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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 | import OAI.Probability.FreeIsing.Trees.InfiniteCanonicalPathResponse
import OAI.Probability.FreeIsing.Trees.InfiniteConeCavity
import OAI.Probability.FreeIsing.Trees.SelectedPathAttenuation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal BigOperators
namespace Problem367
open PlantedSymmetry CavityFieldLimit PathSideBranches
theorem HasFreeIsingLaw.rootResponseProduct_eq_forward_product
{d : ℕ} {β : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh β) μ) (hβ : 0 ≤ β)
(t : ℝ≥0) (v : TreeVertex d) :
∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
rootResponseProduct hβ v (observation t p) =
Real.tanh β ^ v.1.length * ∏ k ∈ Finset.range v.1.length,
responseAttenuation (Real.tanh β) (forwardGaussianField μ t v k p) := by
have hfield (i : Fin v.1.length) :=
hμ.infiniteCavityField_path_eq_forward_posterior hβ t v i i.2
filter_upwards [ae_all_iff.mpr hfield] with p hp
unfold rootResponseProduct
congr 1
apply Finset.prod_congr rfl
intro k hk
exact congrArg (responseAttenuation (Real.tanh β)) (hp ⟨k, Finset.mem_range.mp hk⟩)
theorem HasFreeIsingLaw.integral_rootResponseProduct_sq_le_critical
{d : ℕ} {β : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh β) μ) (hβ : 0 ≤ β) (hd : 3 ≤ d)
(hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh β ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) (v : TreeVertex d) :
(∫ p, rootResponseProduct hβ v (observation t p) ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤
(Real.tanh β ^ v.1.length) ^ 2 *
Real.exp (-(ComponentSignalAttenuation.lossCoefficient (Real.tanh β) *
min (Real.sqrt (t : ℝ)) 1) * (v.1.length : ℝ)) := by
calc
_ = (Real.tanh β ^ v.1.length) ^ 2 *
∫ p, ∏ k ∈ Finset.range v.1.length,
responseAttenuation (Real.tanh β) (forwardGaussianField μ t v k p) ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t) := by
rw [← integral_const_mul]
apply integral_congr_ae
filter_upwards [hμ.rootResponseProduct_eq_forward_product hβ t v] with p hp
rw [hp, mul_pow, Finset.prod_pow]
_ ≤ _ := mul_le_mul_of_nonneg_left
(integral_forwardGaussianField_attenuation_le_exp_critical hμ hβ hd hcritical ht v)
(sq_nonneg _)
theorem HasFreeIsingLaw.integral_gaussianRootResponse_sq_le_critical
{d : ℕ} {β : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh β) μ) (hβ : 0 ≤ β) (hd : 3 ≤ d)
(hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh β ^ 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 β ^ v.1.length) ^ 2 *
Real.exp (-(ComponentSignalAttenuation.lossCoefficient (Real.tanh β) *
min (Real.sqrt (t : ℝ)) 1) * (v.1.length : ℝ)) := by
have he := integral_congr_ae ((hμ.gaussian_rootResponse_eq_canonical_product hβ t v).fun_comp
(fun z : ℝ => z ^ 2))
simp only [Function.comp_apply] at he
rw [he]
exact hμ.integral_rootResponseProduct_sq_le_critical hβ hd hcritical ht v
theorem HasFreeIsingLaw.integral_infiniteGibbsRootResponse_sq_le_critical
{d : ℕ} {β : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh β) μ) (hβ : 0 ≤ β) (hd : 3 ≤ d)
(hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh β ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) (v : TreeVertex d) :
(∫ p, infiniteGibbsResponse β (ConnectedHull.root d) v (observation t p) ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤
(Real.tanh β ^ v.1.length) ^ 2 *
Real.exp (-(ComponentSignalAttenuation.lossCoefficient (Real.tanh β) *
min (Real.sqrt (t : ℝ)) 1) * (v.1.length : ℝ)) := by
have he := integral_congr_ae ((hμ.infiniteGibbsRootResponse_eq_canonical_product hβ t v).fun_comp
(fun z : ℝ => z ^ 2))
simp only [Function.comp_apply] at he
rw [he]
exact hμ.integral_rootResponseProduct_sq_le_critical hβ hd hcritical ht v
end Problem367
end
end OAI
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