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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 | import OAI.Probability.FreeIsing.Estimates.BoundedAverageConcentration
import OAI.Probability.FreeIsing.Gaussian.InwardGaussianCutMean
import OAI.Probability.FreeIsing.Gaussian.InwardProducts
import OAI.Probability.FreeIsing.Trees.CutLevelConcentration
import OAI.Probability.FreeIsing.Trees.InwardCutProducts
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
open PathSideBranches InwardCutChannels CutContinuationChannels
theorem cutContinuationProduct_gaussianInwardConeFactor_le_cut {d k L : ℕ}
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L) :
∀ᵐ z ∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t),
cutContinuationProduct (gaussianInwardConeFactor μ t (Real.tanh beta) z) p w ≤
Real.exp (4 * beta / (1 - Real.tanh beta)) *
inwardContinuationProduct μ t (Real.tanh beta) p w
(Finset.Ico (k + 1) (k + 1 + L)) z := by
have h := inward_attenuation_prod_le_cut hμ hbeta t
(appendCutContinuation p w).1 (k + 1) L (by
rw [(appendCutContinuation p w).2])
simpa only [cutContinuationProduct_gaussianInwardConeFactor,
inwardContinuationProduct_Ico_eq_range] using h
theorem average_gaussianInwardConeFactor_le_cut_average {d k : ℕ}
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (I : Finset (TreeShell d (k + 1))) (L : ℕ) :
∀ᵐ z ∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t),
(∑ v ∈ cutDescendants I L,
rootPathProduct (gaussianInwardConeFactor μ t (Real.tanh beta) z) v.1) /
((cutDescendants I L).card : ℝ) ≤
Real.exp (4 * beta / (1 - Real.tanh beta)) *
((∑ p ∈ I, inwardContinuationAverage μ t (Real.tanh beta) p L
(Finset.Ico (k + 1) (k + 1 + L)) z) / (I.card : ℝ)) := by
have hθ0 := tanh_nonneg_of_beta_nonneg hbeta
have hθ1 := Real.tanh_lt_one beta
have hθ : Real.tanh beta ^ 2 ≤ 1 := by nlinarith
have hcut : ∀ᵐ z ∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t),
∀ p : TreeShell d (k + 1), ∀ w : ReducedTail (cutLastColor p) L,
cutContinuationProduct (gaussianInwardConeFactor μ t (Real.tanh beta) z) p w ≤
Real.exp (4 * beta / (1 - Real.tanh beta)) *
inwardContinuationProduct μ t (Real.tanh beta) p w
(Finset.Ico (k + 1) (k + 1 + L)) z :=
ae_all_iff.mpr (fun p => ae_all_iff.mpr (fun w =>
cutContinuationProduct_gaussianInwardConeFactor_le_cut hμ hbeta t p w))
filter_upwards [hcut] with z hz
refine (average_rootPathProduct_le_continuation_average
(gaussianInwardConeFactor μ t (Real.tanh beta) z)
(fun v => gaussianInwardConeFactor_mem_Icc μ t hθ z v) I L).trans ?_
calc
_ ≤ (∑ p ∈ I, (∑ w : ReducedTail (cutLastColor p) L,
Real.exp (4 * beta / (1 - Real.tanh beta)) *
inwardContinuationProduct μ t (Real.tanh beta) p w
(Finset.Ico (k + 1) (k + 1 + L)) z) /
(((d - 1) ^ L : ℕ) : ℝ)) / (I.card : ℝ) := by
apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
apply Finset.sum_le_sum
intro p hp
apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _)
exact Finset.sum_le_sum (fun w _ => hz p w)
_ = _ := by
simp only [inwardContinuationAverage, card_reducedTail,
← Finset.mul_sum, mul_div_assoc]
theorem measure_gaussian_inward_path_average_ge_le {d k : ℕ}
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) (hd : 2 ≤ d)
(t : ℝ≥0) (I : Finset (TreeShell d (k + 1))) (hI : I.Nonempty) (L : ℕ)
{τ : ℝ} (hτ : 0 ≤ τ)
(hmean : ∀ p ∈ I, (∫ z, inwardContinuationAverage μ t (Real.tanh beta) p L
(Finset.Ico (k + 1) (k + 1 + L)) z
∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) ≤ τ / 4) :
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))
{z | Real.exp (4 * beta / (1 - Real.tanh beta)) * τ ≤
(∑ v ∈ cutDescendants I L,
rootPathProduct (gaussianInwardConeFactor μ t (Real.tanh beta) z) v.1) /
((cutDescendants I L).card : ℝ)} ≤
ENNReal.ofReal (Real.exp (-(I.card : ℝ) * τ ^ 2 / 2)) := by
classical
let := hμ.isProbabilityMeasure
