| import Mathlib |
| import OAI.Probability.CriticalPercolation.Modular.RandomTiltedTransport |
| import OAI.Probability.CriticalPercolation.Modular.PeakClusters |
| import OAI.Probability.CriticalPercolation.Modular.ClusterDiscrepancy |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory |
| open scoped Classical ENNReal BigOperators |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E) |
|
|
| def heavyPeakTransport (N : ℕ) (ω : Set E) (x y : V) : ℝ≥0∞ := |
| if G.modularMass x (G.cluster ω x) = ∞ ∧ G.peakBound N ω x ∧ y∈G.clusterPeaks ω x then 1 else 0 |
|
|
| lemma measurable_heavyPeakTransport (N : ℕ) (x y : V) : |
| Measurable (fun ω : Set E => G.heavyPeakTransport N ω x y) := by |
| apply measurable_const.ite _ measurable_const |
| exact (G.measurableSet_heavyClusterEvent x x).inter |
| ((G.measurable_peakBound N x).and (G.measurable_mem_clusterPeaks x y)).setOf |
|
|
| omit [Countable V] [Countable E] in |
| lemma heavyPeakTransport_aut (N : ℕ) (φ : G.Aut) (ω : Set E) (x y : V) : |
| G.heavyPeakTransport N (φ.edge '' ω) (φ.vertex x) (φ.vertex y) = |
| G.heavyPeakTransport N ω x y := by |
| unfold heavyPeakTransport |
| rw [←φ.cluster_image,φ.modularMass_image,φ.peakBound_iff,←φ.clusterPeaks_image] |
| simp only [Set.mem_image,φ.vertex.injective.eq_iff,exists_eq_right] |
|
|
| omit [Countable V] [Countable E] in |
| lemma heavyPeakTransport_out_bound (N : ℕ) (ω : Set E) (x : V) : |
| (∑' y, G.heavyPeakTransport N ω x y) ≤ N := by |
| by_cases hb : G.peakBound N ω x |
| · obtain ⟨S,hS,hcover⟩ := hb |
| calc |
| _ ≤ ∑' y, if y∈S then (1 : ℝ≥0∞) else 0 := by |
| apply ENNReal.tsum_le_tsum |
| intro y |
| unfold heavyPeakTransport |
| split_ifs with hh hs |
| · rfl |
| · exact False.elim (hs (hcover hh.2.2)) |
| · exact zero_le |
| · rfl |
| _ = (S.card : ℝ≥0∞) := by |
| rw [tsum_eq_sum (s := S) (fun y hy => ite_eq_right hy)] |
| calc |
| _ = ∑ y ∈ S, (1 : ℝ≥0∞) := Finset.sum_congr rfl (fun y hy => ite_eq_left hy) |
| _ = _ := by simp |
| _ ≤ N := by exact_mod_cast hS |
| · simp only [heavyPeakTransport,hb,false_and,and_false,ite_false,tsum_zero,zero_le] |
|
|
| omit [Countable V] [Countable E] in |
| lemma heavyPeakTransport_infinite (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (N : ℕ) (ω : Set E) (y : V) |
| (hh : G.modularMass y (G.cluster ω y) = ∞) (hb : G.peakBound N ω y) |
| (hp : G.clusterPeak ω y) : |
| (∑' x, G.heavyPeakTransport N ω x y * G.stabilizerWeight y x) = ∞ := by |
| apply top_unique |
| rw [←hh] |
| apply ENNReal.tsum_le_tsum |
| intro x |
| by_cases hx : x ∈ G.cluster ω y |
| · have hxy : (G.openGraph ω).Reachable x y := hx.symm |
| have he : G.cluster ω x = G.cluster ω y := G.cluster_eq_of_reachable hxy |
| have hhx : G.modularMass x (G.cluster ω x) = ∞ := by |
| rw [he,G.modularMass_eq_top_change_base hc hl x y] |
| exact hh |
| have hbx : G.peakBound N ω x := (G.peakBound_iff_of_reachable N hxy).mpr hb |
| have hcond : G.modularMass x (G.cluster ω x) = ∞ ∧ G.peakBound N ω x ∧ |
| y∈G.clusterPeaks ω x := ⟨hhx,hbx,⟨hxy,hp⟩⟩ |
| simp only [ite_eq_left hx,heavyPeakTransport,ite_eq_left hcond,one_mul,le_refl] |
| · simp only [ite_eq_right hx,zero_le] |
|
|
| lemma no_heavy_finite_peaks_ae (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (hq : G.QuasiTransitive) (p : unitInterval) : |
| ∀ᵐ ω ∂G.law p, ∀ y : V, ∀ N : ℕ, |
| ¬ (G.modularMass y (G.cluster ω y) = ∞ ∧ G.peakBound N ω y ∧ G.clusterPeak ω y) := by |
| apply ae_all_iff.mpr |
| intro y |
| apply ae_all_iff.mpr |
| intro N |
| have hfin := OrbitRepresentatives.incoming_finite_ae G (G.orbitRepresentatives hq) |
| hc hl p (G.heavyPeakTransport N) (G.measurable_heavyPeakTransport N) |
| (G.heavyPeakTransport_aut N) N (ENNReal.natCast_lt_top N) (G.heavyPeakTransport_out_bound N) y |
| filter_upwards [hfin] with ω hω |
| rintro ⟨hh,hb,hp⟩ |
| rw [G.heavyPeakTransport_infinite hc hl N ω y hh hb hp] at hω |
| exact (lt_irrefl _ hω) |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
|
|
| end OAI |
|
|
