containing $\log|\epsilon|$ or non integral powers of
$|\epsilon|$) in the expansion of $I(\epsilon)$. We conclude that if $\mu_t$
can be written as an expansion in terms of the form $C (\log t)^n
/ t^\alpha$, then the only terms that may appear are those written in
\eqref{position2}.
\item We used the hypothesis
that $\int
\mathrm{d} x\,h_0(x)e^{rx}<\infty$ for some $r>1$ (\textit{i.e.}\@
$\gamma>1$), only once, to get rid of the $\int\mathrm{d} x\,
h_0(x)e^{(1+\epsilon)x}$ term in \eqref{prop22bis}. If we relax this
hypothesis and simply assume $\int\mathrm{d} x\,
h_0(x) xe^{x}<\infty$ (this is needed to reach \eqref{prop22bis}), we would
have at this point that, for $\epsilon<0$,
$\hat\varphi(\epsilon)I(\epsilon)=\int\mathrm{d} x\,
h_0(x)e^{(1+\epsilon)x}+\mathcal P(\epsilon)+\mathcal O(\epsilon^3)$.
We have just shown that, if the position $\mu_t$ of the front is given by
\eqref{position}, then $I(\epsilon)=\mathcal P(\epsilon)+\mathcal O(\epsilon^3)$ for
$\epsilon<0$. We thus see that
\begin{equation}
\mu_t\text{ given by \eqref{position}}\implies
\int\mathrm{d} x\,
h_0(x)e^{(1+\epsilon)x}=\mathcal P(\epsilon)+\mathcal O(\epsilon^3)\text{ for
$\epsilon<0$}\iff \int\mathrm{d} x\, h_0(x) x^3e^{x}<\infty.
\end{equation}
(We omit the proof of the last equivalence.)
Conversely, if $\int\mathrm{d} x\, h_0(x) x^3e^{x}=\infty$, then the
asymptotic expansion for small negative $\epsilon$ of $\int\mathrm{d} x\,
h_0(x)e^{(1+\epsilon)x}$ will feature some singular terms larger than
$\epsilon^3$, and the expression of $\mu_t$ needs to be modified in such
a way that $\hat\varphi(\epsilon)I(\epsilon)$ matches those singular terms.
\end{itemize}
We return to the expression \eqref{expI} of $I(\epsilon)$ without
making any assumption on the sign of
$\epsilon$, and we make the substitution
\begin{equation}
|\epsilon|=-\epsilon+2\epsilon\indic{\epsilon>0},\qquad
|\epsilon|^3=-\epsilon^3+2\epsilon^3\indic{\epsilon>0}.
\end{equation}
We have tuned $b$ and $c$ so that one obtains
$I(\epsilon)=\mathcal P(\epsilon)+\mathcal O(\epsilon)$ for $\epsilon<0$. For $\epsilon$
of either sign,
we have three extra terms multiplied by $\indic{\epsilon>0}$,
corresponding to the three terms with $|\epsilon|$ or $|\epsilon|^3$ in
\eqref{expI}:
\begin{align}
I(\epsilon)&=
\indic{\epsilon>0}e^{(1+\epsilon)a}\epsilon^{3\epsilon}\Big[
2\Gamma(-\tfrac12)\epsilon
-3 \Gamma'(-\tfrac12)\epsilon^2
-4c\Gamma(-\tfrac32)\epsilon^3\log\epsilon
\Big]
+\mathcal P(\epsilon)+\mathcal O(\epsilon^3).
\end{align}
Comparing with \eqref{prop22ter}, we see that
we must have (only for $\epsilon>0$, of course):
\begin{equation}
\Phi(2+2\epsilon)=\hat\varphi(\epsilon)e^{(1+\epsilon)a}\epsilon^{3\epsilon}\Big[
-2\Gamma(-\tfrac12)\epsilon
+3 \Gamma'(-\tfrac12)\epsilon^2
+4c\Gamma(-\tfrac32)\epsilon^3\log\epsilon
\Big]+\mathcal O(\epsilon^3).
\label{Phi1}
\end{equation}
This expression will be, after some transformations, our main result \eqref{Phi}. We now make a small
$\epsilon$ expansion of $\hat\varphi(\epsilon)$.
From the definition~\eqref{defphi} of $\hat\varphi(\epsilon)$ and
the equation \eqref{propomega} followed by $\omega$, one has
\begin{equation}
\hat\varphi(\epsilon)=\int\mathrm{d} z\,F[\omega(z)]e^{(1+\epsilon)z}=
\int\mathrm{d} z\,\big[\omega''(z)+2\omega'(z)+\omega(z)\big]e^{(1+\epsilon)z}
\end{equation}
This function $\hat\phi(\epsilon)$ is analytic around $\epsilon=0$, but we need to assume
$-1<\epsilon<0$ to split the integral into
three terms and integrate by parts. (Recall that $\omega(z)\sim\tilde\alpha z e^{-z}$ as
$z\to\infty$.)
\begin{equation}
\begin{aligned}
\hat\varphi(\epsilon)
&=
\int\mathrm{d} z\,\omega''(z)e^{(1+\epsilon)z}+
2\int\mathrm{d} z\,\omega'(z) e^{(1+\epsilon)z}+
\int\mathrm{d} z\,\omega(z)e^{(1+\epsilon)z},
\\&=
\big[(1+\epsilon)^2-2(1+\epsilon)+1\big]
\int\mathrm{d} z\,\omega(z) e^{(1+\epsilon)z}=
\epsilon^2\int\mathrm{d} z\,\omega(z) e^{(1+\epsilon)z},
\\&= \epsilon^2 e^{-(1+\epsilon)\alpha} \int\mathrm{d} z\,\omega(z-a)
e^{(1+\epsilon)z}\qquad\text{for $-1<\epsilon<0$.}
\end{aligned}
\end{equation}
Recall
\eqref{omegaalphabeta}: for any $q\in(0,p)$,
\begin{equation}
\omega(z-a)= (\alpha z + \beta ) e^{-z}+\mathcal O(e^{-(1+q)z})\quad\text{as
$z\to\infty$},
\end{equation}
Then $\int\mathrm{d} z\,\omega(z-\alpha)e^{(1+\epsilon)z} = \alpha/\epsilon^2-\beta/\epsilon+\mathcal O(1)$ and
\begin{equation}
\hat\varphi(\epsilon)e^{(1+\epsilon)a}= \alpha-\beta\epsilon+\mathcal O(\epsilon^2).
\end{equation}
Even though the intermediate steps are only valid for $\epsilon<0$, the
final result is also valid for $\epsilon>0$ (small enough) by analyticity.
In \eqref{Phi1}, after replacing $c$ and the Gamma functions by their
values \eqref{Gamma} and \eqref{valc}, we obtain
\begin{equation}
\begin{aligned}
\Phi(2+2\epsilon)
&=(\alpha-\beta\epsilon)\epsilon^{3\epsilon}\Big[
4\sqrt\pi\epsilon
-6\sqrt\pi(2-\gamma_E-2\log2)\epsilon^2
+6(5-6\log2)\sqrt\pi\epsilon^3\log\epsilon
\Big]+\mathcal O(\epsilon^3)
\\
&=\sqrt\pi(\alpha-\beta\epsilon)\epsilon^{3\epsilon}\Big[
4\epsilon
-6(2-\gamma_E-2\log2)\epsilon^2
+6(5-6\log2)\epsilon^3\log\epsilon
\Big]+\mathcal O(\epsilon^3)
\end{aligned}
\end{equation}
It remains to develop with the term
$\epsilon^{3\epsilon}=1+3\epsilon\log\epsilon+\frac92\epsilon^2\log^2\epsilon+\cdots$;
only the coefficient of $\epsilon^3\log\epsilon$ requires to combine two
terms: $3\times (-6)(2-\gamma_E-2\log2)+6(5-6\log 2)=6(3\gamma_E-1)$. We obtain.
\begin{equation}
\Phi(2+2\epsilon)=\sqrt\pi(\alpha-\beta\epsilon)
\Big[
4\epsilon+12\epsilon^2\log\epsilon
-6(2-\gamma_E-2\log2)\epsilon^2
+18\epsilon^3\log^2\epsilon
+6(3\gamma_E-1)\epsilon^3\log\epsilon
\Big]+\mathcal O(\epsilon^3)
\end{equation}
The last step is to replace $\epsilon$ by $\epsilon/2$
\begin{equation}
\begin{aligned}
\Phi(2+\epsilon)&=\sqrt\pi\Big(\alpha-\frac\beta2\epsilon\Big)
\Big[ 2\epsilon+3\epsilon^2\log\frac\epsilon2
-3\Big(1-\frac{\gamma_E}2-\log2\Big)\epsilon^2
+\frac94\epsilon^3\log^2\frac\epsilon2
\\&\qquad\qquad\qquad\qquad\qquad\qquad
+\frac34(3\gamma_E-1)\epsilon^3\log\frac\epsilon2
\Big]+\mathcal O(\epsilon^3)
\\&=\sqrt\pi\Big(\alpha-\frac\beta2\epsilon\Big)
\Big[ 2\epsilon+3\epsilon^2\log\epsilon
-3\Big(1-\frac{\gamma_E}2\Big)\epsilon^2
+\frac94\epsilon^3\log^2\epsilon
\\&\qquad\qquad\qquad\qquad\qquad\qquad
+\frac34(3\gamma_E-6\log2-1)\epsilon^3\log\epsilon
\Big]+\mathcal O(\epsilon^3),
\end{aligned}
\end{equation}
which is \eqref{mainresult}. This completes the proof of the
second part of \autoref{mainthm}. It now remains to prove \autoref{prop1t}
to obtain the first part of \autoref{mainthm}.
\section{Proof of \autoref{prop1t}}\label{proofprop1t}
We start by recalling the main results of \cite{Graham.2019}:
\begin{thm}[Cole Graham 2019 \cite{Graham.2019}]\label{thmCole}
Let $h(x,t)$ be the solution to the Fisher-KPP equation~\eqref{FKPP} with
$F(h)=h^2$ and with
initial condition $h_0(x)$.
Assume that $0\le h_0\le 1$ and that $h_0$ is a compact perturbation of the
step function. There exist $\alpha_0$ and $\alpha_1$ in $\mathbb R$
depending on the initial data $h_0$ such that the following holds. For any
$\gamma>0$, there exists $C_\gamma>0$ also depending on $h_0$ such that for
all $x\in\mathbb{R}$ and all $t\ge3$
\begin{equation}
\big|h(\sigma_t + x,t) -U_\text{app}(x,t)\big|\le \frac{C_\gamma (1+|x|)
e^{-x}}{t^{\frac32-\gamma}},
\label{Cole1}
\end{equation}
where
\begin{equation}
\sigma_t = 2t -\frac32\log t+\alpha_0 -\frac{3\sqrt\pi}{\sqrt t}
+\frac98(5-6\log 2)\frac{\log t} t +\frac{\alpha_1}t
\end{equation}
and
\begin{equation}
U_\text{app}(x,t)=\phi(x)+\frac1t \psi(x)+\mathcal
O(t^{\gamma-3/2})\text{\quad locally uniformly in $x$}.
\end{equation}
Here, $\phi(x)$ is the critical travelling wave translated in such a way
that \cite[eqn.~(1.2)]{Graham.2019}
\begin{equation}
\phi(x)=A_0 x e^{-x} + \mathcal O(e^{-(1+q)x})\quad\text{as $x\to\infty$},
\label{phiasymp}
\end{equation}
and
$\psi(x)$ satisfies \cite[Lem.~5 with
$\psi(x)=A_0e^{-x}V_1^-(x)$ as written in the Proof of Thm~3 p.~1985]{Graham.2019}
\begin{equation}
\psi(x)\text{ is bounded},\qquad \psi(x)\sim -\frac{A_0}4 x^3
e^{-x}\quad\text{as $x\to\infty$}.
\label{psiinfty}
\end{equation}
Furthermore, there exist smooth
functions $V^+_1$, $V^+_2$ and $V^+_3$
of $x/\sqrt t$ such that, for $t$ large enough and $\gamma\in(0,3)$,
\begin{equation}
\begin{cases}\displaystyle \Big|U_\text{app}(x,t)-\phi(x)-\frac1t\psi(x)\Big| \le
C_\gamma\frac{\min(1,e^{-x})}{t^{\frac32-\frac23\gamma}} & \text{for $x\le t^{\gamma/6}$},
\\[2ex]\displaystyle \Big| U_\text{app}(x,t) - A_0 e^{-x}\Big(xe^{-x^2/(4t)}+
V^+_1(x/\sqrt t) \\
\qquad\qquad\qquad\qquad+\frac{\log t}{\sqrt t}V^+_2(x/\sqrt
t) +\frac 1 {\sqrt t} V^+_3(x/\sqrt t)\Big)\Big|\le C_\gamma
\dfrac{e^{-x}}{t^{\frac32-\frac12\gamma}}
& \text{for $x>t^{\gamma/6}$}.
\end{cases}
\label{notincole}
\end{equation}
The $V^+_i$ satisfy $V^+_i(0)=V^+_i(\infty)=0$, and so there are
bounded.
\end{thm}
\noindent\textbf{Remarks}\begin{itemize}
\item We introduced in \eqref{propomega} the critical travelling wave $\omega(x)$,
fixing the translational invariance by imposing $\omega(0)=\frac12$. The
functions $\omega$ and $\phi$ are related by
$\omega(x)=\phi\big(\phi^{-1}(\frac12)+x\big)$.
\item \eqref{notincole} is not explicitly written in \cite{Graham.2019},
but it can be pieced together from the proofs:
in the Proof of Theorem~3 p.\,1985, one reads
$U_\text{app}(x,t)=A_0e^{-x}V_\text{app}(x,t)$ and in the proof of
Theorem~9, p.\,1986, one reads
\begin{equation}
V_\text{app}(x,t)=\indic{x<t^\epsilon} V^-(x,t)+\indic{x\ge t^\epsilon}
V^+(x,t)+ K(t) \theta(x t^{-\epsilon}) \varphi(x,t).
\label{V+app}
\end{equation}
At the end of proof (p.\,1995), the author takes $\epsilon=\gamma/6$;
the functions $\varphi$ and $\theta$ are bounded, $K(t)=\mathcal
O(t^{3\epsilon-3/2})$, and $\theta$ is supported on $(0,2)$, see p.\,1986.
