openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /InfiniteEscapeLowerBound.lean
| 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 | |