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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 | import OAI.Probability.FreeIsing.Conditioning.FiniteEntropySmallTime
import OAI.Probability.FreeIsing.Stochastic.NormalizedBrownianVarianceComparison
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped NNReal ENNReal Topology
namespace Problem367.NormalizedCriticalInitialEnergy
variable {d : ℕ} {Ω : Type*} [MeasurableSpace Ω]
def informationConstant (beta : ℝ) : ℝ :=
(1 + Real.tanh beta ^ 2) *
Real.sqrt (1 / ((1 - Real.tanh beta ^ 2) * Real.tanh beta ^ 4))
lemma informationConstant_nonneg (beta : ℝ) : 0 ≤ informationConstant beta := by
unfold informationConstant
positivity
def energyConstant (d : ℕ) (beta : ℝ) : ℝ :=
informationConstant beta / GaussianVarianceLower.varianceLower 1 (1 + (d : ℝ) * beta)
lemma energyConstant_nonneg (d : ℕ) (beta : ℝ) : 0 ≤ energyConstant d beta :=
div_nonneg (informationConstant_nonneg beta) (GaussianVarianceLower.varianceLower_pos _ _).le
theorem lintegral_finite_energy_le
(hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((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))
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D)
(hv : v.1.1 = []) (T : ℝ≥0) (hT : (T : ℝ) ≤ 1) :
(∫⁻ t in Ioc 0 (T : ℝ), ∫⁻ p,
ENNReal.ofReal (∑ w : D, finiteIsingBrownianNormalizedCovariance beta D v w B t p ^ 2)
∂μ.prod (Measure.infinitePi ν)) ≤
ENNReal.ofReal (energyConstant d beta * Real.sqrt (T : ℝ)) := by
classical
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
have h := NormalizedBrownianVarianceComparison.lintegral_energy_le_information
hμ hb ν hB hmB D hD v hv (by norm_num : (0 : ℝ) ≤ 1) T.property hT
refine h.trans (ENNReal.ofReal_le_ofReal ?_)
have hm := div_le_div_of_nonneg_right
(hμ.finiteIsingBrownian_critical_information_upper hd hb hcrit ν hB hmB D hD v T hT)
(GaussianVarianceLower.varianceLower_pos 1 (1 + (d : ℝ) * beta)).le
change _ ≤ informationConstant beta / GaussianVarianceLower.varianceLower 1
(1 + (d : ℝ) * beta) * Real.sqrt (T : ℝ)
convert hm using 1 <;> first | rfl | (dsimp [informationConstant]; ring)
theorem lintegral_infinite_energy_le
(hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((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) (hv : v.1 = []) (T : ℝ≥0) (hT : (T : ℝ) ≤ 1) :
(∫⁻ q, ∑' w, ENNReal.ofReal
(NormalizedInfiniteTimeEnergy.infiniteRow beta v B w q ^ 2)
∂(volume.restrict (Ioc 0 (T : ℝ))).prod (μ.prod (Measure.infinitePi ν))) ≤
ENNReal.ofReal (energyConstant d beta * Real.sqrt (T : ℝ)) := by
classical
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
apply NormalizedInfiniteTimeEnergy.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 (∑ w : D,
finiteIsingBrownianNormalizedCovariance beta D r w B q.1 q.2 ^ 2)
have hR : Measurable R := by
have heq : R = fun q => ∑' w, ENNReal.ofReal
(NormalizedInfiniteTimeEnergy.finiteRow beta D v B w q ^ 2) := by
funext q
exact (NormalizedInfiniteTimeEnergy.tsum_finiteRow_sq beta D r B q).symm
rw [heq]
exact Measurable.tsum (fun w =>
((NormalizedInfiniteTimeEnergy.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
_ ≤ _ := lintegral_finite_energy_le hd hμ hb hcrit ν hB hmB
D (connectedExhaustion_connected v n) r hv T hT
theorem tendsto_infinite_energy_zero
(hd : 3 ≤ d) {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((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) (hv : v.1 = []) :
Tendsto (fun T : ℝ≥0 => ∫⁻ q, ∑' w, ENNReal.ofReal
(NormalizedInfiniteTimeEnergy.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 (energyConstant d beta * Real.sqrt (T : ℝ))) :=
ENNReal.continuous_ofReal.comp
(continuous_const.mul (Real.continuous_sqrt.comp NNReal.continuous_coe))
have hlim : Tendsto (fun T : ℝ≥0 =>
ENNReal.ofReal (energyConstant d beta * Real.sqrt (T : ℝ))) (𝓝 0) (𝓝 0) := by
simpa only [NNReal.coe_zero, Real.sqrt_zero, mul_zero, ENNReal.ofReal_zero] using hc.tendsto 0
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim
(Eventually.of_forall fun _ => bot_le)
filter_upwards [eventually_lt_nhds (by norm_num : (0 : ℝ≥0) < 1)] with T hT
exact lintegral_infinite_energy_le hd hμ hb hcrit ν hB hmB hjoint v hv T
(by exact_mod_cast hT.le)
end Problem367.NormalizedCriticalInitialEnergy
end
end OAI
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