| import Mathlib |
| import OAI.Probability.SKBarriers.Dynamics.ContinuousCrossing |
|
|
| namespace OAI |
|
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| section |
|
|
| noncomputable section |
| open scoped BigOperators Topology |
| open Classical MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def discreteMixedMass {n : ℕ} (β : ℝ) (J : Disorder n) (k : ℕ) : ℝ := |
| ∑x : Config n,if discreteDistance β J x k≤(1/4:ℝ) then gibbs β J x else 0 |
|
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| def continuousMixedMass {n : ℕ} (β : ℝ) (J : Disorder n) (t : ℝ) : ℝ := |
| ∑x : Config n,if continuousDistance β J x t≤(1/4:ℝ) then gibbs β J x else 0 |
|
|
| theorem discreteMixedMass_le_sign {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) (k : ℕ) |
| (v : Config n → Config n) : discreteMixedMass β J k ≤ |
| 4*∑x,gibbs β J x*kernelEvent β J (fun y => overlap (v x) y≤0) k x := by |
| unfold discreteMixedMass |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro x _ |
| by_cases h : discreteDistance β J x k≤(1/4:ℝ) |
| · simp only [ite_eq_left h] |
| have H := discrete_sign_mass_of_tv hn β J x (v x) k h |
| change (1/4:ℝ)≤kernelEvent β J (fun y => overlap (v x) y≤0) k x at H |
| have := mul_le_mul_of_nonneg_left H (gibbs_pos β J x).le |
| linarith only [this] |
| · simp only [ite_eq_right h] |
| exact mul_nonneg (by norm_num) (mul_nonneg (gibbs_pos β J x).le (kernelEvent_nonneg _ _ _ _ _)) |
|
|
| theorem continuousMixedMass_le_sign {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) {t : ℝ} (ht : 0≤t) |
| (v : Config n → Config n) : continuousMixedMass β J t ≤ |
| 4*∑x,gibbs β J x*∑y,if overlap (v x) y≤0 then continuousKernel β J t x y else 0 := by |
| unfold continuousMixedMass |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro x _ |
| by_cases h : continuousDistance β J x t≤(1/4:ℝ) |
| · simp only [ite_eq_left h] |
| have H := continuous_sign_mass_of_tv hn β J x (v x) ht h |
| have := mul_le_mul_of_nonneg_left H (gibbs_pos β J x).le |
| linarith only [this] |
| · simp only [ite_eq_right h] |
| apply mul_nonneg (by norm_num) |
| apply mul_nonneg (gibbs_pos β J x).le |
| apply Finset.sum_nonneg |
| intro y _ |
| split_ifs |
| · exact continuousKernel_nonneg β J ht x y |
| · exact le_rfl |
|
|
| theorem averaged_sign_clock {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) {t : ℝ} (ht : 0≤t) |
| (v : Config n → Config n) : |
| (∑x,gibbs β J x*∑y,if overlap (v x) y≤0 then continuousKernel β J t x y else 0)= |
| ∑' k : ℕ,clockWeight ((n:ℝ)*t) k*(∑x,gibbs β J x*kernelEvent β J (fun y => overlap (v x) y≤0) k x) := by |
| simp_rw [continuous_event_eq_clock hn β J _ ht] |
| simp_rw [← tsum_mul_left] |
| rw [← Summable.tsum_finsetSum (fun x (_ : x∈(Finset.univ : Finset (Config n))) => |
| (clock_kernelEvent_summable hn β J (fun y => overlap (v x) y≤0) (by positivity) x).mul_left (gibbs β J x))] |
| congr 1 |
| funext k |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro x _ |
| ring |
|
|