Then, we have so far, for some $C$,
\begin{equation}
\begin{cases}\displaystyle
\big|U_\text{app}(x,t)-A_0e^{-x}V^-(x,t)\big| \le
C\frac{e^{-x}\indic{x>0} }{t^{\frac32-\frac12\gamma}} & \text{for $x\le t^{\gamma/6}$},
\\[2ex]\displaystyle \big| U_\text{app}(x,t) - A_0 e^{-x}V^+(x,t)\big|\le C
\frac{e^{-x}}{t^{\frac32-\frac12\gamma}}
& \text{for $x>t^{\gamma/6}$}.
\end{cases}
\label{132}
\end{equation}
($C$ is some positive constant independent of $x$ and $t$ which can change
at each occurrence.)
We start with the first line;
the function $V^-$ is given at the top of p.\,1986:
\begin{equation}
V^-(x,t)=V_0^-(x+\zeta_t)+\frac1t V_1^-(x+\zeta_t),
\end{equation}
with
\begin{equation}
V_0^-(x)=A_0^{-1}e^x\phi(x),\qquad
V_1^-(x)=A_0^{-1}e^x\psi(x),\qquad \zeta(t)=\mathcal O(t^{4\epsilon-\frac32})
=\mathcal O(t^{\frac23\gamma-\frac32}).
\label{134}
\end{equation}
(See respectively p.\,1972, proof of Theorem~3 p.\,1985, and bottom of p.\,1985.)
From p.\,1972 and Lemma~5 p.\,1973, we have
$(V_0^-)'(x)\sim 1$ and $(V_1^-)'(x)\sim
-\frac34x^2$ as $x\to\infty$, and $(V_0^-)'(x)=\mathcal O(e^x)$ and
$(V_1^-)'(x)=\mathcal O(e^x)$ as $x\to-\infty$. Thus,
$|(V_0^-)'(x)|$ and $|(V_1^-)'(x)/t|$ are both bounded by $C\min(e^x,1)$ for all
$t>1$ and all $x<\sqrt t$.
This implies that
\begin{equation}
\Big|V^-(x,t) - V_0^-(x)-\frac1t V_1^-(x)\Big|\le
C \min(e^x,1)
\zeta_t\le C\frac{\min(e^{x},1)}{ t^{\frac32-\frac23\gamma}}
\quad\text{for $t>1$ and $x<\sqrt t$}.
\end{equation}
Multiplying by $A_0e^{-x}$ and using \eqref{134},
\begin{equation}
\Big|A_0e^{-x}V^-(x,t) - \phi(x)-\frac1t \psi(x)\Big|\le
C\frac{\min(1,e^{-x})}{ t^{\frac32-\frac23\gamma}}
\quad\text{for $t>1$ and $x<\sqrt t$}.
\label{85}
\end{equation}
Combining with the first line of \eqref{132} under the assumption $\gamma<3$, we obtain the first line of
\eqref{notincole}, as the bounding term in \eqref{132} is small compared to
the bounding term in \eqref{85}.
We now turn to the second line of \eqref{132}.
The function $V^+$, only defined for $x>0$, is given
in~(3.4) p.\,1973 in terms of $\tau=\log t$ and
$\eta=x/\sqrt t$:
\begin{equation}
V^+(x,t)=e^{\tau/2}V_0^+(\eta)+V_1^+(\eta)+\tau
e^{-\tau/2}V_2^+(\eta)+e^{-\tau/2}V_3^+(\eta),
\label{137}
\end{equation}
with
\begin{equation}
V_0^+(\eta)=\eta e^{-\eta^2/4},\qquad\text{and so}\qquad
e^{\tau/2}V_0^+(\eta)=x e^{-x^2/(4t)}.
\label{138}
\end{equation}
(Top of p.\,1974: $V_0^+(\eta)=q_0\phi_0(\eta)$ for some
real $q_0$; middle of p.\,1974: $q_0=1$; bottom of p.\,1973:
$\phi_0(\eta)=\eta e^{-\eta^2/4}$.)
Using \eqref{137} and \eqref{138} in the second line of \eqref{132}
gives the second line of
\eqref{notincole}.
The $V_i^+$ are smooth (they are solutions on some differential equations
written pp.\,1974,~1975), and satisfy $V_i^+(0)=V_i^+(\infty)=0$, see
line after (3.4) p.\,1973.
\end{itemize}
We wrote \eqref{notincole} with the accuracy provided by the proofs of
\cite{Graham.2019}, but we actually need a less precise version, only up to
order $1/t$:
\begin{corr}\label{corrCole}
With the notations and hypotheses of \autoref{thmCole},
for any $\gamma\in(0,1/2]$, if $t$ is large enough,
\begin{equation}
\begin{cases}
\displaystyle
\displaystyle\big|h(\sigma_t+x,t)-\phi(x)\big|\le C_\gamma\frac{(1+|x|^3) e^{-x}}t
&\text{for $x\le t^{\gamma/6}$},
\\[2ex]
\big|h(\sigma_t+x,t)-\phi(x)\big|\le C_\gamma x e^{-x}
&\text{for $x> t^{\gamma/6}$}.
\end{cases}
\label{secondrange}
\end{equation}
\end{corr}
\begin{proof}
Recall from~\eqref{psiinfty} that $\psi$ is bounded and $\psi(x)
\sim C x^3 e^{-x}$ as $x\to\infty$. This implies that
$|\psi(x)|\le
C \min\big(1,(1+|x|^3)e^{-x}\big)$ for some constant $C$. Then, the first line
of \eqref{notincole} implies that
\begin{equation}
\big|U_\text{app}(x,t)-\phi(x)\big| \le
C \frac{\min\big(1,(1+|x|^3)e^{-x}\big)}t \qquad\text{for $x\le
t^{\gamma/6}$}
\label{simple<}
\end{equation}
for some other constant $C$.
With \eqref{Cole1}, this implies the first line of \eqref{secondrange}.
(Recall $\gamma\le\frac12$.)
In the second line of \eqref{notincole}, the quantities $V_i^+$ are
bounded. As $x>t^{\gamma/6}\ge1$, we have
\begin{equation}
\big|U_\text{app}(x,t)\big|\le C x e^{-x}
\qquad\text{for $x> t^{\gamma/6}$}.
\end{equation}
As we also have $\phi(x)\sim A_0 x e^{-x}$, we obtain
$
\big|U_\text{app}(x,t)-\phi(x)\big|\le Cx e^{-x}
\ \text{for $x> t^{\gamma/6}$}.
$
which gives, with \eqref{Cole1}, the second line of
\eqref{secondrange}.
\end{proof}
Unfortunately, \autoref{thmCole} and, consequently,
\autoref{corrCole}
are very imprecise for $x<0$. We will need the following result to
complement \autoref{corrCole}:
\begin{lem}\label{lem4}
With the notations and hypotheses of Theorem 2, there exists $C$
and $t_0$
depending on the initial condition $h_0$ such that, for $t\ge t_0$,
\begin{equation}
\big|h(\sigma_t+x,t)-\phi(x)\big|\le \frac{C}t
\qquad\text{for $x\le 0$}.
\label{firstrange}
\end{equation}
\end{lem}
\begin{proof}
Choose $\alpha\in(0,\frac12)$ and let $x_0=\phi^{-1}(\frac12+\alpha)$.
It suffices to prove $|h(\sigma_t+x,t)-\phi(x)|\le C/t$ for $x\le x_0$:
if $x_0\ge 0$, then \eqref{firstrange} follows; if $x_0<0$, then
\eqref{secondrange} provides the required bound for $x\in[x_0,0]$.
Let
\begin{equation}
\delta(x,t)=h(\sigma_t +x ,t)-\phi(x).
\end{equation}
By substitution, one obtains
\begin{equation}\begin{aligned}
\partial_t \delta
=\frac{\mathrm{d}}{\mathrm{d} t} h(\sigma_t+x,t)
&=\partial_x^2(\phi+\delta)+\dot\sigma_t \partial_x (\phi+\delta)+
(\phi+\delta)-(\phi+\delta)^2,
\\&=\phi''+\dot\sigma_t \phi' +\phi-\phi^2+
\partial_x^2\delta+\dot\sigma_t\partial_x \delta
+\delta-2\delta\phi-\delta^2,
\\&=(\dot\sigma_t-2)\phi'+\partial_x^2\delta+\dot\sigma_t\partial_x
\delta+(1-2\phi -\delta)\delta,
\end{aligned}
\label{eqnr}
\end{equation}
where we used in the last step that $\phi''+2\phi'+\phi-\phi^2=0$.
As $\delta(x,t)$ converges uniformly to 0 \cite{Bramson.1983}, there is
a time $t_0>0$
such that $|\delta(x,t)|\le \alpha$ for all $x$ and all $t\ge t_0$.
Recall that
$\phi(x_0)=\frac12+\alpha$ and $\phi\searrow$. Then
\begin{equation}
1-2\phi(x)+|\delta(x,t)|\le1-2\phi(x_0)+\alpha=-\alpha\qquad\text{for
$x\le x_0$ and $t\ge t_0$}.
\label{c1}
\end{equation}
From respectively \eqref{secondrange} and $|\delta(x,t_0)|\le\alpha$, one can find
$C>0$ such that
\begin{equation}
|\delta(x_0,t)| \le \frac C t \quad\text{for $t\ge t_0$},\qquad
|\delta(x,t_0)|\le \frac C {t_0}\quad\text{for $x\le x_0$}.
\label{c2}
\end{equation}
As $\phi'<0$ is bounded and
$0<2-\dot\sigma_t \sim \frac3{2t}$ for $t$ large enough, one can
increase $t_0$ and $C$ such that, furthermore,
\begin{equation}
\label{c3}
0\le \phi'(x)(\dot\sigma_t-2) \le \alpha\frac C t -\frac
C {t^2}\quad\text{for $t\ge t_0$ and $x\le x_0$}.
\end{equation}
(The reason for the negligible $C/t^2$ term will soon become apparent.)
Let $\hat \delta$ be the solution to
\begin{equation}
\partial_t \hat \delta =\alpha\frac C t-\frac C{t^2}+\partial_x^2\hat
\delta
+\dot\sigma_t \partial_x \hat \delta-\alpha \delta\quad\text{for $x<x_0$,
$t>t_0$},\qquad \hat \delta(x_0,t)=\frac
C t, \qquad \hat \delta (x,t_0)=\frac C {t_0}.
\label{rhat}
\end{equation}
We consider \eqref{eqnr} for $x<x_0$ and $t>t_0$, taking as
``initial'' condition $\delta(x,t_0)$ and as boundary condition
$\delta(x_0,t)$.
Using the comparison principle between $\delta$ and $\hat \delta$, and then between
$-\delta$ and $\hat \delta$, one obtains with \eqref{c1},
\eqref{c2} and \eqref{c3} that
$|\delta(x,t)|\le \hat \delta(x,t)$ for all
$x\le x_0$ and $t\ge t_0$.
But the solution to \eqref{rhat} is $\hat \delta(x,t)=\frac C t$, hence $|\delta(x,t)|\le\frac C t$
for $t\ge t_0$ and $x\le x_0$.
\end{proof}
We can now prove \autoref{prop1t}.
\begin{proof}[Proof of \autoref{prop1t}]
The fact that \eqref{position} holds is already proved in
\cite[Corr.~4]{Graham.2019}, as an easy corollary of \autoref{thmCole},
which states:
\begin{equation}
\mu_t = \sigma_t + \phi^{-1}(\tfrac12)+\mathcal O(\tfrac1t),
\label{musigma}
\end{equation}
so that $a$ in \eqref{position} is given by
$a=\alpha_0+\phi^{-1}(\tfrac12)$.
It remains to prove that \eqref{technical} with $F(h)=h^2$ holds:
\begin{equation}
\left |
\int \mathrm{d} x \, e^{rx} h(\mu_t+x,t)^2
-
\int \mathrm{d} x \, e^{rx} \omega(x)^2
\right| \le \frac C t
\qquad \text{for $t>t_0$ and $r\in U$},
\label{goal1}
\end{equation}
where $C>0$ and $t_0>0$ are some constants, and where $U$ is some real
neighbourhood of $U$. We choose to take
$U=[0,01,1.99]$.
In \eqref{goal1}, make the change of variable $x\to x+\sigma_t-\mu_t$ in
the first integral, and the change $x\to x- \phi^{-1}(\frac12)$ in the
second. Recalling that $\omega(x- \phi^{-1}(\frac12))=\phi(x)$ and
factorizing by $e^{r(\sigma_t-\mu_t)}$, we obtain that \eqref{goal1} is equivalent to
\begin{equation}
e^{r(\sigma_t-\mu_t)}\left |
\int \mathrm{d} x \, e^{rx} h(\sigma_t+x,t)^2
-
e^{r(\mu_t-\sigma_t-\phi^{-1}(\frac12))}
\int \mathrm{d} x \, e^{rx} \phi(x)^2
\right| \le \frac C t
\quad \text{for $t>t_0$ and $r\in U$}.
\end{equation}
The prefactor $e^{r(\sigma_t-\mu_t)}$ is bounded for $r\in U$ and $t>1$,
and can be dropped. As $\int \mathrm{d} x\, e^{rx}\phi(x)^2$ is bounded for
$r\in U$, and since \eqref{musigma} holds, one has for some $C$ and $t_0$:
\begin{equation}
\left|
\int \mathrm{d} x \, e^{rx} \phi(x)^2
-
e^{r(\mu_t-\sigma_t-\phi^{-1}(\frac12))}
\int \mathrm{d} x \, e^{rx} \phi(x)^2
\right| \le \frac C t
\quad \text{for $t>t_0$ and $r\in U$},
\end{equation}
and then \eqref{goal1} is equivalent to
\begin{equation}
\left |
\int \mathrm{d} x \, e^{rx} h(\sigma_t+x,t)^2
-
\int \mathrm{d} x \, e^{rx} \phi(x)^2
\right| \le \frac C t
\qquad \text{for $t>t_0$ and $r\in U$}.
\label{goal2}
\end{equation}
We now show that \eqref{goal2} holds.
First notice that there exists $C>0$ such that, for all $x$ and all
$t$ large enough,
\begin{equation}
h(\sigma_t+x,t)\le C\phi(x).
\label{hphi}
\end{equation}
Indeed,
from \eqref{secondrange}, $|h(\sigma_t+x,t)-\phi(x)|\le 2C_\gamma xe^{-x}$ for
$x\ge1$ and $t$ large enough (we used $x^2\le t$ in the first line, since
$\gamma\le1/2$). Since $\phi(x)\sim A_0xe^{-x}$ for large $x$, this implies
that $h(\sigma_t+x,t)\le C \phi(x)$ for some $C$ is $x\ge1$ and $t$ large
enough. Making $C$ larger if needed so that $C\phi(1)\ge1$ ensures that the
relation also holds for $x\le1$ since $\phi\searrow$ and $h\le1$.
