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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.RelativeModular
import OAI.Probability.CriticalPercolation.Modular.ActionOrbitTransport
import OAI.Probability.CriticalPercolation.Growth.HomogeneousGeometry
import OAI.Probability.CriticalPercolation.Growth.HomogeneousAverage
import OAI.Probability.CriticalPercolation.Criticality.NormalClosureGrowth
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical BigOperators
namespace CriticalPercolation.HomogeneousAnalysis.Geometry
open ActionOrbitTransport
variable {X : Type*} (W : Geometry X)
def HasDegree (d : ℕ) : Prop := ∃ C : ℕ, ∀ R : ℕ, ∃ n≥R, (W.ball n).card≤C*n^d
lemma not_quadratic_exponential_lower
(d : ℕ) (hg : W.HasDegree d) (L : ℕ) (hL : 0<L)
(hlo : ∀ n : ℕ, 2^n ≤ (W.ball (L*n^2)).card) : False := by
obtain ⟨C,hC⟩ := hg
obtain ⟨N,hN⟩ := SparseGrowth.eventually_exponential_dominates (C*(4*L)^d) (2*d)
obtain ⟨r,hr,hbound⟩ := hC (L*(max N 1)^2)
let n := Nat.sqrt (r/L)
have hn : max N 1 ≤ n := Nat.le_sqrt'.mpr ((Nat.le_div_iff_mul_le hL).mpr (by
simpa only [Nat.mul_comm] using hr))
have hn1 : 1 ≤ n := (le_max_right N 1).trans hn
have hnr : L*n^2 ≤ r := by
calc
_ ≤ L*(r/L) := Nat.mul_le_mul_left L (Nat.sqrt_le' (r/L))
_ ≤ r := by simpa only [Nat.mul_comm] using Nat.div_mul_le_self r L
have hrn : r ≤ (4*L)*n^2 := by
have ht := Nat.lt_mul_div_succ r hL
have hs : r/L+1 ≤ (n+1)^2 := Nat.succ_le_of_lt (Nat.lt_succ_sqrt' (r/L))
have hs' : (n+1)^2 ≤ 4*n^2 := by nlinarith
have hj := (Nat.mul_le_mul_left L hs).trans (Nat.mul_le_mul_left L hs')
nlinarith
have hu : 2^n ≤ (C*(4*L)^d)*n^(2*d) := by
calc
_ ≤ (W.ball (L*n^2)).card := hlo n
_ ≤ (W.ball r).card := Finset.card_le_card (W.ball_mono hnr)
_ ≤ C*r^d := hbound
_ ≤ C*((4*L)*n^2)^d := Nat.mul_le_mul_left C (Nat.pow_le_pow_left hrn d)
_ = (C*(4*L)^d)*n^(2*d) := by rw [mul_pow,pow_mul]; ring
exact (not_le_of_gt (hN n ((le_max_left N 1).trans hn))) hu
lemma suborbit_card_relIndex (x y : X) : suborbitCard W.Aut X x y=
(MulAction.stabilizer W.Aut y).relIndex (MulAction.stabilizer W.Aut x) := by
have he : (MulAction.stabilizer W.Aut y).subgroupOf (MulAction.stabilizer W.Aut x)=
MulAction.stabilizer (MulAction.stabilizer W.Aut x) y := by ext g; rfl
rw [Subgroup.relIndex,he,MulAction.index_stabilizer]
rfl
def ratio (x y : X) : ℝ := (suborbitCard W.Aut X x y : ℝ)/(suborbitCard W.Aut X y x : ℝ)
lemma ratio_relativeMass (x y : X) : W.ratio x y=
Subgroup.relativeMass (MulAction.stabilizer W.Aut x) (MulAction.stabilizer W.Aut y) := by
simp only [ratio,W.suborbit_card_relIndex,Subgroup.relativeMass]
lemma ratio_pos (x y : X) : 0<W.ratio x y :=
div_pos (Nat.cast_pos.mpr (W.suborbit_card_pos x y)) (Nat.cast_pos.mpr (W.suborbit_card_pos y x))
lemma ratio_cocycle (x y z : X) : W.ratio x y*W.ratio y z=W.ratio x z := by
simp only [W.ratio_relativeMass]
apply Subgroup.relativeMass_cocycle
all_goals
rw [←W.suborbit_card_relIndex]
exact ne_of_gt (W.suborbit_card_pos _ _)
lemma ratio_self (x : X) : W.ratio x x=1 := by
rw [W.ratio_relativeMass,Subgroup.relativeMass_self]
lemma ratio_aut (g : W.Aut) (x y : X) : W.ratio (g x) (g y)=W.ratio x y := by
change W.ratio (g • x) (g • y)=_
