| 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 |
| |
