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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.BallIncidence
import OAI.Probability.CriticalPercolation.Gluing.OpenTrialBound
namespace OAI
section
noncomputable section
open Set
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma aut_moves_vertex_outside_ball [Infinite V] (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (hq : G.QuasiTransitive) (o : V) (R : ℕ) :
∃ φ : G.Aut, φ.vertex o ∉ G.vertexBall o R := by
obtain ⟨C,hC⟩ := G.orbit_density_radius hc hq o
obtain ⟨v,hv⟩ := (G.vertexBall_finite hl o (R+C)).exists_notMem
obtain ⟨φ,hφ⟩ := hC v
refine ⟨φ,fun h => hv ?_⟩
exact G.vertexBall_add h C (G.vertexBall_comm.mp hφ)
lemma aut_displaces_finite_bonds [Infinite V] (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (hq : G.QuasiTransitive) (S : Set E) (hS : S.Finite) :
∃ φ : G.Aut, Disjoint S (φ.edge '' S) := by
obtain ⟨o⟩ := hc.nonempty
obtain ⟨R,hR⟩ := G.finite_bonds_in_ball hc o hS
obtain ⟨φ,hφ⟩ := G.aut_moves_vertex_outside_ball hc hl hq o (R+R)
refine ⟨φ,(G.inducedEdges_disjoint (G.vertexBalls_disjoint hφ)).mono hR ?_⟩
have hm := Aut.inducedEdges_map G φ hR
rwa [Aut.vertexBall_image] at hm
end CriticalPercolation.BondGraph
end
end
end OAI
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