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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.CoordinateScheme
import OAI.Probability.CriticalPercolation.Gluing.CorridorScaleChoice
import OAI.Probability.CriticalPercolation.Gluing.CoordinateProspective
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
open Corridor
variable {V E : Type*} (G : BondGraph V E)
lemma coordinateMove_sub_const (t z : V → ℝ) (a b : ℝ) (m : ℕ) (e : ℝ) (y : V) (s : ℝ) :
G.coordinateMove (fun x => t x-a) (fun x => z x-b) m e y s=G.coordinateMove t z m e y s := by
unfold coordinateMove
simp only [sub_sub_sub_cancel_right]
have ht : ∀ x, t x-a-(t y-a+1)=t x-(t y+1) := by intro x; ring
have hz : ∀ x, z x-b-(z y-b+s)=z x-(z y+s) := by intro x; ring
simp only [ht,hz]
theorem approximate_moves_contradiction
(hconnected : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(hpc : 0<(G.criticalProbability : ℝ)) (hpc1 : (G.criticalProbability : ℝ)<1)
(hmoves : ∀ e eta : ℝ, 0<e → e<1/10 → 0<eta → eta<1/10 →
∃ m : ℕ, ∀ rho : ℝ, 0<rho → ∃ t z : V → ℝ, ∃ p : unitInterval,
(G.criticalProbability : ℝ)/2≤p ∧ p<G.criticalProbability ∧
(∀ ⦃u v⦄, G.fullGraph.Adj u v → |t u-t v|≤rho) ∧
(∀ ⦃u v⦄, G.fullGraph.Adj u v → |z u-z v|≤rho) ∧
∀ y s, |s|=1 → 1-eta≤(BernoulliProduct.law (fun _ : E => p)).real
(G.coordinateMove t z m e y s)) : False := by
let := G.countable_vertices hconnected hl
let := G.countable_bonds hconnected hl
obtain ⟨o⟩ := hconnected.nonempty
obtain ⟨m,hm⟩ := hmoves ScaleChoice.e ScaleChoice.eta ScaleChoice.e_pos ScaleChoice.e_lt
ScaleChoice.eta_pos ScaleChoice.eta_lt
obtain ⟨D,_hDpos,hD⟩ := G.uniform_incident_bound hq
obtain ⟨M,hM⟩ := G.uniform_ball_bound hq (4*m+2)
obtain ⟨VM,hVM⟩ := G.uniform_ball_bound hq m
obtain ⟨k,R,hk,hR,hRlarge⟩ := ScaleChoice.trial_scales (G.criticalProbability : ℝ) D (M+1) VM m
obtain ⟨rho,hrho,hrhosmall⟩ := ScaleChoice.mesh R m
obtain ⟨t,z,p,hpl,hpu,ht,hz,hmv⟩ := hm rho hrho
let C : CoordinateData V E := {
graph := G, origin := o, time := fun x => t x-t o, height := fun x => z x-z o,
J := ScaleChoice.J, L := ScaleChoice.L, R := R, m := m, D := D, k := k,
M := M+1, VM := VM, e := ScaleChoice.e, rho := rho, kappa := ScaleChoice.kappa,
eta := ScaleChoice.eta, delta := ScaleChoice.delta, contour := ScaleChoice.contour,
pc := G.criticalProbability, p := p }
have hb : C.Bounds := {
connected := hconnected
locallyFinite := hl
time_origin := sub_self _
height_origin := sub_self _
time_step := fun {_ _} h => by simpa only [C,sub_sub_sub_cancel_right] using ht h
height_step := fun {_ _} h => by simpa only [C,sub_sub_sub_cancel_right] using hz h
J_lower := ScaleChoice.J_lower
L_lower := ScaleChoice.L_lower
e_pos := ScaleChoice.e_pos
rho_nonneg := hrho.le
e_small := ScaleChoice.e_small
rho_small := hrhosmall
R_gt := hR
pc_pos := hpc
pc_lt := hpc1
p_lower := hpl
p_upper := hpu.le
degree := hD
M_pos := Nat.succ_pos M
M_bound := fun v => (hM v).trans (Nat.le_succ M)
VM_bound := hVM
kappa_pos := ScaleChoice.kappa_pos
kappa_le := ScaleChoice.kappa_le
eta_nonneg := ScaleChoice.eta_pos.le
eta_small := ScaleChoice.eta_small
k_large := hk.le
R_large := hRlarge
delta_pos := ScaleChoice.delta_pos
root_error := ScaleChoice.root_error
contour_nonneg := ScaleChoice.contour_nonneg
contour_small := ScaleChoice.contour_small
test_error := ScaleChoice.test_error }
apply C.moves_impossible_below_criticality hb hpu
intro y s hs
change 1-ScaleChoice.eta≤(BernoulliProduct.law (fun _ : E => p)).real
(G.coordinateMove (fun x => t x-t o) (fun x => z x-z o) m ScaleChoice.e y s)
rw [coordinateMove_sub_const]
exact hmv y s hs
end CriticalPercolation.BondGraph
end
end
end
end OAI
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