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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.InfiniteEventSubsetHit
import OAI.Probability.CriticalPercolation.Criticality.SharpnessPhi
namespace OAI
section
section CurrentModule8
noncomputable section
open Set MeasureTheory Filter Topology
open scoped Classical BigOperators
namespace CriticalPercolation.Sharpness
variable {E V : Type*} (G : BondGraph V E) [Countable V] [Countable E]
lemma infinite_escape_lower_bound (hl : G.LocallyFinite)
(p q : unitInterval) (hp : 0 < (p : ℝ)) (hpq : p < q) (hq : q < 1)
(o : V) (hflux : ∀ S : Set V, S.Finite → o ∈ S → 1 ≤ phi G p o S) :
((q : ℝ)-p)/(1+q-p) ≤ (G.law q).real (G.infiniteEvent o) := by
let L := (Set.to_countable (Set.univ : Set V)).finiteExhaustion
apply ge_of_tendsto (G.tendsto_real_hit_compl q L o)
apply Eventually.of_forall
intro n
let K := G.incidentEdges (L n)
have hK : K.Finite := finite_incident G hl (L.finite n)
let := hK.fintype
have hf : ∀ R : Set V, (L n)ᶜ ⊆ R → o ∉ R →
1 ≤ boundaryFlux (G.restrictEdges K) (p : ℝ) o R := by
intro R hR ho
have hsub : Rᶜ ⊆ L n := by
simpa only [compl_compl] using Set.compl_subset_compl.mpr hR
have he := phi_eq_boundaryFlux G p o Rᶜ K hK (incident_mono G hsub)
simp only [compl_compl] at he
rw [← he]
exact hflux Rᶜ ((L.finite n).subset hsub) ho
exact (uniform_escape_lower_bound (G.restrictEdges K)
⟨hp, (show (p : ℝ)<q from hpq).trans (show (q : ℝ)<1 from hq)⟩
hpq hq o (L n)ᶜ hf).trans (restricted_escape_le G q o (L n)ᶜ K hK)
theorem subcritical_finite_size (hl : G.LocallyFinite)
(p : unitInterval) (hp : p < G.criticalProbability) (o : V) :
∃ S : Set V, S.Finite ∧ o ∈ S ∧ phi G p o S < 1 := by
by_cases hp0 : (p : ℝ) = 0
· exact ⟨{o},Set.finite_singleton o,Set.mem_singleton o,by simp [phi,hp0]⟩
have hp_pos : 0 < (p : ℝ) := lt_of_le_of_ne p.property.1 (Ne.symm hp0)
by_contra h
have hflux : ∀ S : Set V, S.Finite → o ∈ S → 1 ≤ phi G p o S := by
intro S hS ho
exact le_of_not_gt (fun hh => h ⟨S,hS,ho,hh⟩)
let q : unitInterval := ⟨((p : ℝ)+(G.criticalProbability : ℝ))/2, by
constructor
· linarith [p.property.1,G.criticalProbability.property.1]
· linarith [p.property.2,G.criticalProbability.property.2]⟩
have hpq : p < q := by
change (p : ℝ) < ((p : ℝ)+(G.criticalProbability : ℝ))/2
have hh : (p : ℝ) < G.criticalProbability := hp
linarith
have hqc : q < G.criticalProbability := by
change ((p : ℝ)+(G.criticalProbability : ℝ))/2 < (G.criticalProbability : ℝ)
have hh : (p : ℝ) < G.criticalProbability := hp
linarith
have hq1 : q < 1 := hqc.trans_le G.criticalProbability.property.2
have hb := infinite_escape_lower_bound G hl p q hp_pos hpq hq1 o hflux
have hz : (G.law q).real (G.infiniteEvent o) = 0 := by
apply (measureReal_eq_zero_iff (measure_ne_top _ _)).mpr
apply measure_mono_null (show G.infiniteEvent o ⊆ G.percolates from fun _ hω => ⟨o,hω⟩)
exact G.law_percolates_eq_zero_of_lt_critical hqc
rw [hz] at hb
have hd : 0 < (q : ℝ)-p := sub_pos.mpr hpq
have hden : 0 < 1+(q : ℝ)-p := by linarith
exact (not_le_of_gt (div_pos hd hden)) hb
end CriticalPercolation.Sharpness
end
end CurrentModule8
end
end OAI
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