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135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Dynamics.ClockTail
import OAI.Probability.SKBarriers.Dynamics.BarrierCrossing
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem occupation_mono {n : ℕ} (β : ℝ) (J : Disorder n) (A : Config n → Prop) [DecidablePred A]
{k m : ℕ} (hkm : k ≤ m) (x : Config n) : occupation β J A k x ≤ occupation β J A m x := by
apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.range_mono (Nat.add_le_add_right hkm 1))
intro i _ _
exact kernelEvent_nonneg _ _ _ _ _
theorem clock_kernelEvent_summable {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n)
(A : Config n → Prop) [DecidablePred A] {a : ℝ} (ha : 0 ≤ a) (x : Config n) :
Summable (fun k : ℕ => clockWeight a k*kernelEvent β J A k x) := by
apply Summable.of_nonneg_of_le _ _ (clockWeight_hasSum a).summable
· intro k
exact mul_nonneg (clockWeight_nonneg ha k) (kernelEvent_nonneg _ _ _ _ _)
· intro k
exact mul_le_of_le_one_right (clockWeight_nonneg ha k) (kernelEvent_le_one hn β J A k x)
theorem continuous_event_eq_clock {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n)
(A : Config n → Prop) [DecidablePred A] {t : ℝ} (ht : 0 ≤ t) (x : Config n) :
(∑ y : Config n, if A y then continuousKernel β J t x y else 0) =
∑' k : ℕ, clockWeight ((n:ℝ)*t) k*kernelEvent β J A k x := by
have he (y : Config n) : (if A y then continuousKernel β J t x y else 0) =
∑' k : ℕ, if A y then clockWeight ((n:ℝ)*t) k*discreteKernel β J k x y else 0 := by
by_cases hy : A y
· simp only [ite_eq_left hy,continuousKernel_eq_clock]
· simp only [ite_eq_right hy,tsum_zero]
have hs (y : Config n) : Summable (fun k : ℕ =>
if A y then clockWeight ((n:ℝ)*t) k*discreteKernel β J k x y else 0) := by
by_cases hy : A y
· simp only [ite_eq_left hy]
exact uniformization_summable hn β J ht x y
· simp only [ite_eq_right hy]
exact summable_zero
simp_rw [he]
rw [← Summable.tsum_finsetSum (fun y (_ : y ∈ (Finset.univ : Finset (Config n))) => hs y)]
simp only [kernelEvent,Finset.mul_sum,mul_ite,mul_zero]
theorem continuous_sign_le_stripOccupation_tail {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n)
(v : Config n) {a₀ a₁ : ℝ} (ha₀ : 0 < a₀) (hgap : 2/(n:ℝ) < a₁-a₀)
{t : ℝ} (ht : 0 ≤ t) (m : ℕ) (x : Config n) (hx : a₁ < overlap v x) :
(∑ y : Config n, if overlap v y ≤ 0 then continuousKernel β J t x y else 0) ≤
occupation β J (fun y => a₀ ≤ overlap v y ∧ overlap v y ≤ a₁) m x +
∑' k : ℕ, if m < k then clockWeight ((n:ℝ)*t) k else 0 := by
rw [continuous_event_eq_clock hn β J _ ht]
have ha : 0 ≤ (n:ℝ)*t := by positivity
let O := occupation β J (fun y => a₀ ≤ overlap v y ∧ overlap v y ≤ a₁) m x
have hO : 0 ≤ O := occupation_nonneg _ _ _ _ _
have H := (clock_kernelEvent_summable hn β J (fun y => overlap v y ≤ 0) ha x).tsum_le_tsum
(g := fun k => clockWeight ((n:ℝ)*t) k*O+(if m < k then clockWeight ((n:ℝ)*t) k else 0))
(fun k => show _ ≤ _ from by
by_cases hk : m < k
· simp only [ite_eq_left hk]
have HH := mul_le_of_le_one_right (clockWeight_nonneg ha k) (kernelEvent_le_one hn β J (fun y => overlap v y ≤ 0) k x)
linarith [mul_nonneg (clockWeight_nonneg ha k) hO]
· simp only [ite_eq_right hk,add_zero]
exact mul_le_mul_of_nonneg_left ((signEvent_le_stripOccupation hn β J v ha₀ hgap k x hx).trans
(occupation_mono β J _ (le_of_not_gt hk) x)) (clockWeight_nonneg ha k))
(((clockWeight_hasSum ((n:ℝ)*t)).summable.mul_right O).add (clockWeight_tail_summable ha m))
rw [Summable.tsum_add ((clockWeight_hasSum ((n:ℝ)*t)).summable.mul_right O)
(clockWeight_tail_summable ha m),tsum_mul_right,clockWeight_sum,one_mul] at H
exact H
theorem eighth_le_stripOccupation_of_continuous_tv {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n)
(v : Config n) {a₀ a₁ : ℝ} (ha₀ : 0 < a₀) (hgap : 2/(n:ℝ) < a₁-a₀)
(L : ℝ) (hL : (n:ℝ)*Real.exp (-L) ≤ (1/8:ℝ)) (x : Config n) (hx : a₁ < overlap v x)
(hTV : continuousDistance β J x (Real.exp L) ≤ (1/4:ℝ)) :
(1/8:ℝ) ≤ occupation β J (fun y => a₀ ≤ overlap v y ∧ overlap v y ≤ a₁)
⌈Real.exp (2*L)⌉₊ x := by
have H₁ := continuous_sign_mass_of_tv hn β J x v (le_of_lt (Real.exp_pos L)) hTV
have H₂ := continuous_sign_le_stripOccupation_tail hn β J v ha₀ hgap
(le_of_lt (Real.exp_pos L)) ⌈Real.exp (2*L)⌉₊ x hx
have H₃ := attempt_horizon_tail n L
linarith
theorem mass_continuous_mixed_high_le_strip {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n)
(v : Config n) {a₀ a₁ : ℝ} (ha₀ : 0 < a₀) (hgap : 2/(n:ℝ) < a₁-a₀)
(L : ℝ) (hL : (n:ℝ)*Real.exp (-L) ≤ (1/8:ℝ)) :
(∑ x : Config n, if a₁ < overlap v x ∧ continuousDistance β J x (Real.exp L) ≤ (1/4:ℝ)
then gibbs β J x else 0) ≤ 8*((⌈Real.exp (2*L)⌉₊+1:ℕ):ℝ)*
(∑ y : Config n, if a₀ ≤ overlap v y ∧ overlap v y ≤ a₁ then gibbs β J y else 0) := by
have HH := Finset.sum_le_sum (s := (Finset.univ : Finset (Config n)))
(f := fun x => if a₁ < overlap v x ∧ continuousDistance β J x (Real.exp L) ≤ (1/4:ℝ)
then gibbs β J x else 0)
(g := fun x => 8*(gibbs β J x*occupation β J (fun y => a₀ ≤ overlap v y ∧ overlap v y ≤ a₁)
⌈Real.exp (2*L)⌉₊ x))
(fun x _ => show _ ≤ _ from by
split_ifs with hx
· have H := eighth_le_stripOccupation_of_continuous_tv hn β J v ha₀ hgap L hL x hx.1 hx.2
nlinarith [mul_le_mul_of_nonneg_left H (le_of_lt (gibbs_pos β J x))]
· exact mul_nonneg (by norm_num) (mul_nonneg (le_of_lt (gibbs_pos β J x))
(occupation_nonneg _ _ _ _ _)))
rw [← Finset.mul_sum,occupation_stationary hn] at HH
simpa only [mul_assoc] using HH
end SK.Analytic
end
end
end
end OAI
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