Then, for another constant $C$, for all $t$ large enough and all $r\in U$,
\begin{equation}\begin{aligned}
\left |
\int\mathrm{d} x\, e^{rx} \Big[ h(\sigma+x,t)^2-\phi(x)^2\Big]
\right |,
&\le
\int\mathrm{d} x\, e^{rx} \Big|
h(\sigma+x,t)-\phi(x)\Big|\times\Big(h(\sigma+x,t)+\phi(x)\Big)
\\&
\le
C \int\mathrm{d} x\, e^{rx} \phi(x) \Big| h(\sigma+x,t)-\phi(x)\Big|.
\end{aligned} \end{equation}
We cut the integral in three ranges: $x<0$,
$0<x<t^{\gamma/6}$ and $x>t^{\gamma/6}$. In the first range, we use
$r\ge0.01$, $\phi\le1$ and \eqref{firstrange}. In the two other ranges, we
use $r\le1.99$ and \eqref{secondrange}:
\begin{equation}
\begin{aligned}
\left|\int\mathrm{d} x\, e^{rx} \Big[ h(\sigma+x,t)^2-\phi(x)^2\Big]\right|
&\le C\int_{-\infty}^0\mathrm{d} x\,e^{0.01x}\frac 1 t
+ C\int_0^{t^{\gamma/6}}\mathrm{d} x\,e^{1.99x}\phi(x)\frac {(1+x^3)e^{-x}}t
\\&\qquad\qquad
+ C \int_{t^{\gamma/6}}^\infty\mathrm{d} x\,e^{1.99x}\phi(x) xe^{-x},
\\&
\le \frac C t+ \frac C t + C t^{\gamma/3}e^{-0.01t^{\gamma/6}} \le \frac C t,
\end{aligned}
\end{equation}
where we used $e^{1.99x}\phi(x) xe^{-x}\le C x^2e^{-0.01x}$.
This concludes the proof.
\end{proof}
\section{Conclusion}
In this paper, we study the quantity $\Phi(c)$ appearing in \eqref{Phi},
which describes the behaviour of the solution to the Fisher-KPP equation
\eqref{FKPP} at
time $t$ and position $ct$ for $c>2$. We first showed that \eqref{Phi}
holds for values of $c>2$ satisfying \eqref{Phih0}, and we computed a small $\epsilon=c-2$ expansion of the quantity
$\Phi(c)$ appearing in \eqref{Phi}, up to the order $\mathcal O(\epsilon^3)$, see
\eqref{mainresult}.
The expansion depends on the initial condition and the non-linear term in
\eqref{FKPP}
through two numbers
$\alpha$ and $\beta$ which characterize the shifted travelling wave reached
by the front, see \eqref{Bramson's result} and \eqref{omegaalphabeta}.
Although, $\Phi'(2)$ exists, $\Phi''(2)$ does not. The expansion
\eqref{mainresult} is surprisingly irregular, with several logarithmic corrections.
Our method to reach this result relies on so-called magical relation between the
position $\mu_t$ of the front, the initial condition $h_0$, and the
quantity $\Phi(c)$, see \autoref{prop2}. This approach relates in some way
the large $t$ expansion \eqref{position} of the position $\mu_t$ of the front
and the small $\epsilon$ expansion of $\Phi(2+\epsilon)$.
As explained in the proofs of the present paper and in
\cite{BerestyckiBrunetDerrida.2018}, the magical relation also allows to
predict non-rigorously the coefficients of the large $t$ expansion of the position of the
front for all initial conditions. It would be interesting to turn this
approach into a proof.
We believe that our result is universal; however, the proofs in this paper
rely on knowing the large $t$ expansion of the position of the front, and
on some other technical condition \eqref{technical} which has only been
proved for the Fisher-KPP equation \eqref{FKPP} with the $F(h)=h^2$
non-linearity, and an initial condition which is a compact perturbation of
the step function. Therefore, our result is only proved in that situation.
All the results in this paper could be easily extended to the front studied
in \cite{BerestyckiBrunetDerrida.2017,
BerestyckiBrunetDerrida.2018,BerestyckiBrunetPenington.2019}, where the
non-linearity in the Fisher-KPP equation is replaced by a moving boundary:
$\partial_th=\partial_x^2h+h$ if $x>\mu_t$ and $h(x,t)=1$ if $x\le\mu_t$
with $h$ differentiable at $x=\mu_t$. Then, as can be shown rigorously,
the magical relation \eqref{prop21} still holds with
$\phi(\epsilon,t)=\hat\phi(\epsilon)=1/(1+\epsilon)$, and we believe that
\eqref{mainresult} also holds; the only result missing to prove it with our
method is that the large $t$ expansion of $\mu_t$ is also given by
\eqref{position} for that model. (The technical condition \eqref{technical}
is not needed in that case.)
The magical relation could also be used to compute $\Phi(c)$ for large $c$.
As is clear from inspecting \eqref{prop21}, this would require studying the
early times of the evolution of the front. This point
was already noticed in \cite{DerridaMeersonSasorov.2016}.
Beyond the results themselves, the method used to reach them are,
in our opinion, quite unexpected and interesting. We feel that there
remains many aspects of the Fisher-KPP equation that could be better
understood, and the magical relation might be a useful tool to that
purpose.
\section*{Thanks}
The author wishes to thank Pr.\@ Julien Berestycki for invaluable
discussions.
|
{
"timestamp": "2023-02-21T02:27:14",
"yymm": "2302",
"arxiv_id": "2302.09968",
"language": "en",
"url": "https://arxiv.org/abs/2302.09968"
}
|
\section{Introduction} \label{sec:intro}
While planets around M-dwarf stars are extremely abundant \citep[e.g.][]{dressing:2015, hirano:2018, mulders:2018, hsu:2020}, the vast majority of these planets are smaller than Neptune, particularly around less massive M-dwarfs ($M<0.5 \ensuremath{M_\sun}$). Standard core-accretion formation models have long predicted few Jovian-mass planets around these less massive M-dwarfs \citep[e.g.][who also anticipate a particular scarcity of short-period giant planets]{laughlin:2004}. More recent implementations such as the Bern model \citep{Burn:2021} reproduce the low-mass planet population very well, but predict few gas giants around all M-dwarfs, and cannot produce them around later M-dwarfs with $M<0.5 \ensuremath{M_\sun}$ without fine-tuning of the planetary migration \citep{schlecker:2022}. Even prior to the \textit{Transiting Exoplanet Survey Satellite} mission \citep[\textit{TESS},][]{ricker:2015}, there were discoveries that challenged this (such as Kepler-45 b, \citealt{johnson:2012}; HATS-6 b, \citealt{hartman:2015:hats6}; NGTS-1 b, \citealt{bayliss:2018}). More recently, both \textit{TESS} and radial velocity (RV) surveys have added to the known giant planets orbiting low-mass stars (e.g. GJ 3512 b, \citealt{morales:2019}; TOI-3884 b, \citealt{almenara:2022}), suggesting a potential alternative formation pathway such as gravitational instability \citep[e.g.][]{boss:2006}. However, as noted by \cite{schlecker:2022}, gravitational instability is expected to form very massive planets of $\approx 10 \ensuremath{M_{\rm J}}$ on large orbits, while the planets found to date are mainly of Jupiter mass and many have short orbital periods. Likewise, most of these planets orbit early M-dwarfs, for which the Bern model can, though rarely, produce gas giants; the first, and until now only, exception was TOI-5205 b \citep{kanodia:2022}, which orbits an M4 star. It is also worth noting that the Bern models normally assume a smooth initial gas surface density distribution in the protoplanetary disk; a non-smooth density distribution could modify the migration history and potentially facilitate the formation of these planets.
In this context, the discovery and characterization of giant planets around M-dwarfs, particularly later M-dwarfs, is of paramount importance to planetary formation and migration theory. Transiting planets confirmed by radial velocities, for which both the mass and radius can be measured, are especially valuable. In this letter, we present the transiting gas giant TOI-3235 b, orbiting an M4 star with a period of $\hatcurLCPshort$ days. It is only the second gas giant found to orbit a later M-dwarf on the boundary between partially and fully convective M-dwarfs \citep{chabrier:1997}, and is one of a mere dozen giant planets orbiting M-dwarf stars. The planet was first identified as a candidate by the \textit{TESS} mission, and confirmed with ground-based photometry from HATSouth, MEarth-South, TRAPPIST-South, LCOGT, and ExTrA, and RVs from ESPRESSO.
We present the data in Sect. \ref{sec:obs}. The analysis is described in Sect. \ref{sec:analysis}. Finally, we discuss and summarize our findings in Sect. \ref{sec:disc}.
\section{Observations} \label{sec:obs}
\subsection{Photometry}
\subsubsection{TESS}
TOI-3235 was observed by the \textit{TESS} primary and extended missions, in sectors 11 (23rd April to 20th May 2019) and 38 (29th April to 26th May 2021) respectively. In both cases, it was observed with camera 2 and CCD 4. The long-cadence data (30-minute cadence for sector 11, 10-minute cadence for sector 38) were initially processed by the Quick-Look Pipeline \citep[QLP, ][]{huang:2020, huang:2020b}, which uses full-frame images (FFI) calibrated by the \texttt{tica} package \citep{fausnaugh:2020}. The QLP detected a planet and it was promoted to a TOI following \cite{guerrero:2012}, as noted in the ExoFOP archive \footnote{Located at \url{https://exofop.ipac.caltech.edu/tess/target.php?id=243641947}}. For our analysis, we downloaded the \textit{TESS} PDCSAP light curves \citep{stumpe:2012, smith:2012, stumpe:2014} processed by the \textit{TESS} Science Processing Operation Center pipeline \citep[SPOC,][]{jenkinsSPOC2016} at NASA Ames Research Center, from the TESS-SPOC High Level Science Product on MAST \citep{caldwell:2020}. The SPOC difference image centroiding analysis locates the source of the transit signal to within $3.3 \pm 2.5"$ of the target star \citep{Twicken:DVdiagnostics2018}. The \textit{TESS} light curves are shown in Figure \ref{fig:tess}, and the data listed in Table \ref{tab:phfu}.
\subsubsection{HATSouth}
HATSouth \citep{bakos:2013:hatsouth} is a network of 24 telescopes, distributed in three sites at Las Campanas Observatory (LCO) in Chile, the site of the H.E.S.S. gamma-ray observatory in Namibia, and Siding Spring Observatory (SSO) in Australia. Each telescope has a \tsize{0.18} aperture and \ccdsize{4K} front-illuminated CCD cameras. HATSouth observed TOI-3235 from 11th February 2017 through 15th May 2017, from all three sites. The data were reduced as described in \cite{penev:2013:hats1}. The transit was clearly detected, but was not flagged by the automated search due to the high transit depth and the pre-Gaia poor constraint on the stellar size from J-K magnitudes. The light curve is shown in Figure \ref{fig:toi3235-ground-phot} (left panel), and the data are listed in Table \ref{tab:phfu}.
\subsubsection{MEarth-South}
MEarth-South is an array of eight \tsize{0.4} telescopes at the Cerro Tololo Inter-American Observatory (CTIO) in Chile \citep{nutzman:2008, irwin:2015}. M-Earth observed TOI-3235 with six telescopes on 21st June 2021 in the RG715 filter with $\rm{60\, s}$ exposure time, obtaining a full transit of TOI-3235.01. The light curves are shown in Figure \ref{fig:toi3235-ground-phot} (right panel), where the data from all six telescopes have been plotted together, and the data are listed in Table \ref{tab:phfu}.
\subsubsection{TRAPPIST-South}
TRAPPIST-South \citep{jehin:2011,Gillon2011} is a \tsize{0.6} Ritchey-Chretien robotic telescope at La Silla Observatory in Chile, equipped with a \ccdsize{2K} back-illuminated CCD camera with a pixel scale of 0.65\arcsec/pixel, resulting a field of view of $22\arcmin\times22\arcmin$. A full transit of TOI-3235.01 was observed by TRAPPIST-South on 10th May 2022 in the Sloan-$z'$ filter with an exposure time of 100s. We used the {\tt TESS Transit Finder} tool, which is a customised version of the {\tt Tapir} software package \citep{jensen2013}, to schedule the observations. Data reduction and photometric measurement were performed using the {\tt PROSE}\footnote{\textit{PROSE}: \url{https://github.com/lgrcia/prose}} pipeline \citep{garcia2021}. The light curve is shown in Figure \ref{fig:toi3235-ground-phot} (right panel), and the data are listed in Table \ref{tab:phfu}.
\subsubsection{LCOGT}
The Las Cumbres Observatory global telescope network \citep[LCOGT,][]{brown:2013:lcogt} is a globally distributed network of \tsize{1} telescopes. The telescopes are equipped with $4096\times4096$ SINISTRO cameras having an image scale of $0\farcs389$ per pixel, resulting in a $26\arcmin\times26\arcmin$ field of view. TOI-3235 was observed by LCOGT with the SINISTRO instrument at the South Africa Astronomical Observatory (SAAO) site in the Sloan-$i'$ band on 10th June 2021, and at the Cerro Tololo Inter-American Observatory (CTIO) site in the Sloan-$g'$ band on 1st July 2022, full transits of TOI-3235.01 being obtained in both observations. We used the {\tt TESS Transit Finder}, which is a customized version of the {\tt Tapir} software package \citep{Jensen:2013}, to schedule our transit observations. The images were calibrated by the standard LCOGT {\tt BANZAI} pipeline \citep{McCully:2018}. The differential photometric data were extracted using {\tt AstroImageJ} \citep{Collins:2017}. The light curves are shown in Figure \ref{fig:toi3235-ground-phot} (right panel), and the data listed in Table \ref{tab:phfu}.
\subsubsection{ExTrA}
The ExTrA facility (Exoplanets in Transits and their Atmospheres, \citealt{bonfils:2015}) is composed of a near-infrared (0.85 to 1.55 $\mu$m) multi-object spectrograph fed by three \tsize{0.6} telescopes located at La Silla observatory in Chile. We observed 5 full transits of TOI-3235.01 on 2nd March 2022 (with three telescopes) and on 28th March 2022, 2nd April 2022, 23rd April 2022, and 24th May 2022 (with two telescopes). We observed using the fibers with $8\arcsec$ apertures, used the low resolution mode of the spectrograph (R$\sim20$) and 60-second exposures for all nights. At the focal plane of each telescope, five fiber positioners are used to pick the light from the target and four comparison stars. As comparison stars, we observed 2MASS J13493913-4615443, 2MASS J13515346-4623273, 2MASS J13510825-4612537 and 2MASS J13481046-4615434, with J-magnitude \citep{skrutskie:2006} and $T_{eff}$ \citep{gaiadr2} similar to TOI-3235. The resulting ExTrA data were analyzed using custom data reduction software. The light curves are shown in Figure \ref{fig:toi3235-ground-phot} (right panel), and the data listed in Table \ref{tab:phfu}.