simp only [ratio,smul_suborbitCard]
lemma ratio_pow (g : W.Aut) (x : X) (n : ℕ) : W.ratio x ((g^n) x)=(W.ratio x (g x))^n := by
induction n with
| zero => change W.ratio x x=(W.ratio x (g x))^0; rw [W.ratio_self,pow_zero]
| succ n ih =>
have he : (g^(n+1)) x=(g^n) (g x) := by rw [pow_succ]; rfl
rw [he,←W.ratio_cocycle x ((g^n) x),W.ratio_aut,ih,pow_succ]
lemma aut_pow_ball (g : W.Aut) (x : X) (L : ℕ) (hl : g x∈W.around x L) (n : ℕ) :
(g^n) x∈W.around x (n*L) := by
induction n with
| zero => change x∈W.around x (0*L); simpa only [Nat.zero_mul] using W.self_mem_around x 0
| succ n ih =>
have hstep : (g^n) (g x)∈W.around ((g^n) x) L := by
rw [←W.aut_around_image]
exact Finset.mem_image.mpr ⟨g x,hl,rfl⟩
have hh := W.mem_around_add ih hstep
have he : (g^(n+1)) x=(g^n) (g x) := by rw [pow_succ]; rfl
simpa only [Nat.add_mul,Nat.one_mul,←he] using hh
lemma ratio_power_card_bound (g : W.Aut) (x : X) (L : ℕ) (hl : g x∈W.around x L) (n : ℕ) :
(W.ratio x (g x))^n≤((W.ball (n*L)).card:ℝ) := by
rw [←W.ratio_pow]
have hr : W.ratio x ((g^n) x)≤(suborbitCard W.Aut X x ((g^n) x):ℝ) := by
apply div_le_self (Nat.cast_nonneg _)
exact_mod_cast W.suborbit_card_pos ((g^n) x) x
apply hr.trans
have hsub : stabilizerOrbit W.Aut X x ((g^n) x)⊆(W.around x (n*L):Set X) := by
intro y hy
obtain ⟨k,hk,rfl⟩ := (mem_stabilizerOrbit W.Aut X).mp hy
change k x=x at hk
change k ((g^n) x)∈W.around x (n*L)
simpa only [hk] using W.mem_around.mpr
(GraphTopology.iso_reachBall W.graph k (W.mem_around.mp (W.aut_pow_ball g x L hl n)))
have hc := Set.ncard_le_ncard hsub (W.around x (n*L)).finite_toSet
rw [Set.ncard_coe_finset,W.card_around x (n*L)] at hc
exact_mod_cast hc
lemma ratio_le_one (d : ℕ) (hg : W.HasDegree d) (x y : X) : W.ratio x y≤1 := by
by_contra! hr
obtain ⟨g,hgxy⟩ := W.transitive x y
obtain ⟨k,hk⟩ := pow_unbounded_of_one_lt (2:ℝ) hr
let a := g^k
have ha : 2≤W.ratio x (a x) := by
dsimp [a]
rw [W.ratio_pow,hgxy]
exact hk.le
obtain ⟨p,_,_⟩ := W.connected.exists_path_of_dist x (a x)
let L := max p.length 1
have him : a x∈W.around x L := W.mem_around.mpr ⟨p,le_max_left _ _⟩
apply W.not_quadratic_exponential_lower d hg L (lt_of_lt_of_le Nat.zero_lt_one (le_max_right _ _))
intro n
have hb := W.ratio_power_card_bound a x L him n
have he : (2:ℝ)^n≤((W.ball (n*L)).card:ℝ) :=
(pow_le_pow_left₀ (by norm_num) ha n).trans hb
have hh : 2^n≤(W.ball (n*L)).card := by exact_mod_cast he
apply hh.trans (Finset.card_le_card (W.ball_mono ?_))
have hn : n≤n^2 := by nlinarith
nlinarith
theorem balanced_suborbits (d : ℕ) (hg : W.HasDegree d) (x y : X) :
suborbitCard W.Aut X x y=suborbitCard W.Aut X y x := by
have h1 := W.ratio_le_one d hg x y
have h2 := W.ratio_le_one d hg y x
have hp : (0:ℝ)<suborbitCard W.Aut X y x := by exact_mod_cast W.suborbit_card_pos y x
have hq : (0:ℝ)<suborbitCard W.Aut X x y := by exact_mod_cast W.suborbit_card_pos x y
dsimp [ratio] at h1 h2
have he : (suborbitCard W.Aut X x y:ℝ)=(suborbitCard W.Aut X y x:ℝ) := by
apply le_antisymm
· exact (div_le_one hp).mp h1
· exact (div_le_one hq).mp h2
exact_mod_cast he
end CriticalPercolation.HomogeneousAnalysis.Geometry
end
end
end OAI
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