File size: 6,464 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.ModularClusterMass
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical ENNReal BigOperators
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def bondWeight (o : V) (e : E) : ℝ≥0∞ := ∑ v ∈ (G.ends e).toFinset, G.stabilizerWeight o v
lemma bondWeight_ne_top (o : V) (e : E) : G.bondWeight o e ≠ ∞ := by
apply ENNReal.sum_ne_top.mpr
intro v hv
exact G.stabilizerWeight_ne_top o v
lemma bondWeight_change_base (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(o a : V) (e : E) : G.bondWeight o e = G.stabilizerWeight o a * G.bondWeight a e := by
unfold bondWeight
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro v hv
exact G.stabilizerWeight_change_base hc hl o a v
lemma Aut.bondWeight (φ : G.Aut) (o : V) (e : E) :
G.bondWeight (φ.vertex o) (φ.edge e) = G.bondWeight o e := by
unfold CriticalPercolation.BondGraph.bondWeight
rw [φ.preserves]
have himg : (Sym2.map φ.vertex (G.ends e)).toFinset = (G.ends e).toFinset.image φ.vertex := by
ext v
simp only [Sym2.mem_toFinset,Sym2.mem_map,Finset.mem_image]
rw [himg,Finset.sum_image]
· simp only [Aut.stabilizerWeight]
· exact fun _ _ _ _ h => φ.vertex.injective h
lemma incidence_stabilizerOrbit_subset_ball (v w : V) (n : ℕ) (hw : w∈G.vertexBall v n) :
G.stabilizerOrbit v w ⊆ G.vertexBall v n := by
intro z hz
obtain ⟨φ,hv,hwz⟩ := G.mem_stabilizerOrbit.mp hz
have h := (Aut.vertexBall_mem G φ v w n).mpr hw
simpa only [hv,hwz] using h
lemma stabilizerWeight_le_ball_bound (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(K : ℕ) (hK : ∀ v, (G.vertexBall v 1).ncard≤K) {v w : V}
(_hw : w∈G.vertexBall v 1) (hv : v∈G.vertexBall w 1) :
G.stabilizerWeight v w ≤ K := by
unfold stabilizerWeight modularRatio
apply ENNReal.ofReal_le_of_le_toReal
simp only [ENNReal.toReal_natCast]
apply (div_le_iff₀ (Nat.cast_pos.mpr (G.suborbitCard_pos hc hl v w))).mpr
have hnum : G.suborbitCard w v ≤ K :=
(Set.ncard_le_ncard (G.incidence_stabilizerOrbit_subset_ball w v 1 hv)
(G.vertexBall_finite hl w 1)).trans (hK w)
have hden : (1:ℝ)≤G.suborbitCard v w := by exact_mod_cast G.suborbitCard_pos hc hl v w
exact (Nat.cast_le.mpr hnum).trans (by nlinarith [show (0:ℝ)≤K by positivity])
lemma common_bond_ball_one {v w : V} {e : E} (hv : v∈G.ends e) (hw : w∈G.ends e) :
w∈G.vertexBall v 1 := by
by_cases he : v=w
· subst w
exact G.mem_vertexBall_self v 1
· apply G.vertexBall_step (G.mem_vertexBall_self v 0)
apply (SimpleGraph.fromEdgeSet_adj _).mpr
refine ⟨?_,he⟩
refine ⟨e,Set.mem_univ e,?_⟩
exact (Sym2.mem_and_mem_iff he).mp ⟨hv,hw⟩
lemma bondWeight_le_at_end (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(K : ℕ) (hK : ∀ v, (G.vertexBall v 1).ncard≤K) (o v : V) (e : E)