\subsection{Radial Velocities}
\subsubsection{ESPRESSO}
ESPRESSO \citep[Echelle SPectrograph for Rocky Exoplanets and Stable Spectroscopic Observations,][]{pepe:2021} is an ultra-stable fibre-fed échelle high-resolution spectrograph installed at the incoherent combined Coudé facility of the Very Large Telescope (VLT) in Paranal Observatory, Chile. We observed TOI-3235 with ESPRESSO in HR mode (1 UT, $\mathrm{R \sim 140,000}$) between 2nd and 14th February 2022, obtaining 7 spectra under programme ID 108.22B4.001 aka 0108.C-0123(A). The spectra were reduced with the official ESPRESSO DRS v2.3.5 pipeline \citep{sosnowska:2015, modigliani:2020}, in the EsoReflex environment \citep{freudling:2013}. The RVs and bisector spans are listed in Table \ref{tab:rvs}, and the phase-folded RVs and bisector spans are shown in Figure \ref{fig:toi3235-RV-SED} (left panel). Two of the bisector spans are extreme outliers with values of $<-3000\, \ensuremath{\rm m\,s^{-1}}$, and were excluded from the analysis.
\begin{deluxetable*}{lrrrrl}
\tabletypesize{\small}
\tablewidth{0pc}
\tablecaption{
Light curve data for TOI-3235\label{tab:phfu}.
}
\tablehead{
\colhead{BJD\tablenotemark{a}} &
\colhead{Mag\tablenotemark{b}} &
\colhead{\ensuremath{\sigma_{\rm Mag}}} &
\colhead{Mag(orig)\tablenotemark{c}} &
\colhead{Filter} &
\colhead{Instrument} \\
\colhead{\hbox{~~~~(2,450,000$+$)~~~~}} &
\colhead{} &
\colhead{} &
\colhead{} &
\colhead{} &
\colhead{}
}
\startdata
\input{phfu_tab_short.tex}
\enddata
\tablenotetext{a}{
Barycentric Julian Date computed on the TDB system with correction
for leap seconds.
}
\tablenotetext{b}{
The out-of-transit level has been subtracted. For observations
made with the HATSouth instruments these magnitudes have been corrected for
trends using the EPD and TFA procedures applied {\em prior} to
fitting the transit model. This procedure may lead to an
artificial dilution in the transit depths when used in its plain
mode, instead of the signal reconstruction mode
\citep{kovacs:2005:TFA}. The blend factors for the HATSouth light
curves are listed in Table~\ref{tab:planetparam}. For observations
made with follow-up instruments (anything other than ``HATSouth'' in the
``Instrument'' column), the magnitudes have been corrected for a
quadratic trend in time, and for variations correlated with up to
three PSF shape parameters, fit simultaneously with the transit.
}
\tablenotetext{c}{
Raw magnitude values without correction for the quadratic trend in
time, or for trends correlated with the seeing.
}
\tablecomments{
This table is available in a machine-readable form in the online
journal. A portion is shown here for guidance regarding its form
and content.
}
\end{deluxetable*}
\tabletypesize{\scriptsize}
\begin{deluxetable}{rrrrrr}
\tablewidth{0pc}
\tablecaption{
Relative radial velocities and bisector spans from ESPRESSO for
\hatcurhtr{}. \label{tab:rvs}
}
\tablehead{
\colhead{BJD} &
\colhead{RV\tablenotemark{a}} &
\colhead{\ensuremath{\sigma_{\rm RV}}\tablenotemark{b}} &
\colhead{BS} &
\colhead{\ensuremath{\sigma_{\rm BS}}} &
\colhead{Phase}\\
\colhead{\hbox{(2,450,000$+$)}} &
\colhead{(\ensuremath{\rm m\,s^{-1}})} &
\colhead{(\ensuremath{\rm m\,s^{-1}})} &
\colhead{(\ensuremath{\rm m\,s^{-1}})} &
\colhead{(\ensuremath{\rm m\,s^{-1}})} &
\colhead{}
}
\startdata
\input{rvtable.tex}
\enddata
\tablenotetext{a}{
The zero-point of these velocities is arbitrary. An overall offset
$\gamma$ fitted to the velocities has been subtracted.
}
\tablenotetext{b}{
Internal errors excluding the component of astrophysical jitter
considered in \refsecl{analysis}.
}
\end{deluxetable}
\section{Analysis} \label{sec:analysis}
\begin{figure*}[!ht]
{
\centering
\leavevmode
\includegraphics[width={1.0\linewidth}]{TOI3235-TESS}
}
\caption{
{\em TESS} long-cadence light curves for TOI-3235, for sector 11 (left, 30-minute cadence) and sector 38 (right, 10-minute cadence). For each sector, we show the full un-phased light curve as a function of time ({\em top}), the full phase-folded light curve ({\em second}), the phase-folded light curve zoomed-in on the planetary transit ({\em third}), the residuals from the best-fit model, phase-folded and zoomed-in on the planetary transit ({\em fourth}), and the phase-folded light curve zoomed-in on the secondary eclipse ({\em bottom}). The solid red line in each panel shows the model fit to the light curve. The blue filled circles show the light curve binned in phase with a bin size of 0.002. Other observations included in our analysis of this system are shown in Figures~\ref{fig:toi3235-ground-phot} and ~\ref{fig:toi3235-RV-SED}.
\label{fig:tess}
}
\end{figure*}
\begin{figure*}[!ht]
{
\centering
\leavevmode
\includegraphics[width={0.5\linewidth}]{TOI3235-hs
\hfil
\includegraphics[width={0.5\linewidth}]{TOI3235-lc
}
\caption{
Ground-based photometry for the the transiting planet system TOI-3235. {\em Left:} Phase-folded unbinned full HATSouth light curve (top), light curve zoomed-in on the transit (middle), and residuals from the best-fit model zoomed-in on the transit (bottom). Solid red lines show the best-fit model. Blue circles show the light curves binned in phase with a bin size of 0.002. {\em Right:} Unbinned follow-up transit light curves corrected for instrumental trends fitted simultaneously with the transit model, which is overplotted (left), and residuals to the fit (right). Dates, filters and instruments are indicated. For ExTrA we indicate the midpoint of the spectral range. The error bars represent the photon
and background shot noise, plus the readout noise.
\label{fig:toi3235-ground-phot}
}
\end{figure*}
\begin{figure*}[!ht]
{
\centering
\leavevmode
\includegraphics[width={0.5\linewidth}]{TOI3235-rv
\hfil
\includegraphics[width={0.5\linewidth}]{TOI3235-iso-gk-gabs-isofeh-SED
}
\caption{
{\em Left:}
High-precision RVs from ESPRESSO/VLT phased with respect to the mid-transit time, together with the best-fit model, where the center-of-mass velocity has been subtracted (top); RV $O\!-\!C$ residuals (centre); and bisector spans (bottom). Error bars include the estimated jitter, which is a free parameter in the fitting.
{\em Top Right:} Absolute $G$ magnitude vs.\ the de-reddened $G - K_{S}$ color from Gaia DR2 and 2MASS (filled blue circle) and $1\sigma$ and $2\sigma$ confidence regions, including estimated systematic errors in the photometry (blue lines), compared to theoretical isochrones (black lines, ages listed in Gyr) and stellar evolution tracks (green dashed lines, mass listed in solar masses) from the MIST models interpolated at the best-estimate value for the host metallicity. The red lines show isochrones at higher and lower metallicities than the best-estimate value, labelled with their metallicity and age in Gyr. {\em Bottom Right:} SED as measured via broadband photometry through the listed filters (top), and $O\!-\!C$ residuals from the best-fit model (bottom). We plot the observed magnitudes without correcting for distance or extinction. Overplotted are 200 model SEDs randomly selected from the MCMC posterior distribution produced through the global analysis (gray lines). Black error bars show the catalog errors for the broad-band photometry measurements; red error bars add an assumed 0.02\,mag systematic uncertainty in quadrature to the catalog errors. These latter uncertainties are used in the fit.
\label{fig:toi3235-RV-SED}}
\end{figure*}
We carried out a joint analysis of the photometric, astrometric and RV data for TOI-3235\,b following the methods of \citet{hartman:2019:hats6069} and \citet{bakos:2020:hats71}. We fit the light curve data shown in Figures~\ref{fig:tess} and~\ref{fig:toi3235-ground-phot}, together with the broad-band catalog photometry and {\em Gaia} parallax measurement listed in Table~\ref{tab:stellarobserved}, and the RV data shown in Figure~\ref{fig:toi3235-RV-SED}. The model also makes use of the predicted absolute magnitudes in each bandpass from the MIST isochrones and of the extinction, constrained from the SED. We use a \citet{mandel:2002} transit model with quadratic limb darkening to fit the light curves and assume a Keplerian orbit for fitting the RV measurements. The limb darkening coefficients are allowed to vary, with priors based on the \citet{claret:2012,claret:2013,claret:2018} theoretical models. The stellar parameters are constrained using isochrones from version 1.2 of the MIST theoretical stellar evolution models \citep{paxton:2011,paxton:2013,paxton:2015,choi:2016,dotter:2016}. We allow the line of sight extinction $A_{V}$ to vary in the fit, imposing a maximum of $0.527$\,mag and a Gaussian prior of $0.055 \pm 0.2$\,mag based on the MWDUST 3D Galactic extinction model \citep{bovy:2016}.
We used the ODUSSEAS software \citep{antoniadis:2020}, developed specifically for M-dwarfs, to measure the $\ensuremath{\rm [Fe/H]}$ and $\ensuremath{T_{\rm eff\star}}$ from the ESPRESSO spectra. Although ODUSSEAS was developed for spectra with resolutions from $48\,000$ to $115\,000$, it has been successfully used with ESPRESSO spectra at their original $140\,000$ resolution \citep{lillo-box:2020}. We obtained preliminary values of $\ensuremath{\rm [Fe/H]} = -0.0024 \pm 0.104$, $\ensuremath{T_{\rm eff\star}} = 3196 \pm 67$\,K, which were used as priors for the joint analysis\footnote{An independent estimate of $\ensuremath{T_{\rm eff\star}}=3421\pm53$\,K can be obtained using the absolute G magnitude $M_G$ from Equation~(11) of \citet{rabus:2019}, which is consistent at $\approx 2 \sigma$ with the value inferred from ODUSSEAS.}, in which a combination of the MIST evolution models, the transit-derived stellar bulk density, and the broad-band catalog photometry and parallax are employed to precisely constrain the host star parameters. To determine the spectral type, we used the PyHammer tool \citep{roulston:2020} with the ESPRESSO spectra, which returned an M5 spectral type. However, colour index comparisons with the tables of \cite{pecaut:2013} suggest an earlier spectral type of M3-M4, and visual inspection with the `eyecheck' facility of PyHammer shows an M4 template is also a good match to the spectrum. Therefore, we adopt an M4 spectral type.
We modelled the observations both assuming a circular orbit for the planet, and allowing the orbit to have a non-zero eccentricity. We find that the free-eccentricity model produces an eccentricity consistent with zero ($e \hatcurRVeccentwosiglimeccen{}$ at 95\% confidence). A very low eccentricity is expected, given that we estimate a rapid tidal circularization timescale for this system of $\mathrm{\sim 6\, Myr}$ \citep{hut:1981}. We therefore adopt the parameters that result from assuming a circular orbit. Applying the transit least squares \citep[TLS, ][]{hippke:2019} algorithm to the HATSouth and {\em TESS} light curve residuals to the best-fit model finds no additional transit signals. The stellar parameters derived from the analysis assuming a circular orbit are listed in Table~\ref{tab:stellarderived}, while the planetary parameters are listed in Table~\ref{tab:planetparam}. The best-fit model is shown in Figs. \ref{fig:tess}, \ref{fig:toi3235-ground-phot}, and \ref{fig:toi3235-RV-SED}. We note that the light curve uncertainties are scaled up in the fitting procedure to achieve a reduced $\chi^2$ of unity, but the uncertainties shown in Fig. \ref{fig:toi3235-ground-phot} have not been scaled.
The resulting $\sim 1$\% and $\sim 0.5$\% respective uncertainties on the derived stellar mass and radius are well below the respective $\sim 5$\% and $\sim 4.2$\% estimated systematic uncertainties of \cite{tayar:2022} for these parameters, which stem from inaccuracies in the fundamental observables and stellar evolution models. Likewise, the formal uncertainties of $7.4$\,K on the posterior stellar effective temperature and 0.017\,dex on the metallicity are likely quite a bit smaller than the systematic uncertainties, which we may expect to be closer to the ODUSSEAS-derived uncertainties of $\sim 70$\,K and $\sim 0.1$\,dex, respectively.
However, as described in \citep{eastman:2022}, uncertainties smaller than the general error floors of Tayar et al. (2022) can be achieved for transiting planets by measuring the stellar density $\ensuremath{\rho_\star}$ directly from the transit and employing it in the derivation of other stellar parameters.
Although our fit self-consistently accounts for the relation between the stellar density, transit parameters, $\ensuremath{M_\star}$, $\ensuremath{L_\star}$, $\ensuremath{R_\star}$, and $\ensuremath{T_{\rm eff}}$ throughout the fit, as suggested by \cite{eastman:2022}, and imposes a constraint that each link in the chain must match a stellar evolution model, it does not account for systematic errors in those models when imposing this constraint, and thus the formal uncertainties derived in this analysis are too small. Therefore, we conservatively adopt the error floors of \cite{tayar:2022}, which we report in brackets in Table \ref{tab:stellarderived}; for [Fe/H] we report the ODUSSEAS-derived uncertainty. These systematic uncertainties were formally propagated out to the planetary parameters. Regarding the planetary equilibrium temperature $T_{\rm eq}$ in particular, it is calculated under the assumptions of 0 albedo and full and instantaneous redistribution of heat, which are unlikely to hold completely in reality but provide a useful approximation.
