File size: 4,134 Bytes
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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | 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
|