| import OAI.Probability.InvariantIsing.Haar.HaarTwoSidedTail |
| import OAI.Probability.InvariantIsing.Gaussian.GaussianMedianTail |
|
|
| namespace OAI |
|
|
| /-! |
| The exact Gromov |
| uses the actual polynomial heat operator on SO(N), its proved curvature bound, |
| positive polynomial convolution, and the scalar entropy argument. |
| -/ |
| noncomputable section |
| open Matrix MeasureTheory |
| namespace InvariantIsing |
|
|
| theorem haarConcentration_proved : HaarConcentrationInput := by |
| refine ⟨Real.exp 1,1/192,Real.exp_pos _,by norm_num,?_⟩ |
| intro N hN μ hμ hμinv f hf L hL hLip |
| let : IsProbabilityMeasure μ := hμ |
| let : μ.IsMulLeftInvariant := hμinv |
| obtain ⟨m,hm1,hm2⟩ := measurable_median μ f hf |
| refine ⟨m,hm1,hm2,?_⟩ |
| intro r hr |
| let k := ((N:ℝ)-2)/(8*L^2) |
| have hρ : 0 < (N:ℝ)-2 := by exact_mod_cast (show 0 < (N:ℤ)-2 by omega) |
| have hk : 0 < k := div_pos hρ (by positivity) |
| have hplus (s : ℝ) (hs : 0 < s) : |
| μ.real {U | s ≤ f U-(∫ V, f V ∂μ)} ≤ Real.exp (-k*s^2) := by |
| have hh := haar_lipschitz_upper_tail hN μ f L hL hLip hs |
| convert hh using 1 |
| congr 1 |
| dsimp only [k] |
| ring |
| have hminus (s : ℝ) (hs : 0 < s) : |
| μ.real {U | s ≤ (∫ V, f V ∂μ)-f U} ≤ Real.exp (-k*s^2) := by |
| have hh := haar_lipschitz_lower_tail hN μ f L hL hLip hs |
| convert hh using 1 |
| congr 1 |
| dsimp only [k] |
| ring |
| apply (gaussian_median_tail μ f (∫ V, f V ∂μ) m k hk hm1 hm2 hplus hminus hr).trans |
| apply mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr ?_) (Real.exp_pos _).le |
| have hdim : (N:ℝ) ≤ 3*((N:ℝ)-2) := by |
| have hn : (3:ℝ) ≤ N := by exact_mod_cast hN |
| linarith |
| have hnum : -3*((N:ℝ)-2)*r^2 ≤ -(N:ℝ)*r^2 := by |
| nlinarith [mul_nonneg (sub_nonneg.mpr hdim) (sq_nonneg r)] |
| calc |
| -k*r^2/8 = (-3*((N:ℝ)-2)*r^2)/(192*L^2) := by dsimp only [k]; ring |
| _ ≤ (-(N:ℝ)*r^2)/(192*L^2) := div_le_div_of_nonneg_right hnum (by positivity) |
| _ = -(1/192)*(N:ℝ)*r^2/L^2 := by ring |
|
|
| end InvariantIsing |
|
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| end |
|
|
| end OAI |
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|
