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f205175 | 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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.HitEvent
import OAI.Probability.CriticalPercolation.Modular.ClusterMassFinite
namespace OAI
section
noncomputable section
open Set MeasureTheory Filter
open scoped Classical BigOperators Topology
namespace CriticalPercolation.FiniteClusters
variable {V E : Type*} (G : BondGraph V E)
lemma cluster_inter_eq {ω K : Set E} {v : V}
(h : G.incidentEdges (G.cluster ω v) ⊆ K) :
G.cluster (K ∩ ω) v = G.cluster ω v := by
apply Set.Subset.antisymm (fun _ hr => hr.mono (G.openGraph_mono inter_subset_right))
intro w hw
rw [BondGraph.cluster,Set.mem_ofPred_eq,SimpleGraph.reachable_iff_reflTransGen] at hw ⊢
induction hw with
| refl => exact Relation.ReflTransGen.refl
| @tail u w hu huw ih =>
apply ih.tail
obtain ⟨⟨e,he,hend⟩,hne⟩ := huw
refine ⟨⟨e,⟨h ?_,he⟩,hend⟩,hne⟩
exact ⟨u,((G.openGraph ω).reachable_iff_reflTransGen _ _).mpr hu,by rw [hend]; simp⟩
lemma restrict_cluster (K : Set E) (ω : Set E) (v : V) :
(G.restrictEdges K).cluster (Subtype.val ⁻¹' ω) v = G.cluster (K ∩ ω) v := by
simp only [BondGraph.cluster,G.restrictEdges_openGraph]
lemma restrict_incident (K : Set E) (S : Set V) :
(G.restrictEdges K).incidentEdges S = Subtype.val ⁻¹' (G.incidentEdges S) := rfl
lemma restrict_local (hl : G.LocallyFinite) (K : Set E) : (G.restrictEdges K).LocallyFinite := by
intro v
change (Subtype.val ⁻¹' {e : E | v ∈ G.ends e} : Set K).Finite
exact (hl v).preimage Subtype.val_injective.injOn
lemma restrict_incident_bound (hl : G.LocallyFinite) (K : Set E) (D : ℕ)
(hD : ∀ v, (G.incidentEdges {v}).ncard ≤ D) (v : V) :
((G.restrictEdges K).incidentEdges {v}).ncard ≤ D := by
rw [restrict_incident]
exact (Set.ncard_le_ncard_of_injOn Subtype.val (fun _ h => h)
Subtype.val_injective.injOn (G.finite_incident hl (Set.finite_singleton v))).trans (hD v)
lemma restrict_rootSum_eq {K ω : Set E} {v : V}
(h : G.incidentEdges (G.cluster ω v) ⊆ K) (p : ℝ) :
rootSum (G.restrictEdges K) p (Subtype.val ⁻¹' ω) v = rootSum G p ω v := by
unfold rootSum
rw [restrict_cluster,cluster_inter_eq G h,restrict_incident]
change (∑' e : K, if e.val ∈ G.incidentEdges (G.cluster ω v) then coefficient p ω e.val else 0) = _
apply tsum_subtype_eq_of_support_subset (s := K)
(f := fun e => if e ∈ G.incidentEdges (G.cluster ω v) then coefficient p ω e else 0)
intro e he
by_contra hn
have hi : e ∉ G.incidentEdges (G.cluster ω v) := fun hi => hn (h hi)
exact he (ite_eq_right hi)
lemma restrict_rootAbsolute_eq {K ω : Set E} {v : V}
(h : G.incidentEdges (G.cluster ω v) ⊆ K) (s : ℕ) (p : ℝ) :
rootAbsolute (G.restrictEdges K) s p (Subtype.val ⁻¹' ω) v = rootAbsolute G s p ω v := by
rw [rootAbsolute,restrict_rootSum_eq G h]
unfold rootAbsolute mass size
rw [restrict_cluster,cluster_inter_eq G h]
lemma eventually_incident_subset (hl : G.LocallyFinite)
(K : Set.FiniteExhaustion (Set.univ : Set E)) {ω : Set E} {v : V}
(hC : (G.cluster ω v).Finite) : ∀ᶠ n in atTop, G.incidentEdges (G.cluster ω v) ⊆ K n := by
have hi := G.finite_incident hl hC
