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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | import OAI.Probability.InvariantIsing.Haar.HaarTwoSidedTail
import OAI.Probability.InvariantIsing.Gaussian.GaussianMedianTail
namespace OAI
/-!
The exact Gromov--Milman specialization used by the pressure proof. The proof
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
end
end OAI
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