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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 | import OAI.Probability.FreeIsing.Conditioning.FiniteEntropySmallTime
import OAI.Probability.FreeIsing.Energy.NormalizedInfiniteTimeEnergy
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal ENNReal Topology
namespace Problem367
local instance : Nonempty IsingSpin := ⟨.neg⟩
theorem HasFreeIsingLaw.lintegral_finiteIsingBrownianNormalizedEnergy_le_entropy
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(ν : TreeVertex d → Measure Ω)
{B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hB : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D)
{T : ℝ} (hT : 0 ≤ T) :
(∫⁻ t in Ioc 0 T, ∫⁻ p,
ENNReal.ofReal (∑ u : D, finiteIsingBrownianNormalizedCovariance beta D v u B t p ^ 2)
∂μ.prod (Measure.infinitePi ν)) ≤
ENNReal.ofReal (∫ p, finiteIsingBrownianEntropy beta D v B T p
∂μ.prod (Measure.infinitePi ν)) := by
classical
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
let ρ := μ.prod (Measure.infinitePi ν)
let R := fun t p => ∑ u : D, finiteIsingBrownianNormalizedCovariance beta D v u B t p ^ 2
let E : ℝ → ℝ := fun t => ∫ p, R t p ∂ρ
have hRpos (t : ℝ) (p) : 0 ≤ R t p := Finset.sum_nonneg fun u _ => sq_nonneg _
have hRI (t : ℝ) : Integrable (R t) ρ := by
have hY : Measurable (finiteIsingBrownianObservation B D t) := by
apply Measurable.of_eval
intro w
exact (((measurable_of_finite spinSign).comp
((measurable_pi_apply (w : TreeVertex d)).comp measurable_fst)).const_mul t).add
((hmB w t.toNNReal).comp ((measurable_pi_apply (w : TreeVertex d)).comp measurable_snd))
have hs (eta : D → IsingSpin) : spinSign (eta v) = -1 ∨ spinSign (eta v) = 1 := by
cases eta v <;> simp [spinSign]
have hn : ∃ eta : D → IsingSpin, spinSign (eta v) = -1 := ⟨fun _ => .neg, rfl⟩
have hp : ∃ eta : D → IsingSpin, spinSign (eta v) = 1 := ⟨fun _ => .pos, rfl⟩
apply Integrable.of_bound
((FinitePosterior.continuous_normalizedRowEnergy (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w)) (fun eta => spinSign (eta v))
hs hn hp).measurable.comp hY).aestronglyMeasurable (Fintype.card D)
exact Eventually.of_forall fun p => by
simpa only [Real.norm_eq_abs, Function.comp_def] using
FinitePosterior.abs_normalizedRowEnergy_le_card (finiteIsingEnergy beta D)
(fun (eta : D → IsingSpin) w => spinSign (eta w)) (fun eta => spinSign (eta v))
(finiteIsingBrownianObservation B D t p) hs hn hp
(fun eta w => by cases eta w <;> norm_num [spinSign])
have hinner (t : ℝ) : (∫⁻ p, ENNReal.ofReal (R t p) ∂ρ) = ENNReal.ofReal (E t) :=
(ofReal_integral_eq_lintegral_ofReal (hRI t) (Eventually.of_forall (hRpos t))).symm
have hbound := hμ.integral_finiteIsingBrownianNormalizedEnergy_le hbeta ν hB hmB D hD v hT
have hEI : Integrable E (volume.restrict (Ioc 0 T)) :=
(intervalIntegrable_iff_integrableOn_Ioc_of_le hT).mp hbound.1
change (∫⁻ t in Ioc 0 T, ∫⁻ p, ENNReal.ofReal (R t p) ∂ρ) ≤ _
simp_rw [hinner]
rw [← ofReal_integral_eq_lintegral_ofReal hEI (Eventually.of_forall fun t => integral_nonneg (hRpos t))]
rw [← intervalIntegral.integral_of_le hT]
exact ENNReal.ofReal_le_ofReal hbound.2.1
namespace NormalizedInfiniteTimeEnergy
theorem lintegral_infiniteRow_le_criticalEnvelope
