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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 | import OAI.Probability.FreeIsing.Brownian.EntropyEnergy
import OAI.Probability.FreeIsing.Brownian.PosteriorField
import OAI.Probability.FreeIsing.Gaussian.SiteInformation
import OAI.Probability.FreeIsing.Stochastic.FiniteIsingBrownianPosterior
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal ENNReal
namespace Problem367
local instance finiteEntropySmallTimeNonemptyIsingSpin : Nonempty IsingSpin := ⟨.neg⟩
theorem gaussian_channel_fourth_moment
{Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
{S G : Ω → ℝ} (hS : AEMeasurable S P)
(hSb : ∀ᵐ p ∂P, |S p| ≤ 1) {t : ℝ≥0}
(hG : HasLaw G (gaussianReal 0 t) P) (ht : (t : ℝ) ≤ 1) :
Integrable (fun p => |(t : ℝ) * S p + G p| ^ 4) P ∧
(∫ p, |(t : ℝ) * S p + G p| ^ 4 ∂P) ≤
(8 * (1 + Real.exp 32)) * (t : ℝ) := by
have ht0 := t.coe_nonneg
have hG4 := integrable_abs_pow_of_gaussianLaw hG 4
have hdom : Integrable (fun p => (8 : ℝ) * ((t : ℝ) ^ 4 + |G p| ^ 4)) P :=
((integrable_const _).add hG4).const_mul 8
have hbound : ∀ᵐ p ∂P, |(t : ℝ) * S p + G p| ^ 4 ≤
8 * ((t : ℝ) ^ 4 + |G p| ^ 4) := by
filter_upwards [hSb] with p hp
have hb : |(t : ℝ) * S p + G p| ≤ (t : ℝ) + |G p| := by
calc
_ ≤ |(t : ℝ) * S p| + |G p| := abs_add_le _ _
_ ≤ (t : ℝ) + |G p| := by
rw [abs_mul, abs_of_nonneg ht0]
gcongr
simpa using mul_le_mul_of_nonneg_left hp ht0
exact (pow_le_pow_left₀ (abs_nonneg _) hb 4).trans
(by convert add_pow_le ht0 (abs_nonneg (G p)) 4 using 1; norm_num)
have hi : Integrable (fun p => |(t : ℝ) * S p + G p| ^ 4) P := by
apply hdom.mono' (((hS.const_mul (t : ℝ)).add hG.aemeasurable).abs.pow_const 4).aestronglyMeasurable
filter_upwards [hbound] with p hp
change ‖|(t : ℝ) * S p + G p| ^ 4‖ ≤ _
rw [Real.norm_eq_abs, abs_of_nonneg (by positivity)]
exact hp
have hstd : (∫ x : ℝ, |x| ^ 4 ∂gaussianReal 0 1) ≤ Real.exp 32 := by
calc
_ ≤ ∫ x : ℝ, (1 + |x|) ^ 4 ∂gaussianReal 0 1 :=
integral_mono (integrable_standardGaussian_abs_pow 4)
(integrable_one_add_abs_pow_gaussian 4)
(fun x => pow_le_pow_left₀ (abs_nonneg x) (by linarith) 4)
_ ≤ _ := by
convert integral_one_add_abs_pow_gaussian_le (p := 4) (by decide) using 1; norm_num
have hnoise : (∫ p, |G p| ^ 4 ∂P) ≤ (t : ℝ) ^ 2 * Real.exp 32 := by
have heq := hG.integral_comp (f := fun x : ℝ => |x| ^ 4) (by fun_prop)
change (∫ p, |G p| ^ 4 ∂P) = _ at heq
rw [heq, integral_gaussianReal_abs_pow]
have hs : Real.sqrt (t : ℝ) ^ 4 = (t : ℝ) ^ 2 := by
rw [show (4 : ℕ) = 2 * 2 by decide, pow_mul, Real.sq_sqrt ht0]
rw [hs]
exact mul_le_mul_of_nonneg_left hstd (sq_nonneg _)
have ht2 : (t : ℝ) ^ 2 ≤ (t : ℝ) := by nlinarith
have ht4 : (t : ℝ) ^ 4 ≤ (t : ℝ) := by
have h := pow_le_pow_left₀ (sq_nonneg (t : ℝ)) ht2 2
nlinarith
refine ⟨hi, ?_⟩
calc
_ ≤ ∫ p, (8 : ℝ) * ((t : ℝ) ^ 4 + |G p| ^ 4) ∂P :=
integral_mono_ae hi hdom hbound
_ = 8 * ((t : ℝ) ^ 4 + ∫ p, |G p| ^ 4 ∂P) := by
rw [integral_const_mul, integral_add (integrable_const _) hG4]
simp
_ ≤ 8 * ((t : ℝ) + (t : ℝ) * Real.exp 32) := by
gcongr
exact hnoise.trans (mul_le_mul_of_nonneg_right ht2 (Real.exp_nonneg _))
_ = _ := by ring
theorem finiteIsingBrownianEntropy_le_information
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
(hbeta : 0 ≤ beta) (μ : Measure (TreeVertex d → IsingSpin))
[IsProbabilityMeasure μ] (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)) (v : D) (t : ℝ≥0) (ht : (t : ℝ) ≤ 1) :
(∫ p, finiteIsingBrownianEntropy beta D v B t p
∂μ.prod (Measure.infinitePi ν)) ≤
2 * Real.sqrt
(Real.cosh ((d : ℝ) * beta + 1) ^ 2 *
(∫ p, finiteIsingBrownianMean beta D v B t p ^ 2
∂μ.prod (Measure.infinitePi ν)) +