The formal fit gives a young age of \hatcurISOage Gyr for the host star. However, this is primarily driven by the photometry being somewhat blue compared to the model values (see Fig. \ref{fig:toi3235-RV-SED}, top right), which are known to be uncertain for M-dwarfs. We see no other evidence of youth such as flares. Likewise, the GLS periodogram of the HATSouth photometry shows a significant peak at $44.4264 \pm 0.0010$ days; taking this as the stellar rotation period, the relations of \cite{engle:2018} suggest a much larger age of $\approx 2.7$ Gyr. We also used the BANYAN $\Sigma$ tool \citep{gagne:2018} to check the probability of TOI-3235 belonging to known young stellar associations given its Gaia DR3 \citep{gaiadr3} proper motions and radial velocity, finding it has a 99.9\% probability of being a field star.
Independent estimates of the stellar mass and radius can be obtained from the $\mathrm{K_S}$ magnitude using the mass-radius-luminosity relations of \cite{mann:2018} and \cite{rabus:2019}. Applying these relations leads to a mass of $\ensuremath{M_\star} = 0.3605 \pm 0.087 \ensuremath{M_\sun}$ and a radius of $\ensuremath{R_\star} = 0.37 \pm 0.07 \ensuremath{R_\sun}$. While the radius is fully consistent with that obtained via global modelling, the mass is lower at $1.5\sigma$. We choose to adopt the values from the global modelling, since it accounts for all variables simultaneously. We also note that the planetary mass and radius calculated by employing the values obtained through the mass-radius-luminosity relations remain consistent with those computed from the global modelling values; thus, adopting the lower stellar mass from the mass-radius-luminosity relations would only make this giant planet even more unusual.
\begin{deluxetable*}{lcl}
\tablewidth{0pc}
\tabletypesize{\tiny}
\tablecaption{
Astrometric, Spectroscopic and Photometric parameters for \hatcurhtr{}
\label{tab:stellarobserved}
}
\tablehead{
\multicolumn{1}{c}{~~~~~~~~Parameter~~~~~~~~} &
\multicolumn{1}{c}{Value} &
\multicolumn{1}{c}{Source}
}
\startdata
\noalign{\vskip -3pt}
\sidehead{Astrometric properties and cross-identifications}
~~~~2MASS-ID\dotfill & \hatcurCCtwomass{} & \\
~~~~TIC-ID\dotfill & \hatcurTICID{} & \\
~~~~Gaia~DR3-ID\dotfill & \hatcurCCgaiadrthree{} & \\
~~~~R.A. (J2000)\dotfill & \hatcurCCra{} & Gaia DR3\\
~~~~Dec. (J2000)\dotfill & \hatcurCCdec{} & Gaia DR3\\
~~~~$\mu_{\rm R.A.}$ (\ensuremath{\rm mas\,yr^{-1}}) & \hatcurCCpmra{} & Gaia DR3\\
~~~~$\mu_{\rm Dec.}$ (\ensuremath{\rm mas\,yr^{-1}}) & \hatcurCCpmdec{} & Gaia DR3\\
~~~~parallax (mas) & \hatcurCCparallax{} & Gaia DR3\\
~~~~radial velocity (\ensuremath{\rm km\,s^{-1}}) & $-14.96 \pm 2.72$ & Gaia DR3\\
\sidehead{Spectroscopic properties}
~~~~$\ensuremath{T_{\rm eff\star}}$ (K)\dotfill & \hatcurSMEiteff{} & ODUSSEAS/ESPRESSO\tablenotemark{a} \\
~~~~$\ensuremath{\rm [Fe/H]}$\dotfill & \hatcurSMEizfeh{} & ODUSSEAS/ESPRESSO\tablenotemark{a} \\
~~~~Spectral type \dotfill & M4 & this work\\
\sidehead{Photometric properties\tablenotemark{b}}
~~~~$G$ (mag)\dotfill & \hatcurCCgaiamGthree{} & Gaia DR3 \\
~~~~$BP$ (mag)\dotfill & \hatcurCCgaiamBPthree{} & Gaia DR3 \\
~~~~$RP$ (mag)\dotfill & \hatcurCCgaiamRPthree{} & Gaia DR3 \\
~~~~$J$ (mag)\dotfill & \hatcurCCtwomassJmag{} & 2MASS \\
~~~~$H$ (mag)\dotfill & \hatcurCCtwomassHmag{} & 2MASS \\
~~~~$K_s$ (mag)\dotfill & \hatcurCCtwomassKmag{} & 2MASS \\
~~~~$W1$ (mag)\dotfill & \hatcurCCWonemag{} & WISE \\
~~~~$W2$ (mag)\dotfill & \hatcurCCWtwomag{} & WISE \\
~~~~$W3$ (mag)\dotfill & \hatcurCCWthreemag{} & WISE \\
\enddata
\tablenotetext{a}{
The ODUSSEAS-derived $\ensuremath{T_{\rm eff\star}}$ and $\ensuremath{\rm [Fe/H]}$ are not the final adopted parameters, but are used as priors for the global modelling.
}
\tablenotetext{b}{
The listed uncertainties for the photometry are taken from the catalogs. For the analysis we assume a systematic uncertainty floor of 0.02\,mag.
}
\end{deluxetable*}
\begin{deluxetable}{lc}
\tablewidth{0pc}
\tabletypesize{\footnotesize}
\tablecaption{
Derived stellar parameters for \hatcurhtr{}
\label{tab:stellarderived}
}
\tablehead{
\multicolumn{1}{c}{~~~~~~~~Parameter~~~~~~~~} &
\multicolumn{1}{c}{Value}
}
\startdata
~~~~$\ensuremath{M_\star}$ ($\ensuremath{M_\sun}$)\dotfill & \hatcurISOmlong{} ($\pm 0.020$) \\
~~~~$\ensuremath{R_\star}$ ($\ensuremath{R_\sun}$)\dotfill & \hatcurISOrlong{} ($\pm0.016$) \\
~~~~$\ensuremath{\log{g_{\star}}}$ (cgs)\dotfill & \hatcurISOlogg{} ($\pm 0.063$) \\
~~~~$\ensuremath{\rho_\star}$ (\ensuremath{\rm g\,cm^{-3}})\dotfill & \hatcurLCrho{} \\
~~~~$\ensuremath{L_\star}$ ($\ensuremath{L_\sun}$)\dotfill & \hatcurISOlum{} ($\pm 0.00039$) \\
~~~~$\ensuremath{T_{\rm eff\star}}$ (K)\dotfill & \hatcurISOteff{} ($\pm 68$) \\
~~~~\ensuremath{\rm [Fe/H]}\ (dex)\dotfill & \hatcurISOzfeh{} ($\pm 0.1$) \\
~~~~$A_{V}$ (mag)\dotfill & \hatcurXAv{} \\
~~~~Distance (pc)\dotfill & \hatcurXdistred{}\phn \\
\enddata
\tablecomments{
The listed parameters are those determined through the joint analysis described in Section~\ref{sec:analysis} assuming a circular orbit for the planet. The first uncertainties listed for each parameter are the statistical uncertainties from the fit, not including systematic errors. Values in brackets report the estimated uncertainty floors due to inaccuracies in the fundamental observables and/or the MIST stellar evolution models, where appropriate. These latter floors were formally propagated to the planetary parameter uncertainties.
}
\end{deluxetable}
\begin{deluxetable}{lc}
\tabletypesize{\tiny}
\tablecaption{Adopted orbital and planetary parameters for \hatcurhtr{}\,b\label{tab:planetparam}}
\tablehead{
\multicolumn{1}{c}{~~~~~~~~~~~~~~~Parameter~~~~~~~~~~~~~~~} &
\multicolumn{1}{c}{Value}
}
\startdata
\noalign{\vskip -3pt}
\sidehead{Light curve{} parameters}
~~~$P$ (days) \dotfill & \hatcurLCP{} \\
~~~$T_c$ (${\rm BJD\_{}TDB}$)
\tablenotemark{a} \dotfill & \hatcurLCT{} \\
~~~$T_{14}$ (days)
\tablenotemark{a} \dotfill & $\hatcurLCdur{}$ \\
~~~$T_{12} = T_{34}$ (days)
\tablenotemark{a} \dotfill & $\hatcurLCingdur{}$ \\
~~~$\ensuremath{a/\rstar}$ \dotfill & $\hatcurPPar{}$ \\
~~~$\ensuremath{\zeta/\rstar}$ \tablenotemark{b} \dotfill & $\hatcurLCzeta$\phn \\
~~~$\ensuremath{R_{p}}/\ensuremath{R_\star}$ \dotfill & $\hatcurLCrprstar{}$ \\
~~~$b^2$ \dotfill & $\hatcurLCbsq{}$ \\
~~~$b \equiv a \cos i/\ensuremath{R_\star}$
\dotfill & $\hatcurLCimp{}$\\
~~~$i$ (deg) \dotfill & $\hatcurPPi{}$\phn \\
\sidehead{Dilution factors \tablenotemark{c}}
~~~HAT G701/3 \dotfill & $\hatcurLCiblendA{}$ \\
~~~HAT G701/4 \dotfill & $\hatcurLCiblendB{}$ \\
~~~{\em TESS} Sector 11 \dotfill & $\hatcurLCiblendC{}$ \\
~~~{\em TESS} Sector 38 \dotfill & $\hatcurLCiblendD{}$ \\
\sidehead{Limb-darkening coefficients \tablenotemark{d}}
~~~$c_1,T$ \dotfill & $\hatcurLBiT{}$ \\
~~~$c_2,T$ \dotfill & $\hatcurLBiiT{}$ \\
~~~$c_1,g$ \dotfill & $\hatcurLBig{}$ \\
~~~$c_2,g$ \dotfill & $\hatcurLBiig{}$ \\
~~~$c_1,r$ \dotfill & $\hatcurLBir{}$ \\
~~~$c_2,r$ \dotfill & $\hatcurLBiir{}$ \\
~~~$c_1,z$ \dotfill & $\hatcurLBiz{}$ \\
~~~$c_2,z$ \dotfill & $\hatcurLBiiz{}$ \\
~~~$c_1,RG715$ \dotfill & $\hatcurLBii{}$ \\
~~~$c_2,RG715$ \dotfill & $\hatcurLBiii{}$ \\
\sidehead{RV parameters}
~~~$K$ (\ensuremath{\rm m\,s^{-1}}) \dotfill & $\hatcurRVK{}$\phn\phn \\
~~~$e$ \tablenotemark{e} \dotfill & $\hatcurRVeccentwosiglimeccen$\\
~~~RV jitter ESPRESSO (\ensuremath{\rm m\,s^{-1}}) \dotfill & $\hatcurRVjittertwosiglim{}$\\
\sidehead{Planetary parameters}
~~~$\ensuremath{M_{p}}$ ($\ensuremath{M_{\rm J}}$) \dotfill & $\hatcurPPmlong{}$ \\
~~~$\ensuremath{R_{p}}$ ($\ensuremath{R_{\rm J}}$) \dotfill & $\hatcurPPrlong{}$ \\
~~~$C(\ensuremath{M_{p}},\ensuremath{R_{p}})$
\tablenotemark{g} \dotfill & $\hatcurPPmrcorr{}$ \\
~~~$\ensuremath{\rho_{p}}$ (\ensuremath{\rm g\,cm^{-3}}) \dotfill & $\hatcurPPrho{}$ \\
~~~$\log g_p$ (cgs) \dotfill & $\hatcurPPlogg{}$ \\
~~~$a$ (AU) \dotfill & $\hatcurPParel{}$ \\
~~~$T_{\rm eq}$ (K) \dotfill & $\hatcurPPteff{}$ \\
~~~$\Theta$ \tablenotemark{h} \dotfill & $\hatcurPPtheta{}$ \\
~~~$\log_{10}\langle F \rangle$ (cgs) \tablenotemark{i}
\dotfill & $\hatcurPPfluxavglog{}$ \\
\enddata
\tablecomments{
We adopt a model in which the orbit is assumed to be circular. See the discussion in Section~\ref{sec:analysis}.
}
\tablenotetext{a}{
Times are in Barycentric Julian Date calculated on the Barycentric Dynamical Time (TDB) system.
\ensuremath{T_c}: Reference epoch of
mid transit that minimizes the correlation with the orbital
period.
\ensuremath{T_{14}}: total transit duration, time
between first to last contact;
\ensuremath{T_{12}=T_{34}}: ingress/egress time, time between first
and second, or third and fourth contact.
}
\tablenotetext{b}{
Reciprocal of the half duration of the transit used as a jump parameter in our MCMC analysis in place of $\ensuremath{a/\rstar}$. It is related to $\ensuremath{a/\rstar}$ by the expression $\ensuremath{\zeta/\rstar} = \ensuremath{a/\rstar}(2\pi(1+e\sin\omega))/(P\sqrt{1-b^2}\sqrt{1-e^2})$ \citep{bakos:2010:hat11}.
}
\tablenotetext{c}{
Scaling factor applied to the model transit fit to the HATSouth and {\em TESS} light curves. It accounts for dilution of the transit due to blending from neighboring stars, over-filtering of the light curve, or over-correction of dilution in the {\em TESS} SPOC light curves. These factors are varied in the fit, with independent values adopted for each light curve.
}
\tablenotetext{d}{
Values for a quadratic law. The limb-darkening parameters were directly varied in the fit, using the tabulations from \cite{claret:2012,claret:2013,claret:2018} to place Gaussian priors on their values, assuming a prior uncertainty of $0.2$ for each coefficient.
}
\tablenotetext{e}{
95\% confidence upper limit on the eccentricity determined when $\sqrt{e}\cos\omega$ and $\sqrt{e}\sin\omega$ are allowed to vary in the fit.
}
\tablenotetext{f}{
Term added in quadrature to the formal RV uncertainties for each
instrument. It is a free parameter in the fitting routine.
}
\tablenotetext{g}{
Correlation coefficient between the planetary mass \ensuremath{M_{p}}\ and radius
\ensuremath{R_{p}}\ estimated from the posterior parameter distribution.
}
\tablenotetext{h}{
The Safronov number is given by $\Theta = \frac{1}{2}(V_{\rm
esc}/V_{\rm orb})^2 = (a/\ensuremath{R_{p}})(\ensuremath{M_{p}} / \ensuremath{M_\star} )$
\citep[see][]{hansen:2007}.
}
\tablenotetext{i}{
Incoming flux per unit surface area, averaged over the orbit.
