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f61e547 | 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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import OAI.Probability.FreeIsing.Critical.GlobalEndgame
import OAI.Probability.FreeIsing.Estimates.CoupledSmallLogLoss
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set
namespace Problem367
theorem log_reciprocal_ge_of_le_exp {J t : ℝ} (ht : 0 < t)
(hsmall : t ≤ Real.exp (-32 * J)) :
32 * J ≤ Real.log (1 / t) := by
have hlog : Real.log t ≤ -32 * J := (Real.log_le_iff_le_exp ht).2 hsmall
rw [one_div, Real.log_inv]
linarith
theorem eq_zero_of_coupled_cavity_spatial_bounds
{Ω Ξ : Type*} [MeasurableSpace Ω] [MeasurableSpace Ξ]
{μ : Measure Ω} [IsProbabilityMeasure μ]
{ν : Measure Ξ} [IsProbabilityMeasure ν]
{theta T M J C : ℝ}
{X Y X' Y' H N H' N' : ℝ → Ω → ℝ} {R : ℝ → Ξ → ℝ}
{V S Q L Z : ℝ → ℝ}
(hθ0 : 0 ≤ theta) (hθ1 : theta < 1)
(hT : 0 ≤ T) (hM : 0 ≤ M) (hJ : 0 < J)
(hC : 1 ≤ C) (hcoef : cavityLogVarianceCoefficient theta ≤ C)
(hX : ∀ t, Measurable (X t)) (hX' : ∀ t, Measurable (X' t))
(hH : ∀ t ∈ Icc 0 T, Integrable (fun ω => (H t ω - H' t ω) ^ 2) μ)
(hN : ∀ t ∈ Icc 0 T, Integrable (fun ω => (N t ω - N' t ω) ^ 2) μ)
(hpair : ∀ t ∈ Ioc 0 T, ∀ᵐ ω ∂μ,
X t ω + cavityMessage theta (Y t ω) = H t ω ∧
Y t ω + cavityMessage theta (X t ω) = N t ω ∧
X' t ω + cavityMessage theta (Y' t ω) = H' t ω ∧
Y' t ω + cavityMessage theta (X' t ω) = N' t ω)
(hbound : ∀ t ∈ Icc 0 T, ∀ᵐ ω ∂μ,
|H t ω - H' t ω| ≤ M ∧ |N t ω - N' t ω| ≤ M)
(hVeq : ∀ t, V t = ∫ ω, (H t ω - H' t ω) ^ 2 ∂μ)
(hSeq : ∀ t, S t = ∫ ω,
(Real.log (responseAttenuation theta (X t ω)) -
Real.log (responseAttenuation theta (X' t ω))) ^ 2 ∂μ)
(hNV : ∀ t ∈ Ioc 0 T, (∫ ω, (N t ω - N' t ω) ^ 2 ∂μ) ≤ V t)
(hLaw : ∀ t ∈ Ioc 0 T, IdentDistrib (X t) (R t) μ ν)
(hLaw' : ∀ t ∈ Ioc 0 T, IdentDistrib (X' t) (R t) μ ν)
(hmom : ∀ t ∈ Ioc 0 T, t ≤ Real.exp (-32 * J) →
∀ p : ℕ, 1 ≤ p → Integrable (fun ξ => |R t ξ| ^ p) ν ∧
(∫ ξ, |R t ξ| ^ p ∂ν) ≤
Real.exp (J * (p : ℝ) ^ 2) * t ^ ((p : ℝ) / 4))
(hV : AbsolutelyContinuousOnInterval V 0 T) (hzero : V 0 = 0)
(hspatial : ∀ᵐ t ∂volume.restrict (Ioc 0 T), 0 < V t →
Q t ≤ C * (S t * (((min (Real.sqrt t) 1)⁻¹) ^ 2 +
(Real.log (1 + Real.log ((1 + M ^ 2) / V t))) ^ 2) + V t ^ 2) ∧
L t ≤ C * (V t * ((min (Real.sqrt t) 1)⁻¹ +
Real.log (1 + Real.log ((1 + M ^ 2) / V t))) + V t ^ 2) ∧
Z t ≤ C * (Real.sqrt (S t) *
((min (Real.sqrt t) 1)⁻¹ * Real.sqrt ((min (Real.sqrt t) 1)⁻¹) +
(Real.log (1 + Real.log ((1 + M ^ 2) / V t))) ^ ((3 : ℝ) / 2)) + V t) ∧
