| import Mathlib |
| import OAI.Probability.CriticalPercolation.Criticality.TwoPoint |
| import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls |
| import OAI.Probability.CriticalPercolation.Modular.ModularSuperlevels |
| import OAI.Probability.CriticalPercolation.Modular.CriticalHeavyNull |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory Filter |
| open scoped Classical ENNReal Topology |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def upperEscapeEvent (v : V) (R : ℕ) : Set (Set E) := |
| {ω | ∃ x, x∈G.clusterUpperSet ω v ∧ x∉G.vertexBall v R} |
|
|
| lemma measurableSet_upperEscapeEvent [Countable V] [Countable E] (v : V) (R : ℕ) : |
| MeasurableSet (G.upperEscapeEvent v R) := by |
| exact (Measurable.exists fun x => |
| ((G.measurable_reachable v x).and measurable_const).and measurable_const).setOf |
|
|
| lemma upperEscapeEvent_antitone (v : V) : Antitone (G.upperEscapeEvent v) := by |
| rintro n m h ω ⟨x,hx,hh⟩ |
| exact ⟨x,hx,fun hn => hh (G.vertexBall_mono v h hn)⟩ |
|
|
| lemma not_iInter_upperEscape_of_mass_finite (hc : G.fullGraph.Connected) |
| {ω : Set E} {v : V} (hm : G.modularMass v (G.cluster ω v)≠∞) : |
| ω∉⋂ R : ℕ, G.upperEscapeEvent v R := by |
| obtain ⟨R,hR⟩ := G.finite_subset_vertexBall hc v (G.finite_clusterUpperSet_of_mass_ne_top hm) |
| intro h |
| obtain ⟨x,hx,hout⟩ := Set.mem_iInter.mp h R |
| exact hout (hR hx) |
|
|
| lemma tendsto_upperEscape_zero [Countable V] [Countable E] |
| (hc : G.fullGraph.Connected) (p : unitInterval) (v : V) |
| (hm : ∀ᵐ ω ∂G.law p, G.modularMass v (G.cluster ω v)≠∞) : |
| Tendsto (fun R => (G.law p).real (G.upperEscapeEvent v R)) atTop (𝓝 0) := by |
| have hz : G.law p (⋂ R : ℕ, G.upperEscapeEvent v R)=0 := by |
| simpa only [not_not,Set.ofPred_mem_eq] using |
| (ae_iff.mp (hm.mono (fun _ h => G.not_iInter_upperEscape_of_mass_finite hc h))) |
| have h := tendsto_measure_iInter_atTop (μ := G.law p) |
| (fun R => (G.measurableSet_upperEscapeEvent v R).nullMeasurableSet) |
| (G.upperEscapeEvent_antitone v) ⟨0,measure_ne_top _ _⟩ |
| rw [hz] at h |
| have hh := (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp h |
| change Tendsto (fun R => (G.law p).real (G.upperEscapeEvent v R)) atTop |
| (𝓝 ((0 : ℝ≥0∞).toReal)) at hh |
| simpa only [ENNReal.toReal_zero] using hh |
|
|
| lemma Aut.upperEscapeEvent_preimage (φ : G.Aut) (v : V) (R : ℕ) : |
| (fun ω : Set E => φ.edge '' ω) ⁻¹' G.upperEscapeEvent (φ.vertex v) R = |
| G.upperEscapeEvent v R := by |
| ext ω |
| change (∃ x, (_ ∧ _) ∧ _) ↔ (∃ x, (_ ∧ _) ∧ _) |
| constructor |
| · rintro ⟨x,hx,hout⟩ |
| obtain ⟨y,rfl⟩ := φ.vertex.surjective x |
| refine ⟨y,⟨?_,?_⟩,?_⟩ |
| · exact (SimpleGraph.Iso.reachable_iff (φ := Aut.openGraphIso G φ ω)).mp hx.1 |
| · simpa only [φ.stabilizerWeight] using hx.2 |
| · exact fun hy => hout ((φ.vertexBall_mem G v y R).mpr hy) |
| · rintro ⟨x,hx,hout⟩ |
| refine ⟨φ.vertex x,⟨?_,?_⟩,?_⟩ |
| · exact (SimpleGraph.Iso.reachable_iff (φ := Aut.openGraphIso G φ ω)).mpr hx.1 |
| · simpa only [φ.stabilizerWeight] using hx.2 |
| · exact fun hy => hout ((φ.vertexBall_mem G v x R).mp hy) |
|
|
| lemma Aut.upperEscapeEvent_real [Countable V] [Countable E] |
| (φ : G.Aut) (p : unitInterval) (v : V) (R : ℕ) : |
| (G.law p).real (G.upperEscapeEvent (φ.vertex v) R) = |
| (G.law p).real (G.upperEscapeEvent v R) := by |
| simp only [measureReal_def] |
| congr 1 |
| conv_lhs => rw [←G.law_map_aut p φ] |
| rw [Measure.map_apply (BernoulliProduct.measurable_image_equiv φ.edge) |
