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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 | import OAI.Probability.FreeIsing.Brownian.PosteriorBareResponse
import OAI.Probability.FreeIsing.Critical.CanonicalResponse
import OAI.Probability.FreeIsing.Gaussian.TreeResponseObservedPosterior
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal
namespace Problem367
open PlantedSymmetry
namespace CriticalCanonicalRowEnergy
def rate (beta : ℝ) (t : ℝ≥0) : ℝ :=
ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) *
min (Real.sqrt (t : ℝ)) 1
theorem rate_pos {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) : 0 < rate beta t := by
have htheta : 0 < Real.tanh beta := by
have hn := tanh_nonneg_of_beta_nonneg hb
by_contra h
have hz : Real.tanh beta = 0 := le_antisymm (not_lt.mp h) hn
simp [hz] at hc
exact mul_pos (ComponentSignalAttenuation.lossCoefficient_pos htheta
(Real.tanh_lt_one beta))
(lt_min (Real.sqrt_pos.mpr (show (0 : ℝ) < t from ht)) zero_lt_one)
theorem branching_ratio {d : ℕ} {beta : ℝ}
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) (t : ℝ≥0) :
((d - 1 : ℕ) : ℝ) * (Real.tanh beta ^ 2 * Real.exp (-rate beta t)) =
Real.exp (-rate beta t) := by rw [← mul_assoc, hc, one_mul]
theorem ratio_lt_one {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) :
((d - 1 : ℕ) : ℝ) * (Real.tanh beta ^ 2 * Real.exp (-rate beta t)) < 1 := by
rw [branching_ratio hc]
exact Real.exp_lt_one_iff.mpr (neg_neg_of_pos (rate_pos hb hc ht))
theorem exp_div_one_sub_exp_le_inv {x : ℝ} (hx : 0 < x) :
Real.exp (-x) / (1 - Real.exp (-x)) ≤ x⁻¹ := by
have hq : Real.exp (-x) < 1 := Real.exp_lt_one_iff.mpr (neg_neg_of_pos hx)
have he := mul_le_mul_of_nonneg_right (Real.add_one_le_exp x) (Real.exp_pos (-x)).le
have hcancel : Real.exp x * Real.exp (-x) = 1 := by
rw [← Real.exp_add, add_neg_cancel, Real.exp_zero]
rw [hcancel] at he
apply (div_le_iff₀ (sub_pos.mpr hq)).mpr
rw [inv_mul_eq_div]
exact (le_div_iff₀ hx).mpr (by nlinarith)
end CriticalCanonicalRowEnergy
open CriticalCanonicalRowEnergy
theorem HasFreeIsingLaw.gaussian_rootResponse_critical_radial_bound
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) (v : TreeVertex d) :
(∫ p, gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤
(Real.tanh beta ^ 2 * Real.exp (-rate beta t)) ^ v.1.length := by
have h := hμ.integral_gaussianRootResponse_sq_le_critical hb hd hc ht v
simpa only [rate, mul_pow, ← pow_mul, Nat.mul_comm,
← Real.exp_nat_mul, mul_comm] using h
theorem HasFreeIsingLaw.gaussian_rootResponse_critical_row_energy
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) :
(∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
Summable (fun v => gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2)) ∧
Integrable (fun p => ∑' v, gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2)
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ∧
(∫ p, ∑' v, gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤
1 + (d : ℝ) * Real.tanh beta ^ 2 * Real.exp (-rate beta t) /
(1 - Real.exp (-rate beta t)) := by
have hold := hμ.gaussian_posteriorResponse_row_energy hb ht hc.le
(ConnectedHull.root d) rfl
refine ⟨hold.1, hold.2.1, ?_⟩
have h := integral_tsum_sq_le_of_tree_radial_bound (C := 1)
(fun v => (hμ.gaussian_posteriorResponse_sq_decay hb ht (ConnectedHull.root d) v rfl).1)
(fun v => by simpa only [one_mul] using
hμ.gaussian_rootResponse_critical_radial_bound hb hd hc ht v)
(mul_nonneg (sq_nonneg _) (Real.exp_pos _).le) (ratio_lt_one hb hc ht)
simpa only [one_mul, branching_ratio hc, mul_assoc] using h
theorem HasFreeIsingLaw.integral_gaussian_rootResponse_row_le_inv_rate
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) :
(∫ p, ∑' v, gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤
