File size: 2,332 Bytes
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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.TwoPoint
import OAI.Probability.CriticalPercolation.Modular.LightClusterNumber
import OAI.Probability.CriticalPercolation.GraphGeometry.EncounterGeometry
import OAI.Probability.CriticalPercolation.Modular.LightTwoPointDecay
import OAI.Probability.CriticalPercolation.Modular.MainInfiniteLightBoundary
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma infinitelyManyLightClusters.infinite {ω : Set E}
(h : G.infinitelyManyLightClusters ω) : G.infinitelyManyInfiniteClusters ω := by
intro S
obtain ⟨v,hv,hs⟩ := h S
exact ⟨v,hv.1,hs⟩
theorem main_all_light_or_sparse_boundary [Infinite V]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(hpc : G.criticalProbability<1) (o : V)
(hper : G.law G.criticalProbability G.percolates≠0) :
(¬G.CombinatoriallyUnimodular ∧
(∀ᵐ ω ∂G.law G.criticalProbability, G.infinitelyManyLightClusters ω ∧
∀ v, G.modularMass o (G.cluster ω v)<∞) ∧
(∀ ε : ℝ, 0<ε → ∃ R : ℕ, ∀ x y, y∉G.vertexBall x R →
CriticalPercolation.Sharpness.twoPoint G G.criticalProbability x y<ε) ∧
(∀ k : ℕ, ∃ J : Set E, J.Finite ∧
0<G.law G.criticalProbability (G.encounterEvent J o k))) ∨
(G.SubexponentialGrowth ∧ ∃ d : ℕ, 2≤d ∧
∀ R : ℕ, ∃ n≥R, ∃ v : V, (G.vertexBall v n).ncard≤n^d) := by
let := G.countable_vertices hc hl
let := G.countable_bonds hc hl
rcases G.main_infinite_light_or_sparse_boundary hc hl hq hpc o hper with
⟨hu,hlight,_⟩|hpoly
· have hp : 0<G.criticalProbability := by
by_contra hn
have he : G.criticalProbability=0 := le_antisymm (le_of_not_gt hn) bot_le
exact hper (by rw [he,G.law_zero_percolates])
refine Or.inl ⟨hu,hlight.and (G.nonunimodular_critical_mass_finite_ae hc hl hq hpc hu o),
G.nonunimodular_critical_twoPoint_decay hc hl hq hpc hu,?_⟩
intro k
exact G.exists_positive_encounter hc G.criticalProbability hp o k
(hlight.mono (fun _ h => h.infinite G))
· exact Or.inr hpoly
end CriticalPercolation.BondGraph
end
end
end OAI
|