have hθ0 := tanh_nonneg_of_beta_nonneg hbeta
have hθ1 := Real.tanh_lt_one beta
have hθ : Real.tanh beta ^ 2 ≤ 1 := by nlinarith
choose F hF hb heq using (fun p : TreeShell d (k + 1) =>
exists_inwardContinuationAverage_channel hd μ t hθ p L
(Finset.Ico (k + 1) (k + 1 + L)))
have hmeanF (p : TreeShell d (k + 1)) (hp : p ∈ I) :
(∫ z, F p (globalCone p.1 z)
∂μ.prod (Measure.infinitePi (fun _ : TreeVertex d => gaussianReal 0 t))) ≤ τ / 4 := by
simpa only [heq, PlantedSymmetry.gaussianNoiseLaw] using hmean p hp
have hcon := hμ.measure_cutLevel_average_ge_le
(by linarith : -1 ≤ Real.tanh beta) hθ1.le (k + 1)
(fun _ : TreeVertex d => gaussianReal 0 t) F hF hb I hI hτ hmeanF
change (μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))
{z | τ ≤ (∑ p ∈ I, F p (globalCone p.1 z)) / (I.card : ℝ)} ≤ _ at hcon
refine le_trans ?_ hcon
apply measure_mono_ae
filter_upwards [average_gaussianInwardConeFactor_le_cut_average hμ hbeta t I L]
with z hz
intro h
simp only [heq] at hz
exact (mul_le_mul_iff_right₀ (Real.exp_pos _)).mp (h.trans hz)
theorem measure_gaussian_inward_restricted_path_average_ge_le_double_exp {d : ℕ}
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) (hd : 3 ≤ d)
(hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t)
(forbidden : Finset (Fin d)) (hforbidden : forbidden.card ≤ 1) (k L : ℕ)
{η ℓ : ℝ} (hη0 : 0 ≤ η) (hℓ : 0 ≤ ℓ)
(hη : η ≤ ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) / 4)
(hηlog : η ≤ Real.log ((d : ℝ) - 1) / 8)
(hlevel : ℓ / 2 - 1 ≤ (k + 1 : ℕ)) (htail : ℓ / 2 ≤ (L : ℝ))
(hlength : 8 * Real.log 2 ≤
ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) *
(min (Real.sqrt (t : ℝ)) 1 * ℓ)) :
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t))
{z | Real.exp (4 * beta / (1 - Real.tanh beta)) *
Real.exp (-(η * min (Real.sqrt (t : ℝ)) 1) * ℓ) ≤
(∑ v ∈ cutDescendants (avoidingTreeCut forbidden k) L,
rootPathProduct (gaussianInwardConeFactor μ t (Real.tanh beta) z) v.1) /
((cutDescendants (avoidingTreeCut forbidden k) L).card : ℝ)} ≤
ENNReal.ofReal (Real.exp (-(((d : ℝ) - 1) ^ (ℓ / 4)) /
(2 * ((d : ℝ) - 1)))) := by
have hθ0 : 0 < Real.tanh beta := by
have hn := tanh_nonneg_of_beta_nonneg hbeta
by_contra h
have hz : Real.tanh beta = 0 := le_antisymm (not_lt.mp h) hn
simp [hz] at hcritical
have hθ1 := Real.tanh_lt_one beta
have hc := (ComponentSignalAttenuation.lossCoefficient_pos hθ0 hθ1).le
have hω : 0 ≤ min (Real.sqrt (t : ℝ)) 1 :=
le_min (Real.sqrt_nonneg _) (by norm_num)
have hmean (p : TreeShell d (k + 1)) :
(∫ z, inwardContinuationAverage μ t (Real.tanh beta) p L
(Finset.Ico (k + 1) (k + 1 + L)) z
∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) ≤
Real.exp (-(η * min (Real.sqrt (t : ℝ)) 1) * ℓ) / 4 := by
refine (integral_inwardContinuationAverage_le_exp_critical
hμ hbeta hd hcritical ht p L).trans ?_
calc
_ ≤ Real.exp (-ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) *
(min (Real.sqrt (t : ℝ)) 1 * ℓ) / 2) := by
apply Real.exp_le_exp.mpr
have hm := mul_le_mul_of_nonneg_left htail (mul_nonneg hc hω)
nlinarith
_ ≤ Real.exp (-η * (min (Real.sqrt (t : ℝ)) 1 * ℓ)) / 4 :=
attenuation_le_quarter_threshold (mul_nonneg hω hℓ) hη hlength
_ = _ := by ring_nf
have hq : η * min (Real.sqrt (t : ℝ)) 1 ≤ Real.log ((d : ℝ) - 1) / 8 :=
(mul_le_of_le_one_right hη0 (min_le_right _ _)).trans hηlog
have hd2 : 2 ≤ d := by omega
have hdR : (2 : ℝ) ≤ d := by exact_mod_cast hd2
exact (measure_gaussian_inward_path_average_ge_le hμ hbeta hd2 t
(avoidingTreeCut forbidden k) (avoidingTreeCut_nonempty hd2 forbidden hforbidden k)
L (Real.exp_pos _).le (fun p _ => hmean p)).trans
(ENNReal.ofReal_le_ofReal (BoundedAverageConcentration.exponential_card_bound
(by linarith) hℓ hq (rpow_le_card_avoidingTreeCut hd2 forbidden hforbidden k hlevel)))
end Problem367
end
end OAI
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