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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 | 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
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