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcritical : ((d - 1 : ℕ) : ℝ) * (Real.tanh beta) ^ 2 = 1)
(ν : TreeVertex d → Measure Ω) {B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hB : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2))
(v : TreeVertex d) (T : ℝ≥0) (hT : (T : ℝ) ≤ 1) :
(∫⁻ q, ∑' w, ENNReal.ofReal (infiniteRow beta v B w q ^ 2)
∂(volume.restrict (Ioc 0 (T : ℝ))).prod (μ.prod (Measure.infinitePi ν))) ≤
ENNReal.ofReal (criticalEntropyEnvelope d beta T) := by
let := hμ.isProbabilityMeasure
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
apply lintegral_infiniteRow_le_of_finite hμ hb ν hB hmB hjoint
(connectedExhaustion v) (connectedExhaustion_monotone v)
(connectedExhaustion_connected v)
(fun u => (eventually_mem_connectedExhaustion v u).exists)
v (mem_connectedExhaustion v) T
intro n
let D := connectedExhaustion v n
let r : D := ⟨v, mem_connectedExhaustion v n⟩
let R : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → ℝ≥0∞ :=
fun q => ENNReal.ofReal (∑ u : D,
finiteIsingBrownianNormalizedCovariance beta D r u B q.1 q.2 ^ 2)
have hR : Measurable R := by
have heq : R = fun q => ∑' w, ENNReal.ofReal (finiteRow beta D v B w q ^ 2) := by
funext q
exact (tsum_finiteRow_sq beta D r B q).symm
rw [heq]
exact Measurable.tsum (fun w =>
((measurable_finiteRow beta D v hjoint w).pow_const 2).ennreal_ofReal)
calc
_ = ∫⁻ q, R q
∂(volume.restrict (Ioc 0 (T : ℝ))).prod (μ.prod (Measure.infinitePi ν)) := by
apply lintegral_congr
intro q
exact (ENNReal.ofReal_sum_of_nonneg (fun _ _ => sq_nonneg _)).symm
_ = ∫⁻ t in Ioc 0 (T : ℝ), ∫⁻ p, R (t, p) ∂μ.prod (Measure.infinitePi ν) :=
lintegral_prod R hR.aemeasurable
_ ≤ ENNReal.ofReal (∫ p, finiteIsingBrownianEntropy beta D r B T p
∂μ.prod (Measure.infinitePi ν)) :=
hμ.lintegral_finiteIsingBrownianNormalizedEnergy_le_entropy hb ν hB hmB
D (connectedExhaustion_connected v n) r T.property
_ ≤ _ := ENNReal.ofReal_le_ofReal
(hμ.finiteIsingBrownianEntropy_le_criticalEnvelope hd hb hcritical ν hB hmB
D (connectedExhaustion_connected v n) r T hT)
theorem tendsto_lintegral_infiniteRow_critical_zero
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcritical : ((d - 1 : ℕ) : ℝ) * (Real.tanh beta) ^ 2 = 1)
(ν : TreeVertex d → Measure Ω) {B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hB : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2))
(v : TreeVertex d) :
Tendsto (fun T : ℝ≥0 =>
∫⁻ q, ∑' w, ENNReal.ofReal (infiniteRow beta v B w q ^ 2)
∂(volume.restrict (Ioc 0 (T : ℝ))).prod (μ.prod (Measure.infinitePi ν)))
(𝓝 0) (𝓝 0) := by
have hc : Continuous (fun T : ℝ≥0 =>
ENNReal.ofReal (criticalEntropyEnvelope d beta T)) :=
ENNReal.continuous_ofReal.comp ((continuous_criticalEntropyEnvelope d beta).comp
NNReal.continuous_coe)
have hlim : Tendsto (fun T : ℝ≥0 => ENNReal.ofReal (criticalEntropyEnvelope d beta T))
(𝓝 0) (𝓝 0) := by simpa using hc.tendsto 0
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim
· exact Eventually.of_forall fun _ => zero_le
· filter_upwards [eventually_lt_nhds (by norm_num : (0 : ℝ≥0) < 1)] with T hT
exact lintegral_infiniteRow_le_criticalEnvelope hd hμ hb hcritical ν hB hmB hjoint
v T (by exact_mod_cast hT.le)
end NormalizedInfiniteTimeEnergy
end Problem367
end
end OAI
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