| (G.measurableSet_upperEscapeEvent _ _),φ.upperEscapeEvent_preimage] |
|
|
| lemma uniform_upperEscape_small [Countable V] [Countable E] |
| (hc : G.fullGraph.Connected) (hq : G.QuasiTransitive) (p : unitInterval) |
| (hm : ∀ v, ∀ᵐ ω ∂G.law p, G.modularMass v (G.cluster ω v)≠∞) |
| (ε : ℝ) (hε : 0<ε) : |
| ∃ R : ℕ, ∀ v, (G.law p).real (G.upperEscapeEvent v R)<ε := by |
| obtain ⟨S,hS,hcover⟩ := hq |
| have he (s : V) : ∀ᶠ R in atTop, (G.law p).real (G.upperEscapeEvent s R)<ε := |
| (G.tendsto_upperEscape_zero hc p s (hm s)).eventually (gt_mem_nhds hε) |
| have hh := hS.toFinset.eventually_all.mpr (fun s _ => he s) |
| obtain ⟨R,hR⟩ := hh.exists |
| refine ⟨R,fun v => ?_⟩ |
| obtain ⟨s,hs,φ,rfl⟩ := hcover v |
| rw [φ.upperEscapeEvent_real] |
| exact hR s (hS.mem_toFinset.mpr hs) |
|
|
| lemma stabilizerWeight_ge_one_either (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (x y : V) : 1≤G.stabilizerWeight x y ∨ 1≤G.stabilizerWeight y x := by |
| by_cases hxy : 1≤G.stabilizerWeight x y |
| · exact Or.inl hxy |
| · apply Or.inr |
| by_contra hyx |
| have hprod := G.stabilizerWeight_change_base hc hl x y x |
| rw [G.stabilizerWeight_self] at hprod |
| have hh : (1 : ℝ≥0∞)<1 := calc |
| 1 = G.stabilizerWeight x y * G.stabilizerWeight y x := hprod |
| _ ≤ G.stabilizerWeight x y * 1 := by |
| gcongr |
| exact le_of_not_ge hyx |
| _ = G.stabilizerWeight x y := mul_one _ |
| _ < 1 := lt_of_not_ge hxy |
| exact (lt_irrefl _) hh |
|
|
| lemma twoPoint_symm (p : unitInterval) (x y : V) : |
| CriticalPercolation.Sharpness.twoPoint G p x y= |
| CriticalPercolation.Sharpness.twoPoint G p y x := by |
| unfold CriticalPercolation.Sharpness.twoPoint |
| rw [CriticalPercolation.Sharpness.hit_singleton_eq, |
| CriticalPercolation.Sharpness.hit_singleton_eq] |
| congr 1 |
| ext ω |
| exact SimpleGraph.reachable_comm |
|
|
| theorem uniform_twoPoint_decay_of_finite_mass |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) |
| (p : unitInterval) |
| (hm : ∀ v, ∀ᵐ ω ∂G.law p, G.modularMass v (G.cluster ω v)≠∞) : |
| ∀ ε : ℝ, 0<ε → ∃ R : ℕ, ∀ x y, y∉G.vertexBall x R → |
| CriticalPercolation.Sharpness.twoPoint G p x y<ε := by |
| let := G.countable_vertices hc hl |
| let := G.countable_bonds hc hl |
| intro ε hε |
| obtain ⟨R,hR⟩ := G.uniform_upperEscape_small hc hq p hm ε hε |
| refine ⟨R,fun x y hxy => ?_⟩ |
| have h (x y : V) (hxy : y∉G.vertexBall x R) (hw : 1≤G.stabilizerWeight x y) : |
| CriticalPercolation.Sharpness.twoPoint G p x y<ε := by |
| apply lt_of_le_of_lt ?_ (hR x) |
| unfold CriticalPercolation.Sharpness.twoPoint |
| apply measureReal_mono (μ := G.law p) ?_ (measure_ne_top _ _) |
| rw [CriticalPercolation.Sharpness.hit_singleton_eq] |
| exact fun ω hω => ⟨y,⟨hω,hw⟩,hxy⟩ |
| rcases G.stabilizerWeight_ge_one_either hc hl x y with hw|hw |
| · exact h x y hxy hw |
| · rw [G.twoPoint_symm p x y] |
| exact h y x (fun hx => hxy (G.vertexBall_comm.mpr hx)) hw |
|
|
| theorem nonunimodular_critical_twoPoint_decay |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) |
| (hpc : G.criticalProbability<1) (hu : ¬G.CombinatoriallyUnimodular) : |
| ∀ ε : ℝ, 0<ε → ∃ R : ℕ, ∀ x y, y∉G.vertexBall x R → |
| CriticalPercolation.Sharpness.twoPoint G G.criticalProbability x y<ε := by |
| apply G.uniform_twoPoint_decay_of_finite_mass hc hl hq G.criticalProbability |
| intro v |
| exact (G.nonunimodular_critical_mass_finite_ae hc hl hq hpc hu v).mono (fun ω h => (h v).ne) |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
|
|
| end OAI |
|
|
