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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.GraphModular
namespace OAI
section
noncomputable section
open Set MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def stabilizerWeight (o x : V) : ℝ≥0∞ := ENNReal.ofReal (G.modularRatio x o)
def modularMass (o : V) (S : Set V) : ℝ≥0∞ :=
∑' x : V, if x∈S then G.stabilizerWeight o x else 0
lemma stabilizerWeight_ne_top (o x : V) : G.stabilizerWeight o x ≠ ∞ :=
ENNReal.ofReal_ne_top
lemma stabilizerWeight_pos (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o x : V) : 0 < G.stabilizerWeight o x :=
ENNReal.ofReal_pos.mpr (G.modularRatio_pos hc hl x o)
@[simp] lemma stabilizerWeight_self (o : V) : G.stabilizerWeight o o = 1 := by
simp [stabilizerWeight,modularRatio_self]
lemma stabilizerWeight_change_base (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o a x : V) : G.stabilizerWeight o x =
G.stabilizerWeight o a * G.stabilizerWeight a x := by
unfold stabilizerWeight
rw [←ENNReal.ofReal_mul (G.modularRatio_pos hc hl a o).le]
congr 1
rw [mul_comm,G.modularRatio_cocycle hc hl]
lemma modularMass_change_base (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o a : V) (S : Set V) : G.modularMass o S =
G.stabilizerWeight o a * G.modularMass a S := by
unfold modularMass
rw [←ENNReal.tsum_mul_left]
apply tsum_congr
intro x
split_ifs with hx
· exact G.stabilizerWeight_change_base hc hl o a x
· exact (mul_zero _).symm
lemma modularMass_eq_top_change_base (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o a : V) (S : Set V) : G.modularMass o S = ∞ ↔ G.modularMass a S = ∞ := by
rw [G.modularMass_change_base hc hl o a S,ENNReal.mul_eq_top]
simp [G.stabilizerWeight_ne_top o a,(G.stabilizerWeight_pos hc hl o a).ne']
lemma modularMass_mono (o : V) {S T : Set V} (h : S⊆T) :
G.modularMass o S ≤ G.modularMass o T := by
apply ENNReal.tsum_le_tsum
intro x
split_ifs with hS hT
· rfl
· exact False.elim (hT (h hS))
· exact zero_le
· rfl
lemma modularMass_finite (o : V) {S : Set V} (hS : S.Finite) :
G.modularMass o S < ∞ := by
classical
unfold modularMass
rw [tsum_eq_sum (s := hS.toFinset) (fun x hx => by
simp only [hS.mem_toFinset] at hx
simp only [ite_eq_right hx])]
apply ENNReal.sum_lt_top.mpr
intro x hx
rw [ite_eq_left (hS.mem_toFinset.mp hx)]
exact ENNReal.ofReal_lt_top
lemma infinite_of_modularMass_top (o : V) {S : Set V} (hS : G.modularMass o S = ∞) :
S.Infinite := by
intro hf
have hh := G.modularMass_finite o hf
rw [hS] at hh
exact (lt_irrefl _ hh)
lemma Aut.stabilizerWeight (φ : G.Aut) (o x : V) :
G.stabilizerWeight (φ.vertex o) (φ.vertex x) = G.stabilizerWeight o x := by
simp only [CriticalPercolation.BondGraph.stabilizerWeight,Aut.modularRatio]
lemma Aut.modularMass_image (φ : G.Aut) (o : V) (S : Set V) :
G.modularMass (φ.vertex o) (φ.vertex '' S) = G.modularMass o S := by
unfold modularMass
rw [←φ.vertex.tsum_eq]
apply tsum_congr
intro x
simp only [Set.mem_image,φ.vertex.injective.eq_iff,exists_eq_right,Aut.stabilizerWeight]
def lightInfiniteEvent (o v : V) : Set (Set E) :=
{ω | (G.cluster ω v).Infinite ∧ G.modularMass o (G.cluster ω v) < ∞}
def heavyClusterEvent (o v : V) : Set (Set E) :=
{ω | G.modularMass o (G.cluster ω v) = ∞}
lemma measurable_modularClusterMass [Countable V] [Countable E] (o v : V) :
Measurable (fun ω : Set E => G.modularMass o (G.cluster ω v)) := by
apply Measurable.tsum
intro x
exact measurable_const.ite (G.measurable_reachable v x).setOf measurable_const
lemma measurableSet_lightInfiniteEvent [Countable V] [Countable E] (o v : V) :
MeasurableSet (G.lightInfiniteEvent o v) := by
exact (G.measurable_infinite_cluster v).setOf.inter
(measurableSet_lt (G.measurable_modularClusterMass o v) measurable_const)
lemma measurableSet_heavyClusterEvent [Countable V] [Countable E] (o v : V) :
MeasurableSet (G.heavyClusterEvent o v) :=
measurableSet_eq_fun (G.measurable_modularClusterMass o v) measurable_const
lemma infiniteEvent_eq_light_union_heavy (o v : V) :
G.infiniteEvent v = G.lightInfiniteEvent o v ∪ G.heavyClusterEvent o v := by
ext ω
change (G.cluster ω v).Infinite ↔
((G.cluster ω v).Infinite ∧ _) ∨ _
constructor
· intro h
rcases lt_or_eq_of_le (le_top : G.modularMass o (G.cluster ω v) ≤ ∞) with hl|hh
· exact Or.inl ⟨h,hl⟩
· exact Or.inr hh
· rintro (⟨h,_⟩|h)
· exact h
· exact G.infinite_of_modularMass_top o h
end CriticalPercolation.BondGraph
end
end
end OAI
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