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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Clusters.ClusterTail
import OAI.Probability.CriticalPercolation.Criticality.ExhaustionEvents
import OAI.Probability.CriticalPercolation.Modular.ClusterMassRestriction
import OAI.Probability.CriticalPercolation.Modular.ClusterMassEndpointFinite
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 restrict_edgeDiscrepancy_eq {K ω : Set E} (e : K)
(hx : G.incidentEdges (G.cluster ω (G.ends e.val).out.1) ⊆ K)
(hy : G.incidentEdges (G.cluster ω (G.ends e.val).out.2) ⊆ K)
(s : ℕ) (p : ℝ) :
edgeDiscrepancy (G.restrictEdges K) s p (Subtype.val ⁻¹' ω) e =
edgeDiscrepancy G s p ω e.val := by
have hc := cluster_inter_eq G hx
have hconn : ((G.restrictEdges K).openGraph (Subtype.val ⁻¹' ω)).Reachable
(G.ends e.val).out.1 (G.ends e.val).out.2 ↔
(G.openGraph ω).Reachable (G.ends e.val).out.1 (G.ends e.val).out.2 := by
rw [G.restrictEdges_openGraph]
exact Set.ext_iff.mp hc _
unfold edgeDiscrepancy edgeMass size coefficient
change (if e.val ∈ ω then -(1-p)/p else 1) *
(cappedMass s ((G.restrictEdges K).cluster (Subtype.val ⁻¹' ω) (G.ends e.val).out.1).ncard +
(if ((G.restrictEdges K).openGraph (Subtype.val ⁻¹' ω)).Reachable (G.ends e.val).out.1 (G.ends e.val).out.2 then 0
else cappedMass s ((G.restrictEdges K).cluster (Subtype.val ⁻¹' ω) (G.ends e.val).out.2).ncard)) = _
rw [hconn,restrict_cluster,restrict_cluster,cluster_inter_eq G hx,cluster_inter_eq G hy]
lemma restrict_twoCluster (K : Set E) (ω : Set E) (u v : V) (s : ℕ) :
Subtype.val ⁻¹' ω ∈ (G.restrictEdges K).twoClusterEvent u v s ↔
K ∩ ω ∈ G.twoClusterEvent u v s := by
simp only [BondGraph.twoClusterEvent,Set.mem_inter_iff,BondGraph.clusterTail,
Set.mem_ofPred_eq,restrict_cluster,G.restrictEdges_openGraph]
variable [Countable V] [Countable E]
lemma restrict_twoCluster_real (p : unitInterval) (K : Set E) (u v : V) (s : ℕ) :
((G.restrictEdges K).law p).real ((G.restrictEdges K).twoClusterEvent u v s) =
(G.law p).real {ω | K ∩ ω ∈ G.twoClusterEvent u v s} := by
simp only [BondGraph.law,← BernoulliProduct.law_const]
rw [← BernoulliProduct.law_map_restrict (fun _ : E => p) K]
simp only [measureReal_def]
rw [Measure.map_apply (measurable_restrict K)
((G.restrictEdges K).measurableSet_twoClusterEvent u v s)]
congr 2
ext ω
exact restrict_twoCluster G K ω u v s
lemma restricted_edgeDiscrepancy_integral (p : unitInterval) (K : Set E) (e : K) (s : ℕ) :
(∫ ω, edgeDiscrepancy (G.restrictEdges K) s p (Subtype.val ⁻¹' ω) e ∂G.law p) =
∫ ω, edgeDiscrepancy (G.restrictEdges K) s p ω e ∂(G.restrictEdges K).law p := by
simp only [BondGraph.law,← BernoulliProduct.law_const]
rw [← BernoulliProduct.law_map_restrict (fun _ : E => p) K]
rw [integral_map (measurable_restrict K).aemeasurable
(measurable_edgeDiscrepancy (G.restrictEdges K) s p e).aestronglyMeasurable]
theorem subcritical_endpoint_le_integral (hl : G.LocallyFinite)
(s : ℕ) (hs : 0<s) (p : unitInterval) (hp : (p:ℝ) ∈ Ioo 0 1)
(hpc : p < G.criticalProbability) (e : E) :
(G.law p).real (G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s) ≤
∫ ω, edgeDiscrepancy G s p ω e ∂G.law p := by
let K₀ := (Set.to_countable (Set.univ : Set E)).finiteExhaustion
let K : Set.FiniteExhaustion (Set.univ : Set E) :=
{ toFun := fun n => insert e (K₀ n)
finite' := fun n => (K₀.finite n).insert e
subset_succ' := fun n => Set.insert_subset_insert (K₀.subset_succ n)
iUnion_eq' := by
apply Set.eq_univ_of_forall
intro a
obtain ⟨n,hn⟩ := Set.mem_iUnion.mp (K₀.iUnion_eq.symm ▸ Set.mem_univ a)
exact Set.mem_iUnion.mpr ⟨n,Or.inr hn⟩ }
let en (n : ℕ) : K n := ⟨e,Or.inl rfl⟩
let F := fun n ω => edgeDiscrepancy (G.restrictEdges (K n)) s p (Subtype.val ⁻¹' ω) (en n)
have hb (n : ℕ) : (G.law p).real {ω | K n ∩ ω ∈
G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 s} ≤ ∫ ω, F n ω ∂G.law p := by
rw [← restrict_twoCluster_real G p (K n),show (∫ ω, F n ω ∂G.law p) = _ from
restricted_edgeDiscrepancy_integral G p (K n) (en n) s]
let := (K.finite n).fintype
exact finite_endpoint_le_integral (G.restrictEdges (K n)) s hs p hp (en n)
have ht : Tendsto (fun n => ∫ ω, F n ω ∂G.law p) atTop
(𝓝 (∫ ω, edgeDiscrepancy G s p ω e ∂G.law p)) := by
apply tendsto_integral_of_dominated_convergence (fun _ => 2/(p:ℝ))
· intro n
exact ((measurable_edgeDiscrepancy (G.restrictEdges (K n)) s p (en n)).comp
(measurable_restrict (K n))).aestronglyMeasurable
· exact integrable_const _
· intro n
apply Eventually.of_forall
intro ω
exact edgeDiscrepancy_abs_le (G.restrictEdges (K n)) s hp _ (en n)
· 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 (G.ends e).out.1),
eventually_incident_subset G hl K (hω (G.ends e).out.2)] with n hn hm
exact (restrict_edgeDiscrepancy_eq G (en n) hn hm s p).symm
exact le_of_tendsto_of_tendsto (G.tendsto_real_twoCluster_restrict p _ _ s K) ht
(Eventually.of_forall hb)
end CriticalPercolation.FiniteClusters
end
end
end OAI
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