(hv : v∈G.ends e) : G.bondWeight o e ≤ (2*K : ℕ)*G.stabilizerWeight o v := by
rw [G.bondWeight_change_base hc hl o v]
have hsum : G.bondWeight v e ≤ (2*K : ℕ) := by
calc
_ ≤ ∑ w∈(G.ends e).toFinset, (K : ℝ≥0∞) := by
apply Finset.sum_le_sum
intro w hw
have hw' : w∈G.ends e := Sym2.mem_toFinset.mp hw
exact G.stabilizerWeight_le_ball_bound hc hl K hK
(G.common_bond_ball_one hv hw') (G.common_bond_ball_one hw' hv)
_ = ((G.ends e).toFinset.card * K : ℕ) := by simp [nsmul_eq_mul]
_ ≤ (2*K : ℕ) := by
have hh : (G.ends e).toFinset.card≤2 := by
rw [Sym2.card_toFinset]
split_ifs <;> omega
exact_mod_cast Nat.mul_le_mul_right K hh
calc _ ≤ G.stabilizerWeight o v * (2*K : ℕ) := by gcongr
_ = _ := mul_comm _ _
lemma weighted_incidence_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) (o : V) (S : Set V) :
(∑' e : E, if e∈G.incidentEdges S then G.bondWeight o e else 0) ≤
(D*(2*K) : ℕ)*G.modularMass o S := by
calc
_ ≤ ∑' e : E, ∑' v : V, if v∈S ∧ v∈G.ends e then G.bondWeight o e else 0 := by
apply ENNReal.tsum_le_tsum
intro e
by_cases he : e∈G.incidentEdges S
· rw [ite_eq_left he]
obtain ⟨v,hv,hve⟩ := he
simpa only [ite_eq_left (show v∈S ∧ v∈G.ends e from ⟨hv,hve⟩)] using
(ENNReal.le_tsum v (f := fun v => if v∈S ∧ v∈G.ends e then G.bondWeight o e else 0))
· simp only [ite_eq_right he]; exact zero_le
_ = ∑' v : V, ∑' e : E, if v∈S ∧ v∈G.ends e then G.bondWeight o e else 0 :=
ENNReal.tsum_comm
_ ≤ ∑' v : V, if v∈S then (D*(2*K) : ℕ)*G.stabilizerWeight o v else 0 := by
apply ENNReal.tsum_le_tsum
intro v
by_cases hv : v∈S
· simp only [hv,true_and,ite_true]
have hf : (G.incidentEdges {v}).Finite := by simpa [incidentEdges] using hl v
have hm (e : E) : e∈G.incidentEdges {v} ↔ v∈G.ends e := by
simp [incidentEdges]
have ht : (∑' e : E, if v∈G.ends e then G.bondWeight o e else 0) =
∑ e∈hf.toFinset, G.bondWeight o e := by
rw [tsum_eq_sum (s := hf.toFinset) (fun e he => by
simp only [hf.mem_toFinset,hm] at he
exact ite_eq_right he)]
apply Finset.sum_congr rfl
intro e he
exact ite_eq_left ((hm e).mp (hf.mem_toFinset.mp he))
rw [ht]
calc
_ ≤ ∑ e∈hf.toFinset, (2*K : ℕ)*G.stabilizerWeight o v := by
apply Finset.sum_le_sum
intro e he
exact G.bondWeight_le_at_end hc hl K hK o v e ((hm e).mp (hf.mem_toFinset.mp he))
_ = ((G.incidentEdges {v}).ncard : ℝ≥0∞)*((2*K : ℕ)*G.stabilizerWeight o v) := by
simp only [Finset.sum_const,nsmul_eq_mul,←Set.ncard_eq_toFinset_card _ hf]
_ ≤ (D : ℝ≥0∞)*((2*K : ℕ)*G.stabilizerWeight o v) := by
gcongr
exact_mod_cast hD v
_ = _ := by simp only [Nat.cast_mul,mul_assoc]
· simp only [hv,false_and,ite_false,tsum_zero,le_refl]
_ = _ := by
unfold modularMass
rw [←ENNReal.tsum_mul_left]
apply tsum_congr
intro v
split_ifs <;> simp
end CriticalPercolation.BondGraph
end
end
end OAI
|