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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.GraphGeometry.CutEdges
import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls
import OAI.Probability.CriticalPercolation.ProductMeasure.LevelContactMeasurability
namespace OAI
section
noncomputable section
open Set Filter
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def ballCut (o : V) (n : ℕ) : Set E := G.cutEdges (G.vertexBall o n)
lemma ballCut_endpoints (o : V) (n : ℕ) {e : E} (he : e ∈ G.ballCut o n) :
∃ x y, G.ends e = s(x,y) ∧ x ∈ G.vertexBall o (n+1) \ G.vertexBall o n ∧
y ∈ G.vertexBall o n := by
obtain ⟨x,y,he,hx,hy⟩ := he
have hne : y ≠ x := by rintro rfl; exact hx hy
have hadj : G.fullGraph.Adj y x :=
(SimpleGraph.fromEdgeSet_adj _).mpr ⟨⟨e,Set.mem_univ e,he.trans (Sym2.eq_swap)⟩,hne⟩
exact ⟨x,y,he,⟨G.vertexBall_step hy hadj,hx⟩,hy⟩
lemma ballCut_subset_incident_sphere (o : V) (n : ℕ) :
G.ballCut o n ⊆ G.incidentEdges (G.vertexBall o (n+1) \ G.vertexBall o n) := by
intro e he
obtain ⟨x,y,he,hx,_⟩ := G.ballCut_endpoints o n he
exact ⟨x,hx,by simp [he]⟩
lemma ballCut_finite (hl : G.LocallyFinite) (o : V) (n : ℕ) :
(G.ballCut o n).Finite :=
(G.finite_incident hl ((G.vertexBall_finite hl o (n+1)).sdiff)).subset
(G.ballCut_subset_incident_sphere o n)
lemma ballCut_endpoints_mem (o : V) (n : ℕ) {e : E} (he : e ∈ G.ballCut o n)
{x : V} (hx : x ∈ G.ends e) : x ∈ G.vertexBall o (n+1) := by
obtain ⟨a,b,he,ha,hb⟩ := G.ballCut_endpoints o n he
rw [he] at hx
rcases Sym2.mem_iff.mp hx with rfl|rfl
· exact ha.1
· exact G.vertexBall_mono o (Nat.le_succ n) hb
lemma ballCut_pairwise_disjoint (o : V) : Pairwise (fun n m =>
Disjoint (G.ballCut o n) (G.ballCut o m)) := by
suffices h : ∀ n m : ℕ, n < m → Disjoint (G.ballCut o n) (G.ballCut o m) by
intro n m hnm
rcases lt_or_gt_of_ne hnm with hlt|hlt
· exact h n m hlt
· exact (h m n hlt).symm
intro n m hnm
apply Set.disjoint_left.mpr
intro e hen hem
obtain ⟨x,y,he,hx,hy⟩ := hem
have hmem : x ∈ G.ends e := by simp [he]
exact hx (G.vertexBall_mono o (Nat.succ_le_iff.mpr hnm)
(G.ballCut_endpoints_mem o n hen hmem))
lemma ballCut_card_le (hl : G.LocallyFinite) (o : V) (n D : ℕ)
(hD : ∀ v, (G.incidentEdges {v}).ncard ≤ D) :
(G.ballCut o n).ncard ≤
(G.vertexBall o (n+1) \ G.vertexBall o n).ncard * D := by
exact (Set.ncard_le_ncard (G.ballCut_subset_incident_sphere o n)
(G.finite_incident hl ((G.vertexBall_finite hl o (n+1)).sdiff))).trans
(G.incident_ncard_le _ ((G.vertexBall_finite hl o (n+1)).sdiff) D (fun v _ => hD v))
lemma cluster_subset_ball_of_closed_ballCut (o : V) (n : ℕ) (ω : Set E)
(hc : Disjoint (G.ballCut o n) ω) {v : V} (hv : v ∈ G.vertexBall o n) :
G.cluster ω v ⊆ G.vertexBall o n := by
intro x hx
by_contra hnot
exact G.not_reachable_of_cut_closed (G.vertexBall o n) ω
(fun e he => Set.disjoint_left.mp hc he) hnot hv hx.symm
lemma frequently_small_increments_of_linear_scales (f : ℕ → ℕ) (C : ℕ)
(hf : ∀ R : ℕ, ∃ n≥R, f n ≤ C*n) :
∃ᶠ n in atTop, f (n+1)-f n ≤ C := by
by_contra h
rw [not_frequently] at h
obtain ⟨N,hN⟩ := eventually_atTop.mp h
have hg (k : ℕ) : (C+1)*k ≤ f (N+k) := by
induction k with
| zero => omega
| succ k ih =>
have hh := hN (N+k) (by omega)
have he : N+(k+1)=N+k+1 := by omega
rw [he]
have hinc : f (N+k)+C+1 ≤ f (N+k+1) := by omega
nlinarith
obtain ⟨n,hn,hfn⟩ := hf ((C+1)*N+1)
have hNn : N ≤ n := by nlinarith
have hh := hg (n-N)
rw [Nat.add_sub_cancel' hNn] at hh
have he := Nat.sub_add_cancel hNn
nlinarith
lemma frequently_small_spheres_of_linear_scales (hl : G.LocallyFinite)
(o : V) (C : ℕ) (hf : ∀ R : ℕ, ∃ n≥R, (G.vertexBall o n).ncard ≤ C*n) :
∃ᶠ n in atTop, (G.vertexBall o (n+1) \ G.vertexBall o n).ncard ≤ C := by
have hh := frequently_small_increments_of_linear_scales (fun n => (G.vertexBall o n).ncard) C hf
apply hh.mono
intro n hn
rwa [Set.ncard_sdiff (G.vertexBall_mono o (Nat.le_succ n))
(G.vertexBall_finite hl o n)]
end CriticalPercolation.BondGraph
end
end
end OAI
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