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9b6e014 | 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 | 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
end OAI
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