}
\end{deluxetable}
\section{Discussion and Conclusions}\label{sec:disc}
\begin{figure}[htb]
\centering
\includegraphics[width=1\hsize]{mass_radius_period_radius_mass_starmass_061_5205.pdf}
\caption{{\em Top}: Mass-radius diagram for M-dwarf planets with masses and radii measured to better than 25\%, as reported in TEPCAT. The markers are colour-coded by host star mass. TOI-3235 b and its analogue TOI-5205 b are plotted with star and hexagon symbols respectively and labelled. Theoretical mass-radius curves from \cite{mordasini:2012} are plotted with dashed and dotted lines. {\em Centre}: Period-radius diagram for the same planets. The markers are scaled by planet mass and colour-coded by equilibrium temperature (black when it could not be computed). TOI-3235 b and TOI-5205 b are labelled. {\em Bottom}: Planet mass vs. stellar mass diagram for the same planets. The markers are colour-coded by host star metallicity. TOI-3235 b and TOI-5205 b are plotted with star and hexagon symbols respectively and labelled.}
\label{fig:rad-mass-period}
\end{figure}
TOI-3235 b is a close-in Jupiter with $\ensuremath{M_{p}} = \hatcurPPm \, \ensuremath{M_{\rm J}}$, $\ensuremath{R_{p}} = \hatcurPPr \, \ensuremath{R_{\rm J}}$, orbiting a $\hatcurISOm \, \ensuremath{M_\sun}$ M-dwarf with a period of $\hatcurLCP$ days. To place it in the context of the M-dwarf planet population, in Figure \ref{fig:rad-mass-period} we plot TOI-3235 b together with all other well-characterized planets from the TEPCAT catalogue \citep{southworth:2011} hosted by stars with $\ensuremath{M_\star} \leq 0.61\, \ensuremath{M_\sun}$ (limit chosen to include $\ensuremath{M_\star} \approx 0.6\, \ensuremath{M_\sun}$ stars on K-M boundary). In mass-radius space (Fig. \ref{fig:rad-mass-period}, top panel), TOI-3235 b joins a small cluster of ten giant planets with $0.8\, \ensuremath{R_{\rm J}} \leq \ensuremath{R_{p}} \leq 1.2\, \ensuremath{R_{\rm J}}$, and $0.3\, \ensuremath{M_{\rm J}} \leq \ensuremath{M_{p}} \leq 1.5\, \ensuremath{M_{\rm J}}$. Most of these planets (HATS-6 b, \citealt{hartman:2015:hats6}; HATS-74 b and HATS-75 b, \citealt{jordan:2022}; Kepler-45 b, \citealt{johnson:2012}; TOI-530 b, \citealt{gan:2022:toi530}; TOI-3714 b, \citealt{canas:2022}; WASP-80 b, \citealt{triaud:2013}) are hosted by early M-dwarfs with $\ensuremath{M_\star} \geq 0.5\, \ensuremath{M_\sun}$. The sole other exception, aside from TOI-3235 b itself, is TOI-5205 b, a Jupiter-sized planet transiting a $0.392\, \ensuremath{M_\sun}$ star \citep{kanodia:2022}. Save for TOI-530 b, which has a somewhat longer period of $\mathrm{6.39\,d}$, these planets also cluster together in period-radius space (Fig. \ref{fig:rad-mass-period}, centre panel), forming a group of mid-range close-in Jupiters with periods between 1.63 and 3.33 days. Likewise, all their host stars except WASP-80 are metal-rich (Fig. \ref{fig:rad-mass-period}, bottom panel), in contrast to the wide range of metallicities shown by the host stars of the lower-mass planets. In particular, TOI-3235 has a metallicity of 0.26. These higher metallicities are consistent with previous findings \citep[e.g.][]{johnson:2009,rojas-ayala:2010} that M-dwarfs hosting giant planets tend to be metal-rich. The stellar mass vs planet mass diagram shown in this last panel also highlights the uniqueness of TOI-3235 b and its near twin TOI-5205 b, which inhabit an otherwise empty region of this parameter space.
Despite their clustering in mass-radius space, this group of giant planets spans a fairly wide range of densities, ranging from the very low-density HATS-6 b ($\mathrm{\rho \approx 0.4\, g/cm^3}$) to the Jupiter-analogue TOI-5205 b ($\mathrm{\rho \approx 1.3\, g/cm^3}$) and the high-density HATS-74 b ($\mathrm{\rho \approx 1.6\, g/cm^3}$). TOI-3235 b sits in the centre of the range, with $\mathrm{\rho \approx 0.78\, g/cm^3}$, comparable to the density of Saturn. They are all close to the peak of the theoretical mass-radius relationship derived by \cite{mordasini:2012} (Fig. \ref{fig:rad-mass-period}, top panel, where we show the relationships for both the full synthetic population, and for planets with $\mathrm{a < 1 au}$); save for TOI-530 b, HATS-74 b, and HATS-75 b, which sit on the curve for planets with $\mathrm{a < 1 au}$, and TOI-5205 b, which is consistent with it within error bars, all have larger radii than predicted. However, although all ten of these giants have periods shorter than the typical $\mathrm{10\, d}$ limit taken for hot Jupiters, they have equilibrium temperatures of $\mathrm{\approx 600-900\, K}$(Fig. \ref{fig:rad-mass-period}, centre panel), well below the $\mathrm{1000\, K}$ limit at which the incident flux is expected to begin to inflate the radii \citep{miller:2011, demory:2011, sarkis:2021}. It is also interesting to note that the low-mass planets generally have smaller radii than predicted, suggesting the theoretical relationship - derived from a synthetic population with a fixed stellar mass of $\ensuremath{M_\star} = 1\ensuremath{M_\sun}$ - may not be a good fit to M-dwarf planets overall.
The similarities between these giant planets may point to similar formation and migration histories. However, the differences in host star mass indicate caution; we may be seeing two distinct populations, one corresponding to early M-dwarfs and one corresponding to later M-dwarfs. It is thus particularly interesting and relevant to compare TOI-3235 b to TOI-5205 b. Like TOI-5205, TOI-3235 sits on the edge of the Jao Gap, a narrow gap in the Hertzsprung–Russell Diagram first identified by \cite{jao:2018} in Gaia data and linked by the authors to the transition between partially and fully convective stars, with $\mathrm{M_G = 10.04 \pm 0.95}$ \citep{anders:2022}, and $\mathrm{B_P - R_P \approx 2.6}$ \citep{gaiadr3}. Both these stars are therefore in the transition region between partially and fully convective M-dwarfs, and as such are likely to undergo periodic changeovers from partially to fully convective and vice versa, that alter their radius and luminosity \citep[e.g.][]{vansaders:2012, baraffe:2018}. As noted by \cite{kanodia:2022}, these oscillations may impact the planetary orbital parameters and equilibrium temperature. It is possible that the similar planets of these similar stars may share similar formation and/or evolution histories. \cite{kanodia:2022} studied the disk mass necessary to form TOI-5205 b. Since the host stars have the same mass, we can extrapolate from their analysis; the main difference is that TOI-3235 b is rather less massive than TOI-5205 b. As regards planetary heavy-element mass, using the relations of \cite{thorngren:2016} we find a heavy-element mass of $\mathrm{M_Z \sim 45 M_\oplus}$, corresponding to 75\% of that of TOI-5205 b; therefore, assuming a solid core and scaling the results of \cite{kanodia:2022}, the required disk mass for TOI-3235 b becomes $\sim 2\%- 23\%$ the mass of the host star for $100\%-10\%$ formation efficiency respectively. While lower than the disk mass required to explain TOI-5205 b, given typical disk masses of the order of $\sim 0.1-5\%$ \citep{pascucci:2016}, the formation of TOI-3235 b still requires either an extremely high formation efficiency or a very massive disk.
TOI-3235 b also shows high potential for atmospheric characterization. We compute a Transmission Spectroscopy Metric (TSM, \citealt{kempton:2018}) of $\approx 160$, assuming a scale factor of 1.15. Comparing it to the group of M-dwarf planets it clusters with in mass-period-radius space, TOI-3235 b has the second-highest TSM, surpassed only by WASP-80 b (TSM $\approx 290$), and notably higher than its analogue TOI-5205 b (TSM $\approx 100$). Atmospheric characterization can help place constraints on the formation and migration history \citep[e.g.][]{hobbs:2022, molliere:2022} of this unexpected planet.
\vspace{5mm}
\facilities{\textit{TESS}, HATSouth, MEarth-South, TRAPPIST-South, LCOGT, ExTrA, ESPRESSO, Gaia, Exoplanet Archive}
\software{FITSH \citep{pal:2012}, BLS \citep{kovacs:2002:BLS}, VARTOOLS \citep{hartman:2016:vartools}, CERES \citep{brahm:2017:ceres}, ZASPE \citep{brahm:2017:zaspe}, ODUSSEAS \citep{antoniadis:2020}, AstroImageJ \citep{Collins:2017}, TAPIR \citep{Jensen:2013}}
\vspace{5mm}
We thank the referee for their helpful comments that improved this paper.
This paper includes data collected by the \textit{TESS} mission, which are publicly available from the Mikulski Archive for Space Telescopes (MAST). The specific observations analyzed can be accessed via \dataset[10.17909/mdsd-2297]{https://doi.org/10.17909/mdsd-2297}. Funding for the \textit{TESS} mission is provided by NASA's Science Mission directorate.
This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.
We acknowledge the use of public \textit{TESS} data from pipelines at the \textit{TESS} Science Office and at the \textit{TESS} Science Processing Operations Center.
Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products.
Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme 0108.C-0123(A).
A.J., R.B. and M.H. acknowledge support from ANID - Millennium Science Initiative - ICN12\_009. A.J. acknowledges additional support from FONDECYT project 1210718. R.B. acknowledges support from FONDECYT Project 1120075 and from project IC120009 “Millennium Institute of Astrophysics (MAS)” of the Millenium Science Initiative. This work was funded by the Data Observatory Foundation.
The MEarth Team gratefully acknowledges funding from the David and Lucile Packard Fellowship for Science and Engineering (awarded to D.C.). This material is based upon work supported by the National Science Foundation under grants AST-0807690, AST-1109468, AST-1004488 (Alan T. Waterman Award), and AST-1616624, and upon work supported by the National Aeronautics and Space Administration under Grant No. 80NSSC18K0476 issued through the XRP Program. This work is made possible by a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.
The research leading to these results has received funding from the ARC grant for Concerted Research Actions, financed by the Wallonia-Brussels Federation. TRAPPIST is funded by the Belgian Fund for Scientific Research (Fond National de la Recherche Scientifique, FNRS) under the grant PDR T.0120.21. MG is F.R.S.-FNRS Research Director and EJ is F.R.S.-FNRS Senior Research Associate. Observations were carried out from ESO La Silla Observatory.
The postdoctoral fellowship of KB is funded by F.R.S.-FNRS grant T.0109.20 and by the Francqui Foundation.
This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is funded by NSF.
Based on data collected under the ExTrA project at the ESO La Silla Paranal Observatory. ExTrA is a project of Institut de Plan\'etologie et d'Astrophysique de Grenoble (IPAG/CNRS/UGA), funded by the European Research Council under the ERC Grant Agreement n. 337591-ExTrA.
This work has been supported by a grant from Labex OSUG$@$2020 (Investissements d'avenir -- ANR10 LABX56).
This work has been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation.
This work has been carried out within the framework of the National Centre of Competence in Research PlanetS supported by the Swiss National Science Foundation under grants 51NF40\textunderscore182901 and 51NF40\textunderscore205606. The authors acknowledge the financial support of the SNSF.
This publication benefits from the support of the French Community of Belgium in the context of the FRIA Doctoral Grant awarded to MT.
The contributions at the Mullard Space Science Laboratory by E.M.B. have been supported by STFC through the consolidated grant ST/W001136/1.
|
{
"timestamp": "2023-02-21T02:28:36",
"yymm": "2302",
"arxiv_id": "2302.10008",
"language": "en",
"url": "https://arxiv.org/abs/2302.10008"
}
|
\section{Introduction}
The safe interaction of traffic participants in urban environments is based on different rules like signs or the right of way. Besides static regulations, more dynamic ones like gestures are possible. For example, a police officer manages the traffic~\cite{wiederer2020traffic} by hand gestures, or a pedestrian waves a car through at a crosswalk~\cite{rasouli2019autonomous}. Although the driver intuitively knows the meaning of human gestures, an autonomous vehicle cannot interpret them. To safely integrate the autonomous vehicle into urban traffic, it is essential to understand human gestures.
To detect human gestures, many related approaches only rely on camera data~\cite{baek2022traffic,pham2021efficient}. Camera-based gesture recognition is not always reliable, for instance, due to the weather sensitivity of the camera sensor~\cite{9011192}. One way to mitigate the drawbacks of camera-based approaches is the augmentation of the gesture recognition system by non-optical sensor types. A sensor with less sensitivity to environmental conditions is a radar sensor~\cite{qian2021robust} that has been shown to be suited for gesture recognition. Furthermore, radar sensors are not limited to detecting fine-grained hand gestures but can also predict whole body gestures~\cite{kern2022pointnet+}. Hence, radar sensors are a promising candidate to complement optical sensors for more reliable gesture recognition in automotive scenarios. This has been demonstrated for gesture recognition with camera-radar fusion in close proximity to the sensors, as presented in~\cite{molchanov2015multi}. Furthermore, depending not only on one sensor can increase the system's reliability in case of a sensor failure. In contrast to the prior work, we propose a method for whole-body gesture recognition at larger distances and with a camera-radar fusion approach.
We present a two-stream neural network to realizing camera and radar fusion. Our approach extracts independently a representation for each modality and then fuses them with an additional network module. In the first stream, the features of the unordered radar targets are extracted with a PointNet~\cite{qi2017pointnet} and further prepared for fusion with an spatio-temporal multilayer perceptron (stMLP)~\cite{holzbock2022spatio}. The second stream uses only a stMLP to process the keypoint data. The extracted features of each stream are fused for the classification in an additional stMLP. We train the proposed network with an auxiliary loss function for each modality to improve the feature extraction through additional feedback. For training, we use data containing radar targets of three chirp sequence (CS) radar sensors and human keypoints extracted from camera images. The eight different gestures are presented in Fig.~\ref{fig:gesture_set} and represent common traffic gestures. The performance of our approach is tested in a cross-subject evaluation. We improve performance by 3.8 percentage points compared to the keypoint-only setting and 8.3 percentage points compared to the radar-only setting. Furthermore, we can show robustness against the failure of one sensor.