(16 * (1 + ((d : ℝ) * beta) ^ 2) * (1 + Real.exp 32)) * (t : ℝ)) := by
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
let P := μ.prod (Measure.infinitePi ν)
let M := finiteIsingBrownianMean beta D v B (t : ℝ)
let H := fun p => Real.artanh (M p)
let G := fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω) => B v t (p.2 v)
let S := fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω) => spinSign (p.1 v)
let X := fun p => (t : ℝ) * S p + G p
have hcoord := (measurePreserving_eval_infinitePi ν (v : TreeVertex d)).comp
(measurePreserving_snd (μ := μ) (ν := Measure.infinitePi ν))
have hG : HasLaw G (gaussianReal 0 t) P :=
(hB v).hasLaw_eval t |>.comp ⟨hcoord.measurable.aemeasurable, hcoord.map_eq⟩
have hS : Measurable S := (measurable_of_finite spinSign).comp
((measurable_pi_apply (v : TreeVertex d)).comp measurable_fst)
have hSb : ∀ᵐ p ∂P, |S p| ≤ 1 := Eventually.of_forall fun p => by
cases hp : p.1 v <;> simp [S, hp, spinSign]
have hX4 := gaussian_channel_fourth_moment hS.aemeasurable hSb hG ht
have hspin (eta : D → IsingSpin) : spinSign (eta v) = -1 ∨ spinSign (eta v) = 1 := by
cases eta v <;> simp [spinSign]
have hneg : ∃ eta : D → IsingSpin, spinSign (eta v) = -1 := ⟨fun _ => .neg, rfl⟩
have hpos : ∃ eta : D → IsingSpin, spinSign (eta v) = 1 := ⟨fun _ => .pos, rfl⟩
have hm : Measurable M :=
(FinitePosterior.continuous_fieldMean _ _ _).measurable.comp
(measurable_finiteIsingBrownianObservation hmB D t)
have hart : Measurable Real.artanh := by unfold Real.artanh; fun_prop
have hH : AEMeasurable H P := (hart.comp hm).aemeasurable
have hinterior (p) : M p ∈ Ioo (-1 : ℝ) 1 :=
FinitePosterior.fieldMean_spin_mem_Ioo _ _ _ _ hspin hneg hpos
have htan (p) : Real.tanh (H p) = M p := Real.tanh_artanh (hinterior p)
have hfield : ∀ᵐ p ∂P, |H p - X p| ≤ (d : ℝ) * beta := by
apply Eventually.of_forall
intro p
have hb := finite_tree_gibbs_field_uniform_bound D hbeta
(finiteIsingBrownianObservation B D t p) v
change |H p - finiteIsingBrownianObservation B D t p v| ≤
(d : ℝ) * beta at hb
simpa only [X, S, G, finiteIsingBrownianObservation, Real.toNNReal_coe] using hb
have hsecond := integral_sq_le_tanh_sq_add_fourth
(by positivity : 0 ≤ (d : ℝ) * beta) hH hX4.1 hfield
simp_rw [htan] at hsecond
have hH2 : MemLp H 2 P :=
(memLp_two_iff_integrable_sq hH.aestronglyMeasurable).mpr hsecond.1
have hinfo : (∫ p, H p ^ 2 ∂P) ≤
Real.cosh ((d : ℝ) * beta + 1) ^ 2 * (∫ p, M p ^ 2 ∂P) +
(16 * (1 + ((d : ℝ) * beta) ^ 2) * (1 + Real.exp 32)) * (t : ℝ) := by
apply hsecond.2.trans
have hb := mul_le_mul_of_nonneg_left hX4.2
(by positivity : 0 ≤ 2 * (1 + ((d : ℝ) * beta) ^ 2))
nlinarith
have hE := (finiteIsingBrownianEntropy_moment hbeta μ ν hB hmB D v t).1
calc
_ ≤ ∫ p, 2 * |H p| ∂P := by
apply integral_mono hE ((hH2.integrable (by norm_num)).abs.const_mul 2)
intro p
exact FinitePosterior.fieldEntropy_le_abs_field _ _ _ _ hspin hneg hpos
_ = 2 * ∫ p, |H p| ∂P := integral_const_mul _ _
_ ≤ 2 * Real.sqrt (∫ p, H p ^ 2 ∂P) :=
mul_le_mul_of_nonneg_left (integral_abs_le_sqrt_integral_sq hH hsecond.1) (by norm_num)
_ ≤ _ := mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt hinfo) (by norm_num)
theorem brownianTerminal_information_eq_gaussian
{d : ℕ} {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ]
(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)) (t : ℝ≥0) (v : TreeVertex d) :
(∫ p, ((μ.prod (Measure.infinitePi ν))[fun z => spinSign (z.1 v) |
brownianTerminalSigma B t] p) ^ 2 ∂μ.prod (Measure.infinitePi ν)) =
∫ z, (PlantedSymmetry.posteriorMagnetization
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) t v z) ^ 2
∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) := by
