| import OAI.Probability.FreeIsing.Cavity.Array |
| import OAI.Probability.FreeIsing.Estimates.CoupledSmallLogLoss |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory |
|
|
| namespace Problem367.CavityArray |
|
|
| variable {d : ℕ} {theta : ℝ} |
|
|
| theorem field_identDistrib_of_measurePreserving |
| {Ω : Type*} [MeasurableSpace Ω] {ν : Measure Ω} |
| {μ : Measure (TreeVertex d → ℝ)} {F : Ω → (TreeVertex d → ℝ)} |
| (hF : MeasurePreserving F ν μ) (h0 : 0 ≤ theta) (h1 : theta < 1) |
| (v w : TreeVertex d) : |
| IdentDistrib (fun ω => field h0 h1 (F ω) v w) |
| (fun H => field h0 h1 H v w) ν μ := by |
| have hm := measurable_field_apply h0 h1 v w |
| refine ⟨(hm.comp hF.measurable).aemeasurable, hm.aemeasurable, ?_⟩ |
| rw [show (fun ω => field h0 h1 (F ω) v w) = |
| (fun H => field h0 h1 H v w) ∘ F from rfl, |
| ← Measure.map_map hm hF.measurable, hF.map_eq] |
|
|
| theorem integrable_site_sub_sq_of_bound |
| {ν : Measure (SitePair d)} [IsProbabilityMeasure ν] |
| {M : ℝ} (hM : 0 ≤ M) (v : TreeVertex d) |
| (hbound : ∀ᵐ p ∂ν, |p.1 v - p.2 v| ≤ M) : |
| Integrable (fun p : SitePair d => (p.1 v - p.2 v) ^ 2) ν := by |
| apply (integrable_const (M ^ 2)).mono' |
| ((((measurable_pi_apply v).comp measurable_fst).sub |
| ((measurable_pi_apply v).comp measurable_snd)).pow_const 2).aestronglyMeasurable |
| filter_upwards [hbound] with p hp |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| simpa only [sq_abs, Function.comp_apply, Pi.sub_apply] using |
| (sq_le_sq₀ (abs_nonneg _) hM).2 hp |
|
|
| theorem integral_log_field_sub_sq_le |
| {ν : Measure (SitePair d)} |
| (h0 : 0 ≤ theta) (h1 : theta < 1) |
| (hinv : ∀ g : TreeAut d, MeasurePreserving (pairRelabel g) ν ν) |
| (o v w : TreeVertex d) |
| (hInt : Integrable (fun p : SitePair d => (p.1 o - p.2 o) ^ 2) ν) : |
| (∫ p, (Real.log (responseAttenuation theta (field h0 h1 p.1 v w)) - |
| Real.log (responseAttenuation theta (field h0 h1 p.2 v w))) ^ 2 ∂ν) ≤ |
| cavityLogVarianceCoefficient theta * |
| (∫ p : SitePair d, (p.1 o - p.2 o) ^ 2 ∂ν) := by |
| have hv := (site_sub_identDistrib hinv o v).sq |
| have hw := (site_sub_identDistrib hinv o w).sq |
| exact integral_cavity_log_discrepancy_sq_le h0 h1 |
| ((measurable_field_apply h0 h1 v w).comp measurable_fst) |
| ((measurable_field_apply h0 h1 v w).comp measurable_snd) |
| (hv.integrable_snd hInt) (hw.integrable_snd hInt) |
| (Filter.Eventually.of_forall (fun p => |
| ⟨field_spec h0 h1 p.1 v w, field_spec h0 h1 p.1 w v, |
| field_spec h0 h1 p.2 v w, field_spec h0 h1 p.2 w v⟩)) |
| hv.integral_eq.symm.le hw.integral_eq.symm.le |
|
|
| theorem integral_log_field_sub_sq_small_time |
| {ν : Measure (SitePair d)} [IsProbabilityMeasure ν] |
| {μ : Measure (TreeVertex d → ℝ)} |
| (h0 : 0 ≤ theta) (h1 : theta < 1) |
| (hinv : ∀ g : TreeAut d, MeasurePreserving (pairRelabel g) ν ν) |
| (hfst : MeasurePreserving Prod.fst ν μ) |
| (hsnd : MeasurePreserving Prod.snd ν μ) |
| (o v w : TreeVertex d) {M t c : ℝ} (hM : 0 ≤ M) (ht : 0 < t) |
| (hbound : ∀ᵐ p ∂ν, ∀ z : TreeVertex d, |p.1 z - p.2 z| ≤ M) |
| (hTail : μ {H | t ^ (1 / 8 : ℝ) < |field h0 h1 H v w|} ≤ |
| ENNReal.ofReal (Real.exp (-c * (Real.log (1 / t)) ^ 2))) : |
| (∫ p, (Real.log (responseAttenuation theta (field h0 h1 p.1 v w)) - |
| Real.log (responseAttenuation theta (field h0 h1 p.2 v w))) ^ 2 ∂ν) ≤ |
| cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * |
| (∫ p : SitePair d, (p.1 o - p.2 o) ^ 2 ∂ν) + |
| min (cavityLogVarianceCoefficient theta * |
| (∫ p : SitePair d, (p.1 o - p.2 o) ^ 2 ∂ν)) |
| (cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2)) := by |
| have hInt := integrable_site_sub_sq_of_bound hM o (hbound.mono (fun _ h => h o)) |
| have hv := (site_sub_identDistrib hinv o v).sq |
| have hw := (site_sub_identDistrib hinv o w).sq |
| exact integral_cavity_log_discrepancy_small_time_of_common_law h0 h1 |
| ((measurable_field_apply h0 h1 v w).comp measurable_fst) |
| ((measurable_field_apply h0 h1 v w).comp measurable_snd) |
| (hv.integrable_snd hInt) (hw.integrable_snd hInt) |
| (Filter.Eventually.of_forall (fun p => |
| ⟨field_spec h0 h1 p.1 v w, field_spec h0 h1 p.1 w v, |
| field_spec h0 h1 p.2 v w, field_spec h0 h1 p.2 w v⟩)) |
| hM ht (hbound.mono (fun _ h => ⟨h v, h w⟩)) |
| hv.integral_eq.symm.le hw.integral_eq.symm.le |
| (field_identDistrib_of_measurePreserving hfst h0 h1 v w) |
| (field_identDistrib_of_measurePreserving hsnd h0 h1 v w) hTail |
|
|
| end Problem367.CavityArray |
|
|
| end |
|
|
| end OAI |
|
|
