File size: 5,596 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 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.BallIncidence
import OAI.Probability.CriticalPercolation.Criticality.SubexponentialVolumes
import OAI.Probability.CriticalPercolation.Criticality.WitnessEvents
import OAI.Probability.CriticalPercolation.GraphGeometry.UniformGeometry
import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteAdjacentMeasure
namespace OAI
section
section
noncomputable section
open Set MeasureTheory ProbabilityTheory Filter
open scoped Classical Topology
namespace CriticalPercolation.BondGraph
variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E)
lemma adjacent_bound_of_geometry (hconnected : G.fullGraph.Connected)
(hlocal : G.LocallyFinite) (hg : G.SubexponentialGrowth)
(C M D : ℕ) (hM : ∀ v, (G.vertexBall v C).ncard≤M)
(hDpos : 1≤D) (hD : ∀ v, (G.incidentEdges {v}).ncard≤D)
(hdense : ∀ u v, ∃ φ : G.Aut, φ.vertex u ∈ G.vertexBall v C)
(p : unitInterval) (hp : (p:ℝ)∈Set.Ioo 0 1) (e : E) (K : ℕ) :
(G.law p).real (G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 (4^K)) ≤
8*(M:ℝ)*((D:ℝ)/(p:ℝ)*(6/(2:ℝ)^K)) := by
let a := (G.ends e).out.1
let b := (G.ends e).out.2
let δ := (G.law p).real (G.twoClusterEvent a b (4^K))
let Q := (D:ℝ)/(p:ℝ)*(6/(2:ℝ)^K)
have hQ : 0≤Q := by
dsimp [Q]
exact mul_nonneg (div_nonneg (Nat.cast_nonneg _) hp.1.le) (by positivity)
change δ ≤ 8*(M:ℝ)*Q
by_cases hδ : 0<δ
swap
· exact (not_lt.mp hδ).trans (by positivity)
obtain ⟨J,hJ,hJmass⟩ := G.exists_finite_witnessedTwoCluster p a b (4^K) hδ
obtain ⟨W,hW⟩ := G.finite_bonds_in_ball hconnected a (hJ.insert e)
have hJW : J⊆G.inducedEdges (G.vertexBall a W) := fun f hf => hW (Or.inr hf)
have heW : e∈G.inducedEdges (G.vertexBall a W) := hW (Or.inl rfl)
choose φ hφ using hdense a
obtain ⟨n,_,hn⟩ := G.subexponential_ball_enlargement hlocal hg a (C+W) 0
let B := G.vertexBall a n
let S := G.vertexBall a (n+C+W)
have hBf : B.Finite := G.vertexBall_finite hlocal a n
have hSf : S.Finite := G.vertexBall_finite hlocal a _
have hEf := G.finite_inducedEdges hlocal hSf
let q (f : E) := (G.law p).real {ω | G.inducedEdges S∩ω ∈
G.twoClusterEvent (G.ends f).out.1 (G.ends f).out.2 (4^K)}
have hq : ∀ f, 0≤q f := fun _ => measureReal_nonneg
have hmap : ∀ v∈hBf.toFinset, (φ v).edge e∈hEf.toFinset := by
intro v hv
apply hEf.mem_toFinset.mpr
exact G.translated_bonds_in_ball a a v n C W (φ v) (hBf.mem_toFinset.mp hv)
(hφ v) (Set.singleton_subset_iff.mpr heW) ⟨e,rfl,rfl⟩
have hvbound : ∀ v∈hBf.toFinset, δ/2≤q ((φ v).edge e) := by
intro v hv
have hsub := G.translated_bonds_in_ball a a v n C W (φ v)
(hBf.mem_toFinset.mp hv) (hφ v) hJW
have hm := measureReal_mono (G.witnessedTwoCluster_subset_restrict hsub
((φ v).vertex a) ((φ v).vertex b) (4^K)) (μ := G.law p)
rw [Aut.witnessedTwoCluster_real G (φ v) p J hJ a b (4^K)] at hm
have hend : s((φ v).vertex a,(φ v).vertex b) =
s((G.ends ((φ v).edge e)).out.1,(G.ends ((φ v).edge e)).out.2) := by
calc
_ = G.ends ((φ v).edge e) := by
rw [(φ v).preserves,(G.ends e).out_eq.symm,Sym2.map_mk]
_ = _ := (G.ends ((φ v).edge e)).out_eq.symm
rw [G.twoClusterEvent_sym2 hend (4^K)] at hm
exact hJmass.trans hm
have hfib := G.orbit_edge_fibre_bound hlocal C M hM a e (Sym2.out_fst_mem _)
φ hφ hBf.toFinset
have havg := G.induced_averaged_adjacent hlocal p hp D hDpos hD S hSf K
change (∑ f∈hEf.toFinset, q f) ≤ (S.ncard:ℝ)*Q at havg
have hratio : S.ncard≤2*B.ncard := by simpa only [S,B,Nat.add_assoc] using hn
have hsum : (B.ncard:ℝ)*(δ/2) ≤ (B.ncard:ℝ)*(4*(M:ℝ)*Q) := by
calc
_ = ∑ v∈hBf.toFinset, δ/2 := by
simp only [Finset.sum_const, nsmul_eq_mul]
rw [Set.ncard_eq_toFinset_card B hBf]
_ ≤ ∑ v∈hBf.toFinset, q ((φ v).edge e) := Finset.sum_le_sum hvbound
_ ≤ (2*M:ℕ)*∑ f∈hEf.toFinset, q f :=
sum_comp_le_fibre_bound hBf.toFinset hEf.toFinset (fun v => (φ v).edge e) q
(2*M) hq hmap hfib
_ ≤ (2*M:ℕ)*((S.ncard:ℝ)*Q) :=
mul_le_mul_of_nonneg_left havg (Nat.cast_nonneg _)
_ ≤ (2*M:ℕ)*(((2*B.ncard:ℕ):ℝ)*Q) :=
mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_right (Nat.cast_le.mpr hratio) hQ) (Nat.cast_nonneg _)
_ = _ := by push_cast; ring
have hBpos : (0:ℝ)<B.ncard := Nat.cast_pos.mpr (G.vertexBall_ncard_pos hlocal a n)
have hh := (mul_le_mul_iff_right₀ hBpos).mp hsum
linarith
theorem subexponential_adjacent_bound (hconnected : G.fullGraph.Connected)
(hlocal : G.LocallyFinite) (hquasi : G.QuasiTransitive)
(hg : G.SubexponentialGrowth) :
∃ A : ℝ, 0<A ∧ ∀ (p : unitInterval), (p:ℝ)∈Set.Ioo 0 1 → ∀ (e : E) (K : ℕ),
(G.law p).real (G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 (4^K)) ≤
A/((p:ℝ)*(2:ℝ)^K) := by
obtain ⟨C,hC⟩ := G.uniform_orbit_density_radius hconnected hquasi
obtain ⟨M,hMpos,hM⟩ := G.uniform_positive_ball_bound hquasi C
obtain ⟨D,hDpos,hD⟩ := G.uniform_incident_bound hquasi
have hm : (0:ℝ)<M := by exact_mod_cast hMpos
have hd : (0:ℝ)<D := by exact_mod_cast hDpos
refine ⟨48*(M:ℝ)*D, by positivity, ?_⟩
intro p hp e K
have hh := G.adjacent_bound_of_geometry hconnected hlocal hg C M D hM hDpos hD hC p hp e K
convert hh using 1
ring
end CriticalPercolation.BondGraph
end
end
end
end OAI
|