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
let T : ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) →
(TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) :=
fun p => (p.1, fun i => B i t (p.2 i))
have hT : MeasurePreserving T (μ.prod (Measure.infinitePi ν))
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) :=
(MeasurePreserving.id μ).prod (brownian_array_eval_measurePreserving ν hB hmB t)
have hspin : Measurable (fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) =>
spinSign (z.1 v)) := (measurable_of_finite spinSign).comp
((measurable_pi_apply v).comp measurable_fst)
have hpull := condExp_comp_measurePreserving_comap hT hspin.stronglyMeasurable
(PlantedSymmetry.integrable_plantedSpin _ v)
(PlantedSymmetry.observationSigma_le (TreeVertex d) t)
have hsame : (PlantedSymmetry.observationSigma (TreeVertex d) (t : ℝ)).comap T =
brownianTerminalSigma B t := by
rw [PlantedSymmetry.observationSigma, MeasurableSpace.comap_comp]
change MeasurableSpace.comap
(fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω) =>
fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i)) inferInstance = _
simp only [brownianTerminalSigma, Real.toNNReal_coe]
rw [hsame] at hpull
calc
_ = ∫ p, (PlantedSymmetry.posteriorMagnetization
(μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) t v (T p)) ^ 2
∂μ.prod (Measure.infinitePi ν) := by
apply integral_congr_ae
filter_upwards [hpull] with p hp
exact congrArg (fun x : ℝ => x ^ 2) hp
_ = _ := hT.hasLaw.integral_comp
(((stronglyMeasurable_condExp).mono
(PlantedSymmetry.observationSigma_le (TreeVertex d) t)).pow 2).aestronglyMeasurable
theorem HasFreeIsingLaw.finiteIsingBrownian_critical_information_upper
{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))
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D)
(t : ℝ≥0) (ht : (t : ℝ) ≤ 1) :
(∫ p, finiteIsingBrownianMean beta D v B t p ^ 2
∂μ.prod (Measure.infinitePi ν)) ≤
((1 + (Real.tanh beta) ^ 2) *
Real.sqrt (1 / ((1 - (Real.tanh beta) ^ 2) * (Real.tanh beta) ^ 4))) *
Real.sqrt (t : ℝ) := by
let := hμ.isProbabilityMeasure
have h := hμ.finiteIsingBrownian_information_le ν hB hmB D hD v t
dsimp only at h
rw [brownianTerminal_information_eq_gaussian μ ν hB hmB t v] at h
exact h.trans (hμ.gaussian_site_critical_information_upper hd hb hcritical t ht v)
def criticalEntropyEnvelope (d : ℕ) (beta t : ℝ) : ℝ :=
2 * Real.sqrt
(Real.cosh ((d : ℝ) * beta + 1) ^ 2 *
(((1 + (Real.tanh beta) ^ 2) *
Real.sqrt (1 / ((1 - (Real.tanh beta) ^ 2) * (Real.tanh beta) ^ 4))) *
Real.sqrt t) +
(16 * (1 + ((d : ℝ) * beta) ^ 2) * (1 + Real.exp 32)) * t)
theorem HasFreeIsingLaw.finiteIsingBrownianEntropy_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))
(D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D)
(t : ℝ≥0) (ht : (t : ℝ) ≤ 1) :
(∫ p, finiteIsingBrownianEntropy beta D v B t p
∂μ.prod (Measure.infinitePi ν)) ≤ criticalEntropyEnvelope d beta t := by
let := hμ.isProbabilityMeasure
apply (finiteIsingBrownianEntropy_le_information hb μ ν hB hmB D v t ht).trans
apply mul_le_mul_of_nonneg_left _ (by norm_num : (0 : ℝ) ≤ 2)
apply Real.sqrt_le_sqrt
apply add_le_add_left
exact mul_le_mul_of_nonneg_left
(hμ.finiteIsingBrownian_critical_information_upper hd hb hcritical ν hB hmB D hD v t ht)
(sq_nonneg _)
@[simp] theorem criticalEntropyEnvelope_zero (d : ℕ) (beta : ℝ) :
criticalEntropyEnvelope d beta 0 = 0 := by simp [criticalEntropyEnvelope]
theorem continuous_criticalEntropyEnvelope (d : ℕ) (beta : ℝ) :
Continuous (criticalEntropyEnvelope d beta) := by
unfold criticalEntropyEnvelope
fun_prop
end Problem367
end
end OAI
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