deriv V t ≤ C * (V t + Q t + L t + Real.sqrt (V t) * Z t)) :
∀ t ∈ Icc 0 T, V t = 0 := by
have hcutoff : Real.exp (-32 * J) < 1 := by
rw [Real.exp_lt_one_iff]
linarith
have hVnonneg (t : ℝ) : 0 ≤ V t := by
rw [hVeq]
exact integral_nonneg (fun _ => sq_nonneg _)
have hSnonneg (t : ℝ) : 0 ≤ S t := by
rw [hSeq]
exact integral_nonneg (fun _ => sq_nonneg _)
have hVupper (t : ℝ) (ht : t ∈ Icc 0 T) : V t ≤ 1 + M ^ 2 := by
calc
V t = ∫ ω, (H t ω - H' t ω) ^ 2 ∂μ := hVeq t
_ ≤ ∫ _ : Ω, M ^ 2 ∂μ := by
apply integral_mono_ae (hH t ht) (integrable_const _)
filter_upwards [hbound t ht] with ω hω
have hs := (sq_le_sq₀ (abs_nonneg _) hM).2 hω.1
simpa only [sq_abs] using hs
_ = M ^ 2 := by simp
_ ≤ 1 + M ^ 2 := by linarith
have hnoise (t : ℝ) (ht : t ∈ Ioc 0 T) : S t ≤ C * V t := by
have htcc : t ∈ Icc 0 T := ⟨ht.1.le, ht.2⟩
calc
S t = _ := hSeq t
_ ≤ cavityLogVarianceCoefficient theta * V t := by
exact integral_cavity_log_discrepancy_sq_le hθ0 hθ1 (hX t) (hX' t)
(hH t htcc) (hN t htcc) (hpair t ht) (le_of_eq (hVeq t).symm) (hNV t ht)
_ ≤ C * V t := mul_le_mul_of_nonneg_right hcoef (hVnonneg t)
apply eq_zero_of_ae_critical_spatial_endpoint_bounds
(cutoff := Real.exp (-32 * J)) (C := C) (H := 1 + M ^ 2)
(A := cavityLogVarianceCoefficient theta) (B := cavityLogTailCoefficient theta M)
(c := 1 / (512 * J)) (K := 1 + M ^ 2)
hT (Real.exp_pos _) hcutoff hC (by nlinarith [sq_nonneg M])
(cavityLogVarianceCoefficient_nonneg theta) (cavityLogTailCoefficient_nonneg theta M)
(by positivity) (by nlinarith [sq_nonneg M]) hV hzero
(fun t _ => hVnonneg t) hVupper hVupper
· filter_upwards [hspatial, ae_restrict_mem measurableSet_Ioc] with t hs ht
intro hv
obtain ⟨hq, hl, hz, hd⟩ := hs hv
exact ⟨hSnonneg t, hnoise t ht, hq, hl, hz, hd⟩
· filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
intro htc _
have htcc : t ∈ Icc 0 T := ⟨ht.1.le, ht.2⟩
have hlog := log_reciprocal_ge_of_le_exp ht.1 htc
have htail : ν {ξ | t ^ (1 / 8 : ℝ) < |R t ξ|} ≤
ENNReal.ofReal (Real.exp (-(1 / (512 * J)) * (Real.log (1 / t)) ^ 2)) := by
have he : -(1 / (512 * J)) * (Real.log (1 / t)) ^ 2 =
-((Real.log (1 / t)) ^ 2) / (512 * J) := by ring
rw [he]
exact measure_abs_gt_rpow_le_of_moments hJ ht.1 hlog
(fun p hp => (hmom t ht htc p hp).1) (fun p hp => (hmom t ht htc p hp).2)
rw [hSeq]
exact integral_cavity_log_discrepancy_small_time_of_common_law
hθ0 hθ1 (hX t) (hX' t) (hH t htcc) (hN t htcc) (hpair t ht)
hM ht.1 (hbound t htcc) (le_of_eq (hVeq t).symm) (hNV t ht)
(hLaw t ht) (hLaw' t ht) htail
end Problem367
end
end OAI
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