have hh (e : E) : ∀ᶠ n in atTop, e ∈ K n := by
have : e ∈ ⋃ n, K n := by rw [K.iUnion_eq]; trivial
obtain ⟨n,hn⟩ := Set.mem_iUnion.mp this
exact eventually_atTop.mpr ⟨n,fun m hm => K.mono hm hn⟩
exact (eventually_all_finite hi).mpr (fun e _ => hh e)
variable [Countable V] [Countable E]
omit [Countable E] in
lemma measurable_restrict (K : Set E) :
Measurable (fun ω : Set E => (Subtype.val ⁻¹' ω : Set K)) :=
Measurable.of_eval (fun edge => measurable_set_mem edge.val)
lemma restricted_rootAbsolute_integral (p : unitInterval) (K : Set E) (hK : K.Finite)
(s : ℕ) (v : V) :
(∫ ω, rootAbsolute (G.restrictEdges K) s p (Subtype.val ⁻¹' ω) v ∂G.law p) =
∫ ω, rootAbsolute (G.restrictEdges K) s p ω v ∂(G.restrictEdges K).law p := by
let := hK.fintype
simp only [BondGraph.law,← BernoulliProduct.law_const]
rw [← BernoulliProduct.law_map_restrict (fun _ : E => p) K]
rw [integral_map (measurable_restrict K).aemeasurable
(measurable_rootAbsolute (G.restrictEdges K) s p v).aestronglyMeasurable]
omit [Countable V] [Countable E] in
lemma ae_finite_clusters (p : unitInterval) (hp : p < G.criticalProbability) :
∀ᵐ ω ∂G.law p, ∀ v, (G.cluster ω v).Finite := by
have hh := G.law_percolates_eq_zero_of_lt_critical hp
rw [measure_eq_zero_iff_ae_notMem] at hh
filter_upwards [hh] with ω hω v
exact not_not.mp (fun hc => hω ⟨v,hc⟩)
theorem subcritical_rootAbsolute_bound (hl : G.LocallyFinite)
(D : ℕ) (hDpos : 1≤D) (hD : ∀ v, (G.incidentEdges {v}).ncard ≤ D)
(p : unitInterval) (hp : (p:ℝ) ∈ Ioo 0 1) (hpc : p < G.criticalProbability)
(k : ℕ) (v : V) :
(∫ ω, rootAbsolute G (4^k) p ω v ∂G.law p) ≤ (D:ℝ)/p*(6/(2:ℝ)^k) := by
let : IsProbabilityMeasure (G.law p) := by
unfold BondGraph.law
infer_instance
let K := (Set.to_countable (Set.univ : Set E)).finiteExhaustion
let F := fun n ω => rootAbsolute (G.restrictEdges (K n)) (4^k) p (Subtype.val ⁻¹' ω) v
have hb (n : ℕ) : (∫ ω, F n ω ∂G.law p) ≤ (D:ℝ)/p*(6/(2:ℝ)^k) := by
rw [show (∫ ω, F n ω ∂G.law p) = _ from
restricted_rootAbsolute_integral G p (K n) (K.finite n) (4^k) v]
let := (K.finite n).fintype
exact finite_rootAbsolute_integral (G.restrictEdges (K n)) p hp D hDpos
(restrict_incident_bound G hl (K n) D hD) k v
have ht : Tendsto (fun n => ∫ ω, F n ω ∂G.law p) atTop
(𝓝 (∫ ω, rootAbsolute G (4^k) p ω v ∂G.law p)) := by
apply tendsto_integral_of_dominated_convergence (fun _ => (D:ℝ)/(p:ℝ))
· intro n
exact ((measurable_rootAbsolute (G.restrictEdges (K n)) (4^k) p v).comp
(measurable_restrict (K n))).aestronglyMeasurable
· exact integrable_const _
· intro n
apply Eventually.of_forall
intro ω
dsimp only [F]
rw [Real.norm_eq_abs,abs_of_nonneg (rootAbsolute_nonneg _ _ _ _ _)]
exact rootAbsolute_bound (G.restrictEdges (K n)) (restrict_local G hl (K n)) D
(restrict_incident_bound G hl (K n) D hD) (4^k) hp _ v
· filter_upwards [ae_finite_clusters G p hpc] with ω hω
apply tendsto_const_nhds.congr'
filter_upwards [eventually_incident_subset G hl K (hω v)] with n hn
exact (restrict_rootAbsolute_eq G hn (4^k) p).symm
exact le_of_tendsto ht (Eventually.of_forall hb)
end CriticalPercolation.FiniteClusters
end
end
end OAI
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