| import Mathlib |
| import OAI.Probability.CriticalPercolation.Exploration.Score |
| import OAI.Probability.CriticalPercolation.Modular.ModularClusterCount |
| import OAI.Probability.CriticalPercolation.Modular.HeavyClusterNumber |
| import OAI.Probability.CriticalPercolation.ProductMeasure.ProductConditioning |
| import OAI.Probability.CriticalPercolation.Clusters.FiniteForcing |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory |
| open scoped Classical ENNReal |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| lemma modularMass_union_ne_top (o : V) (S T : Set V) : |
| G.modularMass o (S∪T)≠∞ ↔ G.modularMass o S≠∞ ∧ G.modularMass o T≠∞ := by |
| constructor |
| · intro h |
| exact ⟨ne_top_of_le_ne_top h (G.modularMass_mono o Set.subset_union_left), |
| ne_top_of_le_ne_top h (G.modularMass_mono o Set.subset_union_right)⟩ |
| · rintro ⟨hS,hT⟩ |
| exact ne_top_of_le_ne_top (ENNReal.add_ne_top.mpr ⟨hS,hT⟩) (G.modularMass_union_le o S T) |
|
|
| lemma finiteMassIndicator_insert (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (ω : Set E) (e : E) (a b : V) (he : G.ends e=s(a,b)) : |
| G.finiteMassIndicator (insert e ω) a=G.finiteMassIndicator ω a*G.finiteMassIndicator ω b := by |
| have hh : G.modularMass a (G.cluster (insert e ω) a)≠∞ ↔ |
| G.modularMass a (G.cluster ω a)≠∞ ∧ G.modularMass b (G.cluster ω b)≠∞ := by |
| rw [G.cluster_insert_endpoint ω e a b he,G.modularMass_union_ne_top] |
| exact and_congr_right (fun _ => not_congr (G.modularMass_eq_top_change_base hc hl a b (G.cluster ω b))) |
| by_cases ha : G.modularMass a (G.cluster ω a)=∞ <;> |
| by_cases hb : G.modularMass b (G.cluster ω b)=∞ |
| · have hi : G.modularMass a (G.cluster (insert e ω) a)=∞ := by tauto |
| simp [finiteMassIndicator,ha,hb,hi] |
| · have hi : G.modularMass a (G.cluster (insert e ω) a)=∞ := by tauto |
| simp [finiteMassIndicator,ha,hb,hi] |
| · have hi : G.modularMass a (G.cluster (insert e ω) a)=∞ := by tauto |
| simp [finiteMassIndicator,ha,hb,hi] |
| · have hi := hh.mpr ⟨ha,hb⟩ |
| simp [finiteMassIndicator,ha,hb,hi] |
|
|
| lemma incidentClusterCount_insert_of_ends (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (ω : Set E) (e : E) (a b : V) (he : G.ends e=s(a,b)) : |
| G.incidentClusterCount (insert e ω) e=G.finiteMassIndicator ω a*G.finiteMassIndicator ω b := by |
| rw [G.incidentClusterCount_of_ends hc hl _ e a b he, |
| ite_eq_left (LazyDFS.prop_openEdge_reachable G (insert e ω) (Set.mem_insert e ω) he), |
| G.finiteMassIndicator_insert hc hl ω e a b he] |
|
|
| lemma incidentClusterCount_flip_of_ends (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (ω : Set E) (e : E) (a b : V) (he : G.ends e=s(a,b)) : |
| (G.incidentClusterCount ω e).toReal-(G.incidentClusterCount (insert e ω) e).toReal = |
| if ¬(G.openGraph ω).Reachable a b ∧ |
| ¬(G.modularMass a (G.cluster ω a)=∞ ∧ G.modularMass b (G.cluster ω b)=∞) then 1 else 0 := by |
| rw [G.incidentClusterCount_of_ends hc hl ω e a b he,G.incidentClusterCount_insert_of_ends hc hl ω e a b he] |
| by_cases hr : (G.openGraph ω).Reachable a b |
| · have hf := G.finiteMassIndicator_eq_of_reachable hc hl ω hr |
| rw [ite_eq_left hr,←hf] |
| unfold finiteMassIndicator |
| split_ifs <;> simp_all |
| · simp only [ite_eq_right hr] |
| unfold finiteMassIndicator |
| by_cases ha : G.modularMass a (G.cluster ω a)=∞ <;> |
| by_cases hb : G.modularMass b (G.cluster ω b)=∞ <;> norm_num [ha,hb,hr] |
| |
| lemma forceOn_univ_singleton (e : E) (ω : Set E) : |
| BernoulliProduct.forceOn {e} Set.univ ω=insert e ω := by |
| ext f |
| by_cases hf : f=e <;> simp [hf] |
| |
| lemma forceOn_empty_singleton (e : E) (ω : Set E) : |
| BernoulliProduct.forceOn {e} ∅ ω=ω\{e} := BernoulliProduct.forceOn_empty_eq_sdiff {e} ω |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
|
|
| end OAI |
|
|
