| import Mathlib |
| import OAI.Probability.Ballisticity.Crossings.StripExitBound |
| import OAI.Probability.Ballisticity.Crossings.CrossingGapDecay |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Classical Topology BigOperators |
| namespace DirectionalTransience |
|
|
| lemma strip_box_card_bound {d : ℕ} (m H : ℕ) (hH : 0 < H) : |
| (latticeBox (d := d) (m*H^2)).card ≤ (2*m+1)^d*H^(2*d) := by |
| rw [card_latticeBox] |
| calc |
| _ ≤ ((2*m+1)*H^2)^d := Nat.pow_le_pow_left (by nlinarith [sq_pos_of_pos hH]) _ |
| _ = _ := by rw [mul_pow,← pow_mul] |
|
|
| lemma upward_probability_bound (N : ℕ) (κ : ℝ) (hκ0 : 0 ≤ κ) (hκ1 : κ ≤ 1) : |
| (1-κ*((N+1:ℕ):ℝ)^(-(1/2:ℝ)))^(N+1) ≤ |
| Real.exp (-κ*((N+1:ℕ):ℝ)^(1/2:ℝ)) := by |
| have hn : (0:ℝ) < (N+1:ℕ) := by positivity |
| have h1 : (1:ℝ) ≤ (N+1:ℕ) := by exact_mod_cast Nat.succ_le_succ (Nat.zero_le N) |
| have ha := Real.rpow_le_one_of_one_le_of_nonpos h1 (by norm_num : -(1/2:ℝ) ≤ 0) |
| have hb : κ*((N+1:ℕ):ℝ)^(-(1/2:ℝ)) ≤ 1 := |
| (mul_le_of_le_one_left (by positivity) hκ1).trans ha |
| calc |
| _ ≤ (Real.exp (- (κ*((N+1:ℕ):ℝ)^(-(1/2:ℝ)))))^(N+1) := |
| pow_le_pow_left₀ (by linarith) (Real.one_sub_le_exp_neg _) _ |
| _ = _ := by |
| rw [← Real.exp_nat_mul] |
| congr 1 |
| have hh : ((N+1:ℕ):ℝ)*((N+1:ℕ):ℝ)^(-(1/2:ℝ)) = ((N+1:ℕ):ℝ)^(1/2:ℝ) := by |
| calc |
| _ = ((N+1:ℕ):ℝ)^(1:ℝ)*((N+1:ℕ):ℝ)^(-(1/2:ℝ)) := by rw [Real.rpow_one] |
| _ = _ := by rw [← Real.rpow_add hn]; norm_num |
| nlinarith |
|
|
| lemma stripTime_rapidDecay {d : ℕ} (hd : 2 ≤ d) (ν : Measure (Row d)) |
| [IsProbabilityMeasure ν] (hue : UniformElliptic ν) (e : Direction d) |
| (htrans : DirectionallyTransient ν (realPosition (step e))) : |
| ∃ A : ℕ, 0 < A ∧ RapidDecay (fun N => |
| (annealedLaw ν).real (StripUntil e (N+1) (A*(N+1)^(2*d+1)))) := by |
| obtain ⟨C,ε,hC,hε,hε1,hconf⟩ := spatial_confinement ν hue e htrans |
| obtain ⟨κ,hκ,hωκ⟩ := environment_uniform_elliptic ν hue |
| have hκ1 : κ ≤ 1 := by |
| obtain ⟨ω,hω⟩ := hωκ.exists |
| exact (hω 0 e).trans (row_entry_le_one (ω 0) e) |
| let m : ℕ := ⌈4*C⌉₊ |
| let A : ℕ := (2*m+1)^d |
| have hA : 0 < A := by dsimp [A]; positivity |
| have hm : 4*C ≤ (m:ℝ) := Nat.le_ceil _ |
| have hgeom := ((rapidDecay_geometric (1-ε) (by linarith) (by linarith)).shift |
| (fun _ => by positivity) 1).const_mul (2*d:ℕ) |
| have hbad := (((bad_crossing_rapidDecay hd ν hue e htrans).shift |
| (fun _ => measureReal_nonneg) 1).pow_mul (2*d)).const_mul A |
| have hexp := ((rapidDecay_stretchedExp (1/2) κ (by norm_num) (by exact_mod_cast hκ)).pow_mul (2*d)).const_mul A |
| refine ⟨A,hA,((hgeom.add hbad).add hexp).of_le (fun _ => measureReal_nonneg) ?_⟩ |
| apply Eventually.of_forall |
| intro N |
| let H := N+1 |
| have hH : 0 < H := by dsimp [H]; omega |
| have hH' : (1:ℝ) ≤ H := by exact_mod_cast hH |
| have hM : C*((H:ℝ)+1)^2 ≤ (m*H^2:ℕ) := by |
| push_cast |
| have hs : ((H:ℝ)+1)^2 ≤ 4*(H:ℝ)^2 := by nlinarith |
| have hh := mul_le_mul_of_nonneg_left hs hC.le |
| nlinarith [sq_nonneg (H:ℝ)] |
| have hcard := strip_box_card_bound (d := d) m H hH |
| have hT : (latticeBox (d := d) (m*H^2)).card*H ≤ A*H^(2*d+1) := by |
| simpa only [pow_succ,mul_assoc] using Nat.mul_le_mul_right H hcard |
| have hh := annealed_stripUntil_bound ν e H (A*H^(2*d+1)) H (m*H^2) hH |
| (C*((H:ℝ)+1)^2) hM hT κ hκ1 hωκ |
| have hc := hconf H |
| have hb := mul_le_mul_of_nonneg_right (show ((latticeBox (d := d) (m*H^2)).card:ℝ) ≤ (A*H^(2*d):ℕ) by exact_mod_cast hcard) |
| (show 0 ≤ (environmentLaw ν).real (badCrossingEvent e H (1/2)) from measureReal_nonneg) |
| have he := upward_probability_bound N κ (by positivity) (by exact_mod_cast hκ1) |
| have he' : ((latticeBox (d := d) (m*H^2)).card:ℝ)*(1-(κ:ℝ)*(H:ℝ)^(-(1/2:ℝ)))^H ≤ |
| (A*H^(2*d):ℕ)*Real.exp (-(κ:ℝ)*(H:ℝ)^(1/2:ℝ)) := |
| mul_le_mul (by exact_mod_cast hcard) he (by |
| apply pow_nonneg |
| have ha := Real.rpow_le_one_of_one_le_of_nonpos hH' (by norm_num : -(1/2:ℝ) ≤ 0) |
| have hh := (mul_le_of_le_one_left (by positivity : 0 ≤ (H:ℝ)^(-(1/2:ℝ))) |
| (show (κ:ℝ) ≤ 1 by exact_mod_cast hκ1)).trans ha |
| linarith) (by positivity) |
| have hbnd := hh.trans (add_le_add (add_le_add hc hb) he') |
| simpa only [H,Nat.cast_add,Nat.cast_one,Nat.cast_mul,Nat.cast_pow,mul_assoc] using hbnd |
|
|
| end DirectionalTransience |
|
|
| end |
|
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| end OAI |
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