Overall, we propose a two-stream neural network architecture to fuse keypoint and radar data for gesture recognition. To process the temporal context in the model, we do not rely on recurrent models~\cite{9196702} but instead propose the stMLP fusion. In the training of our two-stream model, we introduce an auxiliary loss for each modality. To the best of our knowledge, we are the first to fuse radar and keypoint data for gesture recognition in autonomous driving.
\newcommand\imgw{0.173}
\begin{figure}[!ht]
\centering
\subfloat[]{
\centering
\begin{minipage}[b]{\imgw\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/img_g0_start.png}%
\hfill
\vspace{1em}
\includegraphics[width=1.0\linewidth]{imgs/img_g0_end.png}%
\end{minipage}%
}
\quad
\subfloat[]{
\begin{minipage}[b]{\imgw\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/img_g1_start.png}%
\hfill
\vspace{1em}
\includegraphics[width=1.0\linewidth]{imgs/img_g1_end.png}%
\end{minipage}%
}
\quad
\subfloat[]{
\begin{minipage}[b]{\imgw\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/img_g2_start.png}%
\hfill
\vspace{1em}
\includegraphics[width=1.0\linewidth]{imgs/img_g2_end.png}%
\end{minipage}%
}
\quad
\subfloat[]{
\begin{minipage}[b]{\imgw\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/img_g3_start.png}%
\hfill
\vspace{1em}
\includegraphics[width=1.0\linewidth]{imgs/img_g3_end.png}%
\end{minipage}%
}
\quad
\par\smallskip
\subfloat[]{
\begin{minipage}[b]{\imgw\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/img_g4_start.png}%
\hfill
\vspace{1em}
\includegraphics[width=1.0\linewidth]{imgs/img_g4_end.png}%
\end{minipage}%
}
\quad
\subfloat[]{
\begin{minipage}[b]{\imgw\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/img_g5_start.png}%
\hfill
\vspace{1em}
\includegraphics[width=1.0\linewidth]{imgs/img_g5_end.png}%
\end{minipage}%
}
\quad
\subfloat[]{
\begin{minipage}[b]{\imgw\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/img_g6_start.png}%
\hfill
\vspace{1em}
\includegraphics[width=1.0\linewidth]{imgs/img_g6_end.png}%
\end{minipage}%
}
\quad
\subfloat[]{
\begin{minipage}[b]{\imgw\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{imgs/img_g7_start.png}%
\hfill
\vspace{1em}
\includegraphics[width=1.0\linewidth]{imgs/img_g7_end.png}%
\end{minipage}%
}
\caption{Visualization of the characteristic poses of the eight gestures. (a)~Fly. (b)~Come closer. (c)~Slow down. (d)~Wave. (e)~Push away. (f)~Wave through. (g)~Stop. (h)~Thank you.}
\label{fig:gesture_set}
\end{figure}
\section{Related Work}
In the following, we discuss other methods regarding gesture recognition in general as well as in autonomous driving. We give an overview of approaches relying only on keypoint or radar data. Besides the single sensor gesture recognition, we present methods using a combination of different sensors.
\paragraph{\textbf{Gesture Recognition with Camera Data}}
Gesture recognition is applied to react to humans outside the vehicle~\cite{xu2021action,pham2021efficient} and to the passenger's desires~\cite{holzbock2022spatio,wharton2021coarse}. For gesture recognition of police officers, related work processes camera image snippets~\cite{baek2022traffic} directly. Due to the advances in human body pose estimation~\cite{9667074,belagiannis2014holistic}, related approaches extract the body skeleton data from the images and perform gesture recognition on it~\cite{wang2021simple,pham2021efficient,mishra2021authorized}. For processing skeletons to predict the gesture, recurrent neural networks~\cite{wang2021simple} or convolutional neural networks~\cite{pham2021efficient} are applied. Besides the police officer gesture recognition, the actions of other human traffic participants like cyclists~\cite{xu2021action} or pedestrians~\cite{geng2020using} are also analyzed in literature. Similar to pedestrian gesture recognition is the pedestrian intention prediction~\cite{quintero2017pedestrian,abughalieh2020predicting}, where the pedestrian's intention to cross the street should be recognized. Changing the view to the interior of the car, there are approaches to recognize the driver's activities in order to check if the driver is focused on the traffic. For this purpose, methods like attention-based neural networks~\cite{wharton2021coarse} or models only built on multi-layer perceptrons~\cite{holzbock2022spatio} are developed. Compared to our work, lighting and weather conditions influence the performance of camera-based approaches. Due to the fusion with the radar sensor data, our method mitigate these factors.
\paragraph{\textbf{Gesture Recognition with Radar Data}}
Besides radar sensors' insensitivity to adverse weather and lighting, they also evoke fewer privacy issues than cameras. As a result, radar-based gesture recognition has received increased attention in recent years, with research efforts mainly devoted to human-machine interaction in the consumer electronics area \cite{lien2016soli}. For the control of devices with hand gestures, gesture recognition algorithms based on a wide range of neural networks have been proposed, involving, e.g., 2D-CNNs \cite{kim2016hand}, 2D-CNNs with LSTMs \cite{wang2016interacting}, or 3D-CNNs with LSTM \cite{zhang2018latern}. These approaches exploit spectral information in the form of micro-Doppler spectrograms \cite{kim2016hand} or range-Doppler spectra \cite{wang2016interacting}, but it is also possible to distinguish between gestures \cite{kern2022pointnet+} and activities \cite{singh2019radhar} using radar point clouds. The latter are a more compact representation of the radar observations and are obtained by finding valid targets in the radar data. Point clouds facilitate the application of geometrical transformations \cite{singh2019radhar} as well as the inclusion of additional information \cite{kern2022pointnet+}. While most research considers small-scale gesture recognition close to the radar sensor, reliable macro gesture recognition at larger distances has been also shown to be feasible with radar sensors for applications such as smart homes \cite{ninos2021real} or traffic scenarios \cite{kern2020robust,kern2022pointnet+}. While radar-only gesture recognition has shown promising results, augmenting it by camera data can further improve classification accuracy, as demonstrated in this paper. This is particularly important in safety-critical applications such as autonomous driving.
\paragraph{\textbf{Gesture Recognition with Sensor Fusion}}
By combining data of multiple sensors, sensor fusion approaches can overcome the drawbacks of the individual sensor types, like the environmental condition reliance of the camera or its missing depth information. Sensor fusion has been applied e.g. for gesture-based human machine interaction in vehicles, where touchless control can increase safety ~\cite{molchanov2015hand}. Besides the image information for gesture recognition, the depth data contains essential knowledge. Therefore, other methods fuse camera data with the data of a depth sensor~\cite{ohn2014hand} and process the data with a 3D convolutional neural network~\cite{molchanov2015hand,zengeler2018hand}. Molchanov et al.~\cite{molchanov2015multi} run a short-range radar sensor next to the RGB camera and depth camera and process the recorded data with a 3D convolutional neural network for a more robust gesture recognition. Another approach fuses the short-range radar data with infrared sensor data instead of RGB images~\cite{skaria2021radar}. Current fusion approaches are not only limited to short ranges and defined environments but can also be applied in more open scenarios like surveillance~\cite{de2021classification} or smart home applications~\cite{vandersmissen2020indoor}. Here, camera images and radar data, converted to images, are used to classify the gesture, while we use the skeletons extracted from the images and the radar targets described as unstructured point clouds. Moreover, contrary to~\cite{vandersmissen2020indoor}, our fusion approach enhances the gesture recognition accuracy not only in cases where one modality is impaired but also in normal operation.
\section{Fusion Method}
We present a fusion technique for radar and keypoint data for robust gesture recognition. To this end, we develop a neural network ${\mathbf{\hat{y}} = f((\mathbf{x}_R, \mathbf{x}_K), \theta)}$ defined by its parameters $\theta$. The output prediction $\mathbf{\hat{y}}$ is defined as one-hot vector ${\mathbf{\hat{y}}_i \in \{0,1\}^{C}}$, such that $\sum_{c=1}^{C}\mathbf{\hat{y}}_i(c)=1$ for a C-category classification problem. The radar input data is represented by ${\mathbf{x}_R \in \mathbb{R}^{T \times 5 \times 300}}$, where T is the number of time steps and 300 is the number of sampled radar targets in each time step, each of which is described by 5 target parameters. For the keypoint stream, 17 2D keypoints are extracted and flattened in each time step, such that the keypoint data is described by ${\mathbf{x}_K \in \mathbb{R}^{T \times 34}}$. We use the ground truth class label $\mathbf{y}$ for the training of $f(.)$ to calculate the loss of the model prediction $\mathbf{\hat{y}}$. In the following, we first present the architecture of our neural network and afterward show the training procedure.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.53\textwidth]{imgs/architecture_overview_stmlp.pdf}
\caption{Architecture of the proposed neural network for gesture recognition with radar and keypoint data. The blue layers correspond to the inference neural network while the red layers are only used to compute the auxiliary losses.}
\label{fig:overview_model}
\end{figure}
\subsection{Neural Network Architecture}
\label{subsec:architecture}
The neural network inputs are the radar data $\mathbf{x}_R$ and the keypoint data $\mathbf{x}_K$. The network consists of two different streams, one for each modality, to extract the features of the different modalities. The information of the two streams is concatenated and fed to a joint network which does the fusion and the gesture classification. Additionally, auxiliary outputs are added to the model for the training procedure. An overview of the proposed network architecture is given in Fig.~\ref{fig:overview_model}.
The feature extraction from the unordered radar data ${\mathbf{x}_R}$ is done with a PointNet~\cite{qi2017pointnet} that extracts the features for each time step ${t \in \{1, 2, \dots, T\}}$. The features of the time steps are concatenated to one feature tensor ${\mathbf{x}_{R,PN} \in \mathbb{R}^{T \times 512}}$. The PointNet does not process the data in the temporal dimension but only in the point dimension. For the temporal processing, we use the stMLP model~\cite{holzbock2022spatio}, which replaces a standard method for temporal data processing, e.g. a long short-term memory model (LSTMs)~\cite{hochreiter1997long}. Unlike LSTMs, the stMLP is based solely on multilayer perceptrons (MPLs) and does not have any recurrent parts. The stMLP processes the radar data $\mathbf{x}_{R,PN}$ extracted with the PointNet in the temporal and feature dimensions and outputs mixed radar features ${\mathbf{\tilde{x}}_{R} \in \mathbb{R}^{T \times H/2}}$, where $H$ is the hidden dimension. Due to the ordered structure of the keypoint data $\mathbf{x}_K$, the mixed keypoint features ${\mathbf{\tilde{x}}_K \in \mathbb{R}^{T \times H/2}}$ are extracted only with an stMLP model, which processes the extracted keypoint features in both dimensions, namely the spatial and the temporal dimension. The extracted features of both modalities, $\mathbf{\tilde{x}}_R$ and $\mathbf{\tilde{x}}_K$, are concatenated to a single feature tensor ${\mathbf{\tilde{x}} \in \mathbb{R}^{T \times H}}$ and fed into another stMLP model. This model performs a spatial and temporal fusion of the radar and keypoint features to produce a meaningful representation for the gesture classification step. The gesture classification is performed by a Layer Normalization~\cite{ba2016layer} and a single linear layer for each time step.
Additionally, we add for each modality an auxiliary output during training~\cite{szegedy2015going}. The auxiliary output of both modalities is built on a Layer Normalization and a linear layer for the auxiliary classification. In Fig.~\ref{fig:overview_model}, the parts of the auxiliary outputs and the corresponding layers are drawn in red. For the gesture prediction, each auxiliary output only uses one modality and no information from the other. This means that the auxiliary radar output $\mathbf{\hat{y}}_R$ gets the extracted radar features $\mathbf{\tilde{x}}_R$ for the prediction and the auxiliary keypoint output $\mathbf{\hat{y}}_K$ the keypoint data features $\mathbf{\tilde{x}}_K$. For the model, the auxiliary outputs give additional specialized feedback for updating the network parameters in the radar and the keypoint stream. After training, the layers of the auxiliary outputs can be removed.
\subsection{Model Training}
For the training of the neural network introduced in Sec.~\ref{subsec:architecture} we utilize the data presented in Sec.~\ref{subsec:dataset}. During training, we use the fused output of the model $\mathbf{\hat{y}}$ and the auxiliary outputs derived from the radar $\mathbf{\hat{y}}_R$ and keypoint data $\mathbf{\hat{y}}_K$. For each output, we calculate the cross entropy loss $\mathcal{L}_{CE}$, which can be formulated as
\begin{equation}
\mathcal{L}_{CE} = - \sum^{C}_{c=0}y_c \log (\hat{y}_c),
\end{equation}
where $y$ is the ground truth label, $\hat{y}$ the network prediction, and $C$ the number of gesture classes. We use a weighted sum of the different sub-losses to get the overall loss $\mathcal{L}$ which can be expressed by
\begin{equation}
\label{eq:loss}
\mathcal{L} = \mathcal{L}_F + \mu * ( \mathcal{L}_R + \mathcal{L}_K ).
\end{equation}
In the overall loss, $\mathcal{L}_F$ is the loss of the fused output, $\mathcal{L}_R$ of the auxiliary radar output, and $\mathcal{L}_K$ of the auxiliary keypoint output. The auxiliary losses are weighted with the auxiliary loss weight $\mu$.
\section{Results}
We evaluate our approach in two different settings. First, we test our approach with two modalities, assuming no issue with the sensors and getting data from both. Second, we use only one modality to evaluate the model. Only having one modality can be motivated by adverse weather conditions, e.g. the camera sensor is completely covered by snow or technical problems with one sensor.
\subsection{Dataset}
\label{subsec:dataset}
For the development of robust gesture recognition with the fusion of radar and keypoint data in the context of autonomous driving, we need a dataset that contains both modalities captured synchronously and at sufficient ranges. Consequently, the custom traffic gesture dataset introduced in~\cite{kern2022pointnet+} with both camera and radar data is used. Following, we specify the setup and the data processing of the dataset.