1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t := by
apply (hμ.gaussian_rootResponse_critical_row_energy hb hd hc ht).2.2.trans
have h := mul_le_mul_of_nonneg_left
(exp_div_one_sub_exp_le_inv (rate_pos hb hc ht))
(mul_nonneg (Nat.cast_nonneg d) (sq_nonneg (Real.tanh beta)))
simpa only [mul_div_assoc, div_eq_mul_inv, mul_assoc] using
add_le_add (show (1 : ℝ) ≤ 1 from le_rfl) h
theorem HasFreeIsingLaw.gaussian_rootResponse_critical_row_tail
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) (L : ℕ) :
(∫ p, ∑' v : TreeVertex d, if L < v.1.length then
gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2 else 0
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤
(d : ℝ) * Real.tanh beta ^ 2 * Real.exp (-rate beta t) *
Real.exp (-rate beta t) ^ L / (1 - Real.exp (-rate beta t)) := by
have h := integral_tsum_sq_cutoff_le_of_tree_radial_bound (C := 1)
(fun v => (hμ.gaussian_posteriorResponse_sq_decay hb ht (ConnectedHull.root d) v rfl).1)
(fun v => by simpa only [one_mul] using
hμ.gaussian_rootResponse_critical_radial_bound hb hd hc ht v)
(mul_nonneg (sq_nonneg _) (Real.exp_pos _).le) (ratio_lt_one hb hc ht) L
simpa only [one_mul, branching_ratio hc, mul_assoc] using h
theorem HasFreeIsingLaw.lintegral_gaussian_rootResponse_row_le_inv_rate
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) :
(∫⁻ p, ∑' v, ENNReal.ofReal
(gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2)
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤
ENNReal.ofReal (1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t) := by
have hrow := hμ.gaussian_rootResponse_critical_row_energy hb hd hc ht
have he := hrow.1.mono fun p hp =>
ENNReal.ofReal_tsum_of_nonneg (fun v => sq_nonneg
(gaussianPosteriorResponse μ t (ConnectedHull.root d) v p)) hp
rw [← lintegral_congr_ae he,
← ofReal_integral_eq_lintegral_ofReal hrow.2.1
(Eventually.of_forall fun p => tsum_nonneg (fun v => sq_nonneg _))]
exact ENNReal.ofReal_le_ofReal
(hμ.integral_gaussian_rootResponse_row_le_inv_rate hb hd hc ht)
theorem HasFreeIsingLaw.brownian_rootResponse_critical_row_energy
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((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)) {t : ℝ≥0} (ht : 0 < t) :
(∀ᵐ p ∂μ.prod (Measure.infinitePi ν),
Summable (fun v => brownianPosteriorResponse μ ν B t (ConnectedHull.root d) v p ^ 2)) ∧
Integrable (fun p => ∑' v,
brownianPosteriorResponse μ ν B t (ConnectedHull.root d) v p ^ 2)
(μ.prod (Measure.infinitePi ν)) ∧
(∫ p, ∑' v, brownianPosteriorResponse μ ν B t (ConnectedHull.root d) v p ^ 2
∂μ.prod (Measure.infinitePi ν)) ≤
1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
have hT := (MeasurePreserving.id μ).prod
(brownian_array_eval_measurePreserving ν hB hmB t)
have hg := hμ.gaussian_rootResponse_critical_row_energy hb hd hc ht
have heq := ae_all_iff.mpr (fun v => brownian_past_posteriorResponse_eq_gaussian
μ ν hB hmB t (ConnectedHull.root d) v)
have he : (fun p => ∑' v,
brownianPosteriorResponse μ ν B t (ConnectedHull.root d) v p ^ 2) =ᵐ[
μ.prod (Measure.infinitePi ν)]
(fun q => ∑' v, gaussianPosteriorResponse μ t (ConnectedHull.root d) v q ^ 2) ∘
Prod.map id (fun p i => B i t (p i)) := by
filter_upwards [heq] with p hp
exact tsum_congr fun v => congrArg (fun x : ℝ => x ^ 2) (hp v)
refine ⟨?_, (hT.integrable_comp_of_integrable hg.2.1).congr he.symm, ?_⟩
· filter_upwards [hT.quasiMeasurePreserving.ae hg.1, heq] with p hp hpeq
exact hp.congr fun v => congrArg (fun x : ℝ => x ^ 2) (hpeq v).symm
· rw [integral_congr_ae he, hT.hasLaw.integral_comp hg.2.1.aestronglyMeasurable]
exact hμ.integral_gaussian_rootResponse_row_le_inv_rate hb hd hc ht
end Problem367
end
end OAI
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