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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.RandomTiltedTransport
import OAI.Probability.CriticalPercolation.Modular.ModularClusterCount
import OAI.Probability.CriticalPercolation.Modular.ModularIncidence
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical ENNReal BigOperators
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma Aut.mem_ends_iff (φ : G.Aut) (e : E) (v : V) :
φ.vertex v∈G.ends (φ.edge e) ↔ v∈G.ends e := by
rw [φ.preserves]
simp only [Sym2.mem_map,φ.vertex.injective.eq_iff,exists_eq_right]
lemma Aut.incident_cluster_iff (φ : G.Aut) (ω : Set E) (x : V) (e : E) :
φ.edge e∈G.incidentEdges (G.cluster (φ.edge '' ω) (φ.vertex x)) ↔
e∈G.incidentEdges (G.cluster ω x) := by
rw [←φ.cluster_image]
constructor
· rintro ⟨v,⟨w,hw,rfl⟩,hwe⟩
exact ⟨w,hw,(Aut.mem_ends_iff G φ e w).mp hwe⟩
· rintro ⟨v,hv,hve⟩
exact ⟨φ.vertex v,⟨v,hv,rfl⟩,(Aut.mem_ends_iff G φ e v).mpr hve⟩
lemma measurable_incidentCluster [Countable V] [Countable E] (x : V) (e : E) :
Measurable (fun ω : Set E => e∈G.incidentEdges (G.cluster ω x)) :=
Measurable.exists (fun v => (G.measurable_reachable x v).and measurable_const)
def normalizedSelectedIncidence (H : (Set E) → E → ℝ≥0∞) (ω : Set E) (x : V) : ℝ≥0∞ :=
(∑' e, if e∈G.incidentEdges (G.cluster ω x) then H ω e*G.bondWeight x e else 0)*
G.reciprocalClusterMass ω x
def normalizedIncidenceTransport (H : (Set E) → E → ℝ≥0∞) (ω : Set E) (x y : V) : ℝ≥0∞ :=
∑' e, if e∈G.incidentEdges (G.cluster ω x) ∧ y∈G.ends e then
H ω e*G.stabilizerWeight x y*G.reciprocalClusterMass ω x else 0
lemma measurable_normalizedSelectedIncidence [Countable V] [Countable E]
(H : (Set E) → E → ℝ≥0∞) (hm : ∀ e, Measurable (fun ω => H ω e)) (x : V) :
Measurable (fun ω : Set E => G.normalizedSelectedIncidence H ω x) := by
apply Measurable.mul _ (G.measurable_reciprocalClusterMass x)
exact Measurable.tsum (fun e => ((hm e).mul_const _).ite
(G.measurable_incidentCluster x e).setOf measurable_const)
lemma measurable_normalizedIncidenceTransport [Countable V] [Countable E]
(H : (Set E) → E → ℝ≥0∞) (hm : ∀ e, Measurable (fun ω => H ω e)) (x y : V) :
Measurable (fun ω : Set E => G.normalizedIncidenceTransport H ω x y) := by
apply Measurable.tsum
intro e
exact (((hm e).mul_const _).mul (G.measurable_reciprocalClusterMass x)).ite
((G.measurable_incidentCluster x e).and measurable_const).setOf measurable_const
lemma Aut.normalizedIncidenceTransport (H : (Set E) → E → ℝ≥0∞)
(ha : ∀ (φ : G.Aut) ω e, H (φ.edge '' ω) (φ.edge e)=H ω e)
(φ : G.Aut) (ω : Set E) (x y : V) :
G.normalizedIncidenceTransport H (φ.edge '' ω) (φ.vertex x) (φ.vertex y)=
G.normalizedIncidenceTransport H ω x y := by
unfold CriticalPercolation.BondGraph.normalizedIncidenceTransport
rw [←φ.edge.tsum_eq]
simp only [φ.incident_cluster_iff,φ.mem_ends_iff,ha,φ.stabilizerWeight,φ.reciprocalClusterMass]
lemma normalizedIncidenceTransport_out (H : (Set E) → E → ℝ≥0∞) (ω : Set E) (x : V) :
(∑' y, G.normalizedIncidenceTransport H ω x y)=G.normalizedSelectedIncidence H ω x := by
unfold normalizedIncidenceTransport normalizedSelectedIncidence
rw [ENNReal.tsum_comm,←ENNReal.tsum_mul_right]
apply tsum_congr
intro e
by_cases he : e∈G.incidentEdges (G.cluster ω x)
· simp only [he,true_and,ite_true]
