| import Mathlib |
| import OAI.Probability.CriticalPercolation.Model |
| import OAI.Probability.CriticalPercolation.Criticality.FiniteSeedTree |
| import OAI.Probability.CriticalPercolation.Gluing.CollarDistance |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| lemma inward_path (hc : G.fullGraph.Connected) {B : Set V} (hB : B.Nonempty) |
| (v : V) (l : ℕ) (hl : l ≤ G.distanceTo B v) : |
| ∃ (y : V) (P : Set E), P.Finite ∧ P.ncard ≤ l ∧ |
| G.distanceTo B y+l=G.distanceTo B v ∧ y∈G.vertexBall v l ∧ |
| P⊆G.inducedEdges (G.vertexBall v l∩{z | G.distanceTo B z ≤ G.distanceTo B v}) ∧ |
| P⊆(G.inducedEdges {z | G.distanceTo B v ≤ G.distanceTo B z})ᶜ ∧ |
| (G.openGraph P).Reachable v y := by |
| induction l generalizing v with |
| | zero => |
| exact ⟨v,∅,Set.finite_empty,by simp,by omega,rfl,Set.empty_subset _,Set.empty_subset _, |
| SimpleGraph.Reachable.refl _⟩ |
| | succ l ih => |
| obtain ⟨w,hw,hr⟩ := G.distanceTo_predecessor hc hB v (by omega) |
| obtain ⟨y,P,hP,hcard,hy,hyball,hends,hfresh,hreach⟩ := ih w (by omega) |
| have hwball : w∈G.vertexBall v 1 := G.vertexBall_step (G.mem_vertexBall_self v 0) hw |
| obtain ⟨⟨e,_,he⟩,hne⟩ := hw |
| refine ⟨y,insert e P,hP.insert _,?_,by omega,?_,?_,?_,?_⟩ |
| · exact (Set.ncard_insert_le _ _).trans (by omega) |
| · simpa only [Nat.add_comm 1] using G.vertexBall_add hwball l hyball |
| · intro f hf z hz |
| rcases hf with rfl|hf |
| · rw [he] at hz |
| have hz' : z=v ∨ z=w := Sym2.mem_iff.mp hz |
| rcases hz' with hz'|hz' |
| · subst z |
| exact ⟨G.mem_vertexBall_self v (l+1),show G.distanceTo B v ≤ G.distanceTo B v from le_rfl⟩ |
| · subst z |
| exact ⟨G.vertexBall_mono v (by omega) hwball,by change G.distanceTo B w ≤ G.distanceTo B v; omega⟩ |
| · have hh := hends hf z hz |
| have hh2 : G.distanceTo B z ≤ G.distanceTo B w := hh.2 |
| refine ⟨?_,by change G.distanceTo B z ≤ G.distanceTo B v; omega⟩ |
| simpa only [Nat.add_comm 1] using G.vertexBall_add hwball l hh.1 |
| · intro f hf hf' |
| rcases hf with rfl|hf |
| · have hh := hf' w (by rw [he]; simp) |
| change G.distanceTo B v ≤ G.distanceTo B w at hh |
| omega |
| · apply hfresh hf |
| intro z hz |
| have hh := hf' z hz |
| change G.distanceTo B v ≤ G.distanceTo B z at hh |
| change G.distanceTo B w ≤ G.distanceTo B z |
| omega |
| · apply (show (G.openGraph (insert e P)).Adj v w from |
| ⟨⟨e,Set.mem_insert e P,he⟩,hne⟩).reachable.trans |
| exact hreach.mono (G.openGraph_mono (Set.subset_insert e P)) |
|
|
| lemma inward_seed_trial (hc : G.fullGraph.Connected) (hlocal : G.LocallyFinite) |
| {B : Set V} (hB : B.Nonempty) (v : V) (m : ℕ) (hm : m<G.distanceTo B v) : |
| ∃ (y : V) (F : Set E), F.Finite ∧ |
| F.ncard ≤ m+1+(G.vertexBall y m).ncard ∧ |
| G.distanceTo B y+(m+1)=G.distanceTo B v ∧ |
| F⊆G.inducedEdges (G.vertexBall v (2*m+1)∩{z | G.distanceTo B z ≤ G.distanceTo B v}) ∧ |
| F⊆(G.inducedEdges {z | G.distanceTo B v ≤ G.distanceTo B z})ᶜ ∧ |
| ∀ z∈G.vertexBall y m, (G.openGraph F).Reachable v z := by |
| obtain ⟨y,P,hP,hcard,hy,hyball,hends,hfresh,hreach⟩ := G.inward_path hc hB v (m+1) (by omega) |
| obtain ⟨Q,hQ,hQends,hQcard,hQreach⟩ := G.finite_seed_tree hlocal y m |
| refine ⟨y,P∪Q,hP.union hQ,?_,hy,?_,?_,?_⟩ |
| · exact (Set.ncard_union_le P Q).trans (by omega) |
| · intro e he z hz |
| rcases he with he|he |
| · have hh := hends he z hz |
| exact ⟨G.vertexBall_mono v (by omega) hh.1,hh.2⟩ |
| · have hh := hQends he z hz |
| have hle := G.distanceTo_ball hc hB hh |
| refine ⟨?_,by change G.distanceTo B z ≤ G.distanceTo B v; omega⟩ |
| have := G.vertexBall_add hyball m hh |
| simpa only [show m+1+m=2*m+1 by omega] using this |
| · intro e he he' |
| rcases he with he|he |
| · exact hfresh he he' |
| · obtain ⟨⟨x,z⟩,hxz⟩ := Sym2.mk_surjective (G.ends e) |
| have hx : x∈G.ends e := by rw [← hxz]; simp |
| have hh := G.distanceTo_ball hc hB (hQends he x hx) |
| have hh' := he' x hx |
| change G.distanceTo B v ≤ G.distanceTo B x at hh' |
| omega |
| · intro z hz |
| exact (hreach.mono (G.openGraph_mono Set.subset_union_left)).trans |
| ((hQreach z hz).mono (G.openGraph_mono Set.subset_union_right)) |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
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|
