| import OAI.Probability.InvariantIsing.Haar.HaarLipschitzMoment |
| import OAI.Probability.InvariantIsing.Haar.HaarLipschitzContinuity |
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| namespace OAI |
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| /-! Gaussian tails about the Haar mean for arbitrary Lipschitz observables. -/ |
| noncomputable section |
| open Matrix MeasureTheory Set |
| namespace InvariantIsing |
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| theorem haar_lipschitz_chernoff {N : ℕ} (hN : 3 ≤ N) |
| (μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant] |
| (f : SpecialOrthogonal N → ℝ) (L : ℝ) (hL : 0 < L) |
| (hLip : ∀ U V, |f U-f V| ≤ L*frobeniusDistance U V) |
| (r β : ℝ) (hβ : 0 < β) : |
| μ.real {U | r ≤ f U-(∫ V, f V ∂μ)} ≤ |
| Real.exp ((4*L^2)/(2*((N:ℝ)-2))*β^2-β*r) := by |
| have hf := continuous_of_frobenius_lipschitz f L hLip |
| let m := ∫ V, f V ∂μ |
| let A := (4*L^2)/(2*((N:ℝ)-2)) |
| let F (U : SpecialOrthogonal N) := Real.exp (β*(f U-m)) |
| have hfi : Integrable F μ := continuous_haar_integrable μ _ |
| (Real.continuous_exp.comp (continuous_const.mul (hf.sub continuous_const))) |
| have hmoment := haar_lipschitz_moment_bound hN μ f hf L hL hLip hβ |
| have hI : (∫ U, F U ∂μ) ≤ Real.exp (A*β^2) := by |
| have he : (∫ U, F U ∂μ) = Real.exp (-β*m)*(∫ U, Real.exp (β*f U) ∂μ) := by |
| rw [← integral_const_mul] |
| apply integral_congr_ae |
| exact ae_of_all μ fun U => by |
| change Real.exp (β*(f U-m)) = Real.exp (-β*m)*Real.exp (β*f U) |
| rw [← Real.exp_add] |
| congr 1 |
| ring |
| rw [he] |
| calc |
| _ ≤ Real.exp (-β*m)*Real.exp (m*β+A*β^2) := |
| mul_le_mul_of_nonneg_left hmoment (Real.exp_pos _).le |
| _ = Real.exp (A*β^2) := by |
| rw [← Real.exp_add] |
| congr 1 |
| ring |
| have hset : {U | Real.exp (β*r) ≤ F U} = {U | r ≤ f U-m} := by |
| ext U |
| simp only [mem_ofPred_eq,F,Real.exp_le_exp] |
| exact mul_le_mul_iff_right₀ hβ |
| have hm := mul_meas_ge_le_integral_of_nonneg (ae_of_all μ fun U => (Real.exp_pos _).le) |
| hfi (Real.exp (β*r)) |
| rw [hset] at hm |
| have hb : μ.real {U | r ≤ f U-m} ≤ Real.exp (A*β^2)/Real.exp (β*r) := by |
| apply (le_div_iff₀ (Real.exp_pos _)).mpr |
| nlinarith [hm] |
| simpa only [Real.exp_sub] using hb |
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| theorem haar_lipschitz_upper_tail {N : ℕ} (hN : 3 ≤ N) |
| (μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant] |
| (f : SpecialOrthogonal N → ℝ) (L : ℝ) (hL : 0 < L) |
| (hLip : ∀ U V, |f U-f V| ≤ L*frobeniusDistance U V) {r : ℝ} (hr : 0 < r) : |
| μ.real {U | r ≤ f U-(∫ V, f V ∂μ)} ≤ |
| Real.exp (-((N:ℝ)-2)*r^2/(8*L^2)) := by |
| have hρ : 0 < (N:ℝ)-2 := by exact_mod_cast (show 0 < (N:ℤ)-2 by omega) |
| have hC : 0 < 4*L^2 := by positivity |
| have hb : 0 < ((N:ℝ)-2)*r/(4*L^2) := div_pos (mul_pos hρ hr) hC |
| have h := haar_lipschitz_chernoff hN μ f L hL hLip r (((N:ℝ)-2)*r/(4*L^2)) hb |
| convert h using 1 |
| congr 1 |
| field_simp |
| ring |
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| end InvariantIsing |
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| end |
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| end OAI |
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