\paragraph{\textbf{Setup}}
The gesture dataset comprises measurements of eight different gestures shown in Fig.~\ref{fig:gesture_set} for 35 participants. The measurements are conducted on a small street on the campus of Ulm University as well as inside a large hall resembling a car park. For each participant, data recording is repeated multiple times under different orientations, and the measurements are labeled by means of the camera data. For the measurements, a setup consisting of three chirp sequence (CS) radar sensors and a RGB camera as illustrated by the sketch in Fig.~\ref{fig:sensor_setup} are used. The radar sensors operate in the automotive band at \SI{77}{\giga\hertz}. Each sensor has a range and velocity resolution of \SI{4.5}{\centi\meter} and \SI{10.7}{\centi\meter\per\second}, respectively, and eight receive channels for azimuth angle estimation. The camera in the setup has a resolution of 1240$\times$1028, and keypoints obtained from the camera serve as input to the keypoint stream of the proposed gesture recognition algorithm. All sensors are mounted on a rail and synchronized with a common trigger signal with \SI{30}{\fps}.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.7\linewidth]{imgs/overview_schematic.png}
\caption{Measurement system consisting of the RGB camera and three CS radar sensors. All sensors receive a common trigger signal for time synchronization.}
\label{fig:sensor_setup}
\end{figure}
\paragraph{\textbf{Data Processing}}
From the gesture recordings, per-frame radar point clouds and 2D skeletal keypoints are computed. The radar responses recorded by the sensors are processed sensor-wise to obtain range-Doppler maps \cite{Winkler.2007}. The Ordered Statistics CFAR algorithm \cite{Rohling.1983} is applied to extract valid targets and thereby compress the information in the range-Doppler maps. For each radar target, its azimuth angle is estimated by digital beamforming \cite{Vasanelli.2020}. The target parameters are normalized by the radar sensors' unambiguous ranges. Each target in the resulting target lists is described by its range, velocity, azimuth angle, its reflected power in \si{\decibel}, and the index ${i_n \in {0,1,2}}$ of the radar sensor that detected it. Since the number of detected targets varies from frame to frame, target lists with a constant number $N_R$ of targets are sampled randomly for frames whose target count exceeds $N_R$. Contrary, for frames where the number of targets is less than $N_R$, zero-padding is applied to fill the target lists. The target lists of the radar sensors are stacked, such that the final target list contains ${3N_R}$ targets. After repeating the processing for all $T_M$ frames in the measurement, the radar observations are described by input data of shape ${\mathbf{x}_R \in \mathbb{R}^{T_M \times 5 \times 300}}$, when setting $N_R$ to 100.
The camera data is processed by Detectron2~\cite{wu2019detectron2} to extract 17 keypoints in the COCO keypoint format~\cite{Lin.2014}. The extracted keypoints are normalized with the image width and height to restrict the keypoint values for the neural network to the range between 0 and 1. The keypoints' 2D pixel positions over the frames are summarized in the camera observation tensor ${\mathbf{o}_K \in \mathbb{R}^{T_M \times 2 \times 17}}$, which is flattened to the keypoint data ${\mathbf{x}_K \in \mathbb{R}^{T_M \times 34}}$. After the signal processing, the radar and camera data of the measurements are downsampled to \SI{15}{\fps} and segmented into smaller snippets with ${T=30}$ time steps each, corresponding to \SI{2}{\second} of observation. Finally, 15700 samples are available for training the gesture recognition model. In the cross-subject evaluation, we use the data of 7 subjects for testing and the data of the remaining subjects as training and validation data. We train the model 5 times, using different subjects for testing in each run and average over the 5 runs for the final performance.
\subsection{Experimental Setup}
\label{subsec:setup}
The neural network is implemented with the PyTorch~\cite{NEURIPS2019_9015} framework, and we use PointNet\footnote{https://github.com/fxia22/pointnet.pytorch} and stMLP\footnote{https://github.com/holzbock/st\_mlp} as a base for our new fusion architecture. In the PointNet, we apply the feature transformation and deactivate the input transformation. In each stMLP structure we use 4 mixer blocks and set the stMLP hyperparameters as follows: the hidden input dimension to 256, the hidden spatial-mixing dimension to 64, and the hidden temporal-mixing dimension to 256. The loss is calculated with the function defined in Eq.~\ref{eq:loss}, where we set the auxiliary weight $\mu$ to 0.5. We train our model for 70 epochs with a batch size of 32 and calculate the gradients with the SGD optimizer that uses a learning rate of 0.003, a momentum of 0.95, and a weight decay of 0.001. To get the best training result, we check the performance during the training on the validation set and use the best validation epoch for testing on the test set. The model performance is measured with the accuracy metric. To prepare the neural network for missing input data, we skip in 30\% of the training samples the radar or the keypoint data, which we call skipped-modality training (SM-training). When reporting the results with a single-modality model, we remove the layers for the other modality.
\subsection{Results}
During testing with optimal data we assume that we get the keypoint data from the camera sensor and the targets from the radar sensors, i.e., we have no samples in the train and test set that only contain one modality's data. The results are shown in Tab.~\ref{table:correct_data}, where we compare with a model using an LSTM instead of an stMLP for temporal processing. In the \textit{Single Modality} part of the table, we show the performance of our architecture trained and tested on only one modality. In these cases, the layers belonging to the other modality are removed. As it can be seen, the LSTM and the stMLP model are on par when training only with the radar data, and the stMLP model is better than the LSTM model trained only with the keypoint data.
In the \textit{Fusion} part of Tab.~\ref{table:correct_data}, we show the fusion performance of our architecture. We first train the stMLP and the LSTM fusion with all the training data and then apply SM-training with a ratio of 30\%. The SM-training means that we skip the radar or the keypoint data in 30\% of the training samples. The stMLP fusion performs better in both cases compared to the LSTM fusion. Furthermore, skipping randomly one modality in 30\% of the training data slightly benefits the test performance. Overall, the stMLP architecture performs better in the fusion of the radar and the keypoint data than the LSTM architecture. Compared to the single-modality model, the fusion improves the performance in both architectures (LSTM and stMLP) by over 4 percentage points. This shows that the fusion of the keypoint and radar data for gesture recognition benefits the performance compared to single-modality gesture recognition.
\setlength{\tabcolsep}{4pt}
\begin{table}[ht]
\begin{center}
\caption{Performance of our model trained with both modalities. \textit{Single Modality} is a model that only contains layers for one modality and is trained and tested only with this modality. The \textit{Fusion} rows use our proposed model. \textit{SM-training} means that we skip one modality in 30\% of the training samples during the training.}
\label{table:correct_data}
\begin{tabular}{lC{1.5cm}C{1.4cm}C{1.4cm}C{1.4cm}C{1.4cm}C{1.4cm}}
\toprule
\multirow{2}{*}{Model} & Temporal & \multicolumn{2}{c}{Test data} & SM- & Accuracy \\
& Processing & Radar & Keypoint & training & in \% \\
\midrule
\multirow{4}{*}{Single Modality} & LSTM & \xmark & \cmark & \xmark & 86.2 \\
& stMLP & \xmark & \cmark & \xmark & 89.9 \\
& LSTM & \cmark & \xmark & \xmark & 85.7 \\
& stMLP & \cmark & \xmark & \xmark & 85.4 \\
\midrule
\multirow{4}{*}{Fusion} & LSTM & \cmark & \cmark & \xmark & 90.2 \\
& stMLP & \cmark & \cmark & \xmark & 93.7 \\
& LSTM & \cmark & \cmark & \cmark & 90.7 \\
& stMLP & \cmark & \cmark & \cmark & 93.8 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\subsection{Results with Single Modality}
\label{subsec:corrupted_data}
When we test our architecture with single modality, during testing only the radar or the keypoint data are fed into the model, but not both. This can be compared with one fully corrupted sensor due to adverse weather or a technical problem. Contrary, during training we either use all training data (no SM-training) or set the SM-training to a ratio of 30\%, such that in 30\% of samples one modality is skipped. In the first part of Tab.~\ref{table:corrupted_data}, results are shown for the model trained without the skipped modalities, which means that the model has not learned to handle single-modality samples. Despite the missing modality in the evaluation, the model can still classify the gestures, but with a decreased accuracy. Comparing the LSTM and stMLP variants, the LSTM variant performs better with only the radar data and the stMLP with only the keypoint data.
In the second part of Tab.~\ref{table:corrupted_data}, we show the performance of our model trained with skipped modalities in 30\% of the training samples (SM-training is 30\%). When the model is trained with the skipped modalities, it learns better to handle missing modalities. This results in a better performance in the single-modality evaluation, and we can improve the accuracy by a minimum of 3 percentage points compared to the model trained without the skipped modalities. Overall, the fusion can improve the reliability of gesture recognition in cases where one sensor fails e.g. due to adverse weather conditions or a technical sensor failure.
\setlength{\tabcolsep}{4pt}
\begin{table}[ht]
\begin{center}
\caption{Performance of our model with single-modality data. The \textit{Fusion} rows use our proposed model. \textit{SM-training} means that we skip one modality in 30\% of the training samples during the training.}
\label{table:corrupted_data}
\begin{tabular}{lC{1.5cm}C{1.4cm}C{1.4cm}C{1.4cm}C{1.4cm}C{1.4cm}}
\toprule
\multirow{2}{*}{Model} & Temporal & \multicolumn{2}{c}{Test data} & SM- & Accuracy \\
& Processing & Radar & Keypoint & training & in \% \\
\midrule
\multirow{4}{*}{Fusion} & LSTM & \xmark & \cmark & \xmark & 69.7 \\
& LSTM & \cmark & \xmark & \xmark & 82.6 \\
& stMLP & \xmark & \cmark & \xmark & 87.7 \\
& stMLP & \cmark & \xmark & \xmark & 78.3 \\
\midrule
\multirow{4}{*}{Fusion} & LSTM & \xmark & \cmark & \cmark & 80.8 \\
& LSTM & \cmark & \xmark & \cmark & 85.0 \\
& stMLP & \xmark & \cmark & \cmark & 91.2 \\
& stMLP & \cmark & \xmark & \cmark & 83.0 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\section{Ablation Studies}
The performance of our model depends on different influence factors, which we evaluate in the ablation studies. As standard setting, we choose the hyperparameters defined in Sec.~\ref{subsec:setup} and change the ratio of the SM-training and the auxiliary loss weight during the ablations.
\subsection{Amount of Single-Modality Training Samples}
The training with missing data teaches the model to handle singe modality input data, as shown in Sec.~\ref{subsec:corrupted_data}. In our standard setting for the SM-training, we randomly skip one modality in 30\% of the training samples. In this ablation study, the SM-training ratio ranges from 0\% to 60\% during training and shows the influence of this parameter on the performance.
We test the model with different SM-training ratios with the test data containing both modalities and show the result in the \textit{Fusion} row of Tab.~\ref{table:ablation_sm}. Additionally, we test with the data of only one modality and present the results in the \textit{Only Keypoints} and \textit{Only Radar} row. The experiment shows that the SM-training does not significantly influence the performance when testing with both modalities. This is explainable because in the testing set no samples have lacking modalities. Testing with only one modality, the amount of skipped-modality samples during the training influences the accuracy. Here, the performance increases until the SM-training ratio reaches 30\% and then stays constant. In this case, the amount of skipped-modality samples during training influences the adaptation of the model to single-modality samples.
\setlength{\tabcolsep}{4pt}
\begin{table}[ht]
\begin{center}
\caption{Influence of the SM-training ratio on the overall accuracy.}
\label{table:ablation_sm}
\begin{tabular}{lC{.8cm}C{.8cm}C{.8cm}C{.8cm}C{.8cm}C{.8cm}C{.8cm}}
\toprule
Skip modality in \% & 0 & 10 & 20 & 30 & 40 & 50 & 60 \\
\midrule
Fusion & 93.7 & 92.8 & 92.1 & 93.8 & 94.1 & 93.2 & 93.6 \\
Only Keypoints & 87.7 & 89.9 & 89.8 & 91.2 & 91.4 & 91.0 & 91.5 \\
Only Radar & 78.3 & 81.7 & 81.1 & 83.0 & 83.0 & 80.7 & 83.2 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\subsection{Loss Function}
Besides the ratio of the SM-training, the auxiliary loss weight $\mu$ is an essential parameter in training. In our standard evaluation, we set $\mu$ to 0.5, while we vary the loss weight in this ablation study from 0 to 3.
The results of the different $\mu$ are shown in Tab.~\ref{table:ablation_lw}. In the first row, we deliver the results when testing with both modalities (\textit{Fusion}). In the \textit{Only Keypoints} and \textit{Only Radar} row, we present the performance when evaluating only one modality. As we can see in Tab.~\ref{table:ablation_lw}, the fusion results behave equally to the single-modality results. For the Fusion as well as for the single modality, a higher $\mu$ increases the performance, and the best accuracy is reached with $\mu = 0.8$. Further increasing $\mu$ leads to a decreasing performance in gesture recognition. Comparing the results of the best $\mu = 0.8$ with the model without the auxiliary loss ($\mu = 0.0$) shows that with the auxiliary loss, the fusion performance stays equal but the single-modality accuracy increases. This indicates that the model benefits from the additional feedback of the auxiliary loss during training.
\setlength{\tabcolsep}{4pt}
\begin{table}[ht]
\begin{center}
\caption{Influence of the loss weights on the overall performance.}
\label{table:ablation_lw}
\begin{tabular}{lC{.8cm}C{.8cm}C{.8cm}C{.8cm}C{.8cm}C{.8cm}C{.8cm}}
\toprule
Loss weight & 0.0 & 0.2 & 0.5 & 0.8 & 1.0 & 2.0 & 3.0 \\
\midrule
Fusion & 94.2 & 92.9 & 93.8 & 94.1 & 93.7 & 92.4 & 93.4 \\
Only Keypoints & 89.2 & 90.8 & 91.2 & 92.1 & 91.0 & 90.1 & 91.7 \\
Only Radar & 84.3 & 81.2 & 83.0 & 84.7 & 82.2 & 81.3 & 82.8 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\setlength{\tabcolsep}{1.4pt}
\section{Conclusion}
We present a novel two-stream neural network architecture for the fusion of radar and keypoint data to reliably classify eight different gestures in autonomous driving scenarios. The proposed fusion method improves the classification accuracy over the values obtained with single sensors, while enhancing the recognition robustness in cases of technical sensor failure or adverse environmental conditions. In the model, we first process the data of each modality on its own and then fuse them for the final classification. We propose a stMLP fusion which applies besides the fusion of the features of both modalities also temporal processing. Furthermore, for a better overall performance of our approach, we introduce an auxiliary loss in the training that provides additional feedback to each modality stream. The evaluation of our method on the radar-camera dataset, and we show that even with missing modalities, the model can reach a promising classification performance. In the ablation studies, we demonstrate the influence of the SM-training ratio and the auxiliary loss weight.
\section*{Acknowledgment}
Part of this work was supported by INTUITIVER (7547.223-3/4/), funded by State Ministry of Baden-Württemberg for Sciences, Research and Arts and the State Ministry of Transport Baden-Württemberg.
\clearpage
\bibliographystyle{splncs04}
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