rw [tsum_eq_sum (s := (G.ends e).toFinset) (fun y hy => by
simpa only [Sym2.mem_toFinset] using (ite_eq_right (by simpa only [Sym2.mem_toFinset] using hy) :
(if y∈G.ends e then H ω e*G.stabilizerWeight x y*G.reciprocalClusterMass ω x else 0)=0))]
have heq : (∑ y∈(G.ends e).toFinset, if y∈G.ends e then
H ω e*G.stabilizerWeight x y*G.reciprocalClusterMass ω x else 0) =
∑ y∈(G.ends e).toFinset, H ω e*G.stabilizerWeight x y*G.reciprocalClusterMass ω x := by
apply Finset.sum_congr rfl
intro y hy
exact ite_eq_left (Sym2.mem_toFinset.mp hy)
rw [heq,←Finset.sum_mul,←Finset.mul_sum]
rfl
· simp only [he,false_and,ite_false,tsum_zero,zero_mul]
lemma normalizedIncidenceTransport_in (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(H : (Set E) → E → ℝ≥0∞) (ω : Set E) (y : V) :
(∑' x, G.normalizedIncidenceTransport H ω x y*G.stabilizerWeight y x)=
∑' e, if y∈G.ends e then H ω e*G.incidentClusterCount ω e else 0 := by
unfold normalizedIncidenceTransport
simp only [←ENNReal.tsum_mul_right]
rw [ENNReal.tsum_comm]
apply tsum_congr
intro e
by_cases hy : y∈G.ends e
· simp only [hy,and_true,ite_true,incidentClusterCount,←ENNReal.tsum_mul_left]
apply tsum_congr
intro x
by_cases he : e∈G.incidentEdges (G.cluster ω x)
· simp only [ite_eq_left he]
calc _ = (H ω e*G.reciprocalClusterMass ω x)*
(G.stabilizerWeight x y*G.stabilizerWeight y x) := by ac_rfl
_ = _ := by rw [G.stabilizerWeight_mul_reverse hc hl,mul_one]
· simp only [ite_eq_right he,zero_mul,mul_zero]
· simp only [hy,and_false,ite_false,zero_mul,tsum_zero]
lemma normalizedSelectedIncidence_le (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(D K : ℕ) (hD : ∀ v, (G.incidentEdges {v}).ncard≤D)
(hK : ∀ v, (G.vertexBall v 1).ncard≤K)
(H : (Set E) → E → ℝ≥0∞) (hH : ∀ ω e, H ω e≤1) (ω : Set E) (x : V) :
G.normalizedSelectedIncidence H ω x≤(D*(2*K) : ℕ) := by
unfold normalizedSelectedIncidence
by_cases hm : G.modularMass x (G.cluster ω x)=∞
· simp [reciprocalClusterMass,hm]
· calc
_ ≤ ((D*(2*K) : ℕ)*G.modularMass x (G.cluster ω x))*G.reciprocalClusterMass ω x := by
apply mul_le_mul_left
apply le_trans _ (G.weighted_incidence_le hc hl D K hD hK x _)
apply ENNReal.tsum_le_tsum
intro e
split_ifs
· exact mul_le_of_le_one_left zero_le (hH ω e)
· rfl
_ = _ := by
rw [reciprocalClusterMass,mul_assoc,ENNReal.mul_inv_cancel
(ne_of_gt (zero_lt_one.trans_le (G.one_le_modularClusterMass ω x))) hm,mul_one]
theorem expected_normalizedSelectedIncidence [Countable V] [Countable E]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(p : unitInterval) (H : (Set E) → E → ℝ≥0∞)
(hm : ∀ e, Measurable (fun ω => H ω e))
(ha : ∀ (φ : G.Aut) ω e, H (φ.edge '' ω) (φ.edge e)=H ω e) :
(∑ x∈(G.orbitRepresentatives hq).roots, ∫⁻ ω, G.normalizedSelectedIncidence H ω x ∂G.law p)=
∑ y∈(G.orbitRepresentatives hq).roots, ∫⁻ ω,
∑' e, if y∈G.ends e then H ω e*G.incidentClusterCount ω e else 0 ∂G.law p := by
simpa only [G.normalizedIncidenceTransport_out,G.normalizedIncidenceTransport_in hc hl] using
OrbitRepresentatives.random_tilted_transport G (G.orbitRepresentatives hq) hc hl p
(G.normalizedIncidenceTransport H) (G.measurable_normalizedIncidenceTransport H hm)
(fun φ => Aut.normalizedIncidenceTransport G H ha φ)
end CriticalPercolation.BondGraph
end
end
end OAI
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