| 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 |
|
|
