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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.GrowthAlternatives
import OAI.Probability.CriticalPercolation.Criticality.SubexponentialVolumes
import OAI.Probability.CriticalPercolation.Modular.GraphModular
namespace OAI
section
noncomputable section
open Set
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma stabilizerOrbit_subset_ball {x y : V} {n : ℕ} (hy : y ∈ G.vertexBall x n) :
G.stabilizerOrbit x y ⊆ G.vertexBall x n := by
intro z hz
obtain ⟨φ,hx,rfl⟩ := G.mem_stabilizerOrbit.mp hz
rw [←hx]
exact (Aut.vertexBall_mem G φ x y n).mpr hy
lemma Aut.pow_vertex_ball (φ : G.Aut) (x : V) (C : ℕ)
(hx : φ.vertex x ∈ G.vertexBall x C) (n : ℕ) :
(φ^n).vertex x ∈ G.vertexBall x (n*C) := by
induction n with
| zero =>
rw [pow_zero,Nat.zero_mul]
exact G.mem_vertexBall_self x 0
| succ n ih =>
have hstep := (Aut.vertexBall_mem G (φ^n) x (φ.vertex x) C).mpr hx
have hh := G.vertexBall_add ih C hstep
have heq : (φ^(n+1)).vertex x = (φ^n).vertex (φ.vertex x) := by rw [pow_succ]; rfl
simpa only [Nat.add_mul,Nat.one_mul,←heq] using hh
lemma Aut.modularRatio_pow (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(φ : G.Aut) (x : V) (n : ℕ) :
G.modularRatio x ((φ^n).vertex x) = (G.modularRatio x (φ.vertex x))^n := by
induction n with
| zero =>
rw [pow_zero,pow_zero]
exact G.modularRatio_self x
| succ n ih =>
have heq : (φ^(n+1)).vertex x = (φ^n).vertex (φ.vertex x) := by rw [pow_succ]; rfl
rw [heq,← G.modularRatio_cocycle hc hl x ((φ^n).vertex x),Aut.modularRatio,ih,pow_succ]
lemma modularRatio_le_suborbitCard (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(x y : V) : G.modularRatio x y ≤ (G.suborbitCard x y : ℝ) := by
apply div_le_self (Nat.cast_nonneg _)
exact_mod_cast G.suborbitCard_pos hc hl y x
lemma modular_power_log_bound (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(φ : G.Aut) (x : V) (C : ℕ) (hx : φ.vertex x ∈ G.vertexBall x C) (n : ℕ) :
(n:ℝ)*Real.log (G.modularRatio x (φ.vertex x)) ≤
Real.log ((G.vertexBall x (n*C)).ncard : ℝ) := by
have hp := G.modularRatio_pos hc hl x (φ.vertex x)
have hh : (G.modularRatio x (φ.vertex x))^n ≤ (G.vertexBall x (n*C)).ncard := by
rw [←Aut.modularRatio_pow G hc hl φ x n]
apply (G.modularRatio_le_suborbitCard hc hl x _).trans
exact_mod_cast Set.ncard_le_ncard
(G.stabilizerOrbit_subset_ball (Aut.pow_vertex_ball G φ x C hx n))
(G.vertexBall_finite hl x (n*C))
have hlog := Real.log_le_log (pow_pos hp n) hh
rwa [Real.log_pow] at hlog
lemma subexponential_modularRatio_le_one (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (hg : G.SubexponentialGrowth) (φ : G.Aut) (x : V) :
G.modularRatio x (φ.vertex x) ≤ 1 := by
by_contra hn
have hr : 1 < G.modularRatio x (φ.vertex x) := lt_of_not_ge hn
have hlog : 0 < Real.log (G.modularRatio x (φ.vertex x)) := Real.log_pos hr
obtain ⟨C₀,hC₀⟩ := G.finite_subset_vertexBall hc x (Set.finite_singleton (φ.vertex x))
let C := C₀+1
have hC : 0 < C := by omega
have hx : φ.vertex x ∈ G.vertexBall x C :=
G.vertexBall_mono x (Nat.le_succ _) (hC₀ (Set.mem_singleton _))
let a := Real.log (G.modularRatio x (φ.vertex x))
have hCr : (0:ℝ)<C := Nat.cast_pos.mpr hC
obtain ⟨N,hN⟩ := hg x (a/(2*C)) (div_pos hlog (by positivity))
let n := N+1
have hn : 0<n := by omega
have hnC : N ≤ n*C := (Nat.le_succ N).trans (Nat.le_mul_of_pos_right n hC)
have hlo := G.modular_power_log_bound hc hl φ x C hx n
have hhi := hN (n*C) hnC
have heq : (a/(2*(C:ℝ))) * ((n*C:ℕ):ℝ) = (n:ℝ)*a/2 := by
push_cast
field_simp
change (n:ℝ)*a ≤ _ at hlo
rw [heq] at hhi
have hnr : (0:ℝ)<n := Nat.cast_pos.mpr hn
nlinarith [mul_pos hnr hlog]
lemma subexponential_combinatoriallyUnimodular (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (hg : G.SubexponentialGrowth) : G.CombinatoriallyUnimodular := by
intro φ x
have h1 := G.subexponential_modularRatio_le_one hc hl hg φ x
have h2 := G.subexponential_modularRatio_le_one hc hl hg φ⁻¹ (φ.vertex x)
have heq : (φ⁻¹).vertex (φ.vertex x) = x := φ.vertex.symm_apply_apply x
rw [heq] at h2
unfold modularRatio at h1 h2
have hp1 : (0:ℝ)<G.suborbitCard x (φ.vertex x) := Nat.cast_pos.mpr (G.suborbitCard_pos hc hl x (φ.vertex x))
have hp2 : (0:ℝ)<G.suborbitCard (φ.vertex x) x := Nat.cast_pos.mpr (G.suborbitCard_pos hc hl (φ.vertex x) x)
have ha := (div_le_one hp2).mp h1
have hb := (div_le_one hp1).mp h2
exact_mod_cast le_antisymm ha hb
lemma exponential_of_not_combinatoriallyUnimodular [Nonempty V]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(hu : ¬ G.CombinatoriallyUnimodular) :
∃ a : ℝ, 0<a ∧ ∃ N : ℕ, ∀ n≥N, ∀ v : V,
a*n ≤ Real.log ((G.vertexBall v n).ncard : ℝ) := by
rcases G.growth_alternative hc hl hq with hg | hg
· exact False.elim (hu (G.subexponential_combinatoriallyUnimodular hc hl hg))
· exact hg
end CriticalPercolation.BondGraph
end
end
end OAI
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