| import Mathlib |
| import OAI.Probability.CriticalPercolation.Model |
| import OAI.Probability.CriticalPercolation.Modular.PeakClusters |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory |
| open scoped Classical ENNReal |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| lemma stabilizerWeight_le_one_iff (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (o a x : V) : G.stabilizerWeight a x ≤ 1 ↔ G.stabilizerWeight o x ≤ G.stabilizerWeight o a := by |
| rw [G.stabilizerWeight_change_base hc hl o a x] |
| conv_rhs => rhs; rw [←mul_one (G.stabilizerWeight o a)] |
| exact (ENNReal.mul_le_mul_iff_right (G.stabilizerWeight_pos hc hl o a).ne' |
| (G.stabilizerWeight_ne_top o a)).symm |
| |
| def clusterUpperSet (ω : Set E) (v : V) : Set V := |
| {x | x ∈ G.cluster ω v ∧ 1 ≤ G.stabilizerWeight v x} |
|
|
| lemma mem_clusterUpperSet_self (ω : Set E) (v : V) : v∈G.clusterUpperSet ω v := by |
| exact ⟨SimpleGraph.Reachable.refl v,by simp⟩ |
|
|
| lemma finite_clusterUpperSet_of_mass_ne_top {ω : Set E} {v : V} |
| (hm : G.modularMass v (G.cluster ω v) ≠ ∞) : (G.clusterUpperSet ω v).Finite := by |
| apply (ENNReal.finite_const_le_of_tsum_ne_top hm (ε := 1) one_ne_zero).subset |
| intro x hx |
| change 1 ≤ if x∈G.cluster ω v then G.stabilizerWeight v x else 0 |
| rw [ite_eq_left hx.1] |
| exact hx.2 |
|
|
| lemma finite_clusterUpperSet_has_peak (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (ω : Set E) (v : V) (hf : (G.clusterUpperSet ω v).Finite) : |
| ∃ y∈G.cluster ω v, G.clusterPeak ω y ∧ |
| G.peakBound hf.toFinset.card ω y := by |
| obtain ⟨y,hy,hmax⟩ := hf.toFinset.exists_max_image (G.stabilizerWeight v) |
| ⟨v,hf.mem_toFinset.mpr (G.mem_clusterUpperSet_self ω v)⟩ |
| have hy' := hf.mem_toFinset.mp hy |
| have hbound (z : V) (hz : z∈G.cluster ω v) : G.stabilizerWeight v z ≤ G.stabilizerWeight v y := by |
| by_cases h : 1≤G.stabilizerWeight v z |
| · exact hmax z (hf.mem_toFinset.mpr ⟨hz,h⟩) |
| · exact (le_of_not_ge h).trans hy'.2 |
| have hpeak : G.clusterPeak ω y := by |
| intro z hz |
| exact (G.stabilizerWeight_le_one_iff hc hl v y z).mpr (hbound z (hy'.1.trans hz)) |
| refine ⟨y,hy'.1,hpeak,hf.toFinset,le_rfl,?_⟩ |
| intro z hz |
| apply hf.mem_toFinset.mpr |
| refine ⟨hy'.1.trans hz.1,?_⟩ |
| exact hy'.2.trans ((G.stabilizerWeight_le_one_iff hc hl v z y).mp (hz.2 y hz.1.symm)) |
|
|
| lemma clusterPeaks_finite_of_mass_ne_top (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (ω : Set E) (v : V) (hm : G.modularMass v (G.cluster ω v) ≠ ∞) : |
| (G.clusterPeaks ω v).Finite := by |
| apply (G.finite_clusterUpperSet_of_mass_ne_top hm).subset |
| intro x hx |
| refine ⟨hx.1,?_⟩ |
| have hh := (G.stabilizerWeight_le_one_iff hc hl v x v).mp (hx.2 v hx.1.symm) |
| simpa only [G.stabilizerWeight_self] using hh |
|
|
| lemma clusterPeaks_nonempty_of_mass_ne_top (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (ω : Set E) (v : V) (hm : G.modularMass v (G.cluster ω v) ≠ ∞) : |
| (G.clusterPeaks ω v).Nonempty := by |
| obtain ⟨y,hy,hp,_⟩ := G.finite_clusterUpperSet_has_peak hc hl ω v |
| (G.finite_clusterUpperSet_of_mass_ne_top hm) |
| exact ⟨y,hy,hp⟩ |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
|
|
| end OAI |
|
|