| theorem averaged_kernelEvent_nonneg {n : ℕ} (β : ℝ) (J : Disorder n) (A : Config n → Config n → Prop) (k : ℕ) : |
| 0≤∑x,gibbs β J x*kernelEvent β J (A x) k x := by |
| apply Finset.sum_nonneg |
| intro x _ |
| exact mul_nonneg (gibbs_pos β J x).le (kernelEvent_nonneg _ _ _ _ _) |
|
|
| theorem averaged_kernelEvent_le_one {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) (A : Config n → Config n → Prop) (k : ℕ) : |
| (∑x,gibbs β J x*kernelEvent β J (A x) k x)≤1 := by |
| rw [← gibbs_sum β J] |
| apply Finset.sum_le_sum |
| intro x _ |
| exact mul_le_of_le_one_right (gibbs_pos β J x).le (kernelEvent_le_one hn β J _ _ _) |
|
|
| theorem averaged_sign_clock_bound {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) {t : ℝ} (ht : 0≤t) |
| (v : Config n → Config n) (m : ℕ) {E : ℝ} (hE : 0≤E) |
| (h : ∀k ≤ m,(∑x,gibbs β J x*kernelEvent β J (fun y => overlap (v x) y≤0) k x)≤E) : |
| (∑x,gibbs β J x*∑y,if overlap (v x) y≤0 then continuousKernel β J t x y else 0)≤ |
| E+∑' k : ℕ,if m<k then clockWeight ((n:ℝ)*t) k else 0 := by |
| rw [averaged_sign_clock hn β J ht v] |
| have ha : 0≤(n:ℝ)*t := by positivity |
| have hs : Summable (fun k : ℕ => clockWeight ((n:ℝ)*t) k* |
| (∑x,gibbs β J x*kernelEvent β J (fun y => overlap (v x) y≤0) k x)) := by |
| apply Summable.of_nonneg_of_le _ _ (clockWeight_hasSum ((n:ℝ)*t)).summable |
| · intro k; exact mul_nonneg (clockWeight_nonneg ha k) (averaged_kernelEvent_nonneg _ _ _ _) |
| · intro k; exact mul_le_of_le_one_right (clockWeight_nonneg ha k) (averaged_kernelEvent_le_one hn _ _ _ _) |
| have HB := hs.tsum_le_tsum (g := fun k => clockWeight ((n:ℝ)*t) k*E+(if m<k then clockWeight ((n:ℝ)*t) k else 0)) |
| (fun k => by |
| by_cases hk : m<k |
| · simp only [ite_eq_left hk] |
| have H := mul_le_of_le_one_right (clockWeight_nonneg ha k) (averaged_kernelEvent_le_one hn β J (fun x y => overlap (v x) y≤0) k) |
| linarith only [H,mul_nonneg (clockWeight_nonneg ha k) hE] |
| · simp only [ite_eq_right hk,add_zero] |
| exact mul_le_mul_of_nonneg_left (h k (le_of_not_gt hk)) (clockWeight_nonneg ha k)) |
| (((clockWeight_hasSum ((n:ℝ)*t)).summable.mul_right E).add (clockWeight_tail_summable ha m)) |
| simpa only [Summable.tsum_add ((clockWeight_hasSum ((n:ℝ)*t)).summable.mul_right E) (clockWeight_tail_summable ha m),tsum_mul_right,clockWeight_sum,one_mul] using HB |
|
|
| theorem mixedMass_bounds_of_crossing {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) (L E : ℝ) |
| (hL : 0≤L) (hE : 0≤E) (v : Config n → Config n) |
| (h : ∀k≤⌈Real.exp (2*L)⌉₊,(∑x,gibbs β J x*kernelEvent β J (fun y => overlap (v x) y≤0) k x)≤E) : |
| continuousMixedMass β J (Real.exp L)≤4*(E+(n:ℝ)*Real.exp (-L)) ∧ |
| discreteMixedMass β J ⌊Real.exp L⌋₊≤4*E := by |
| constructor |
| · refine (continuousMixedMass_le_sign hn β J (Real.exp_pos L).le v).trans ?_ |
| apply mul_le_mul_of_nonneg_left _ (by norm_num) |
| exact (averaged_sign_clock_bound hn β J (Real.exp_pos L).le v _ hE h).trans |
| (add_le_add le_rfl (attempt_horizon_tail n L)) |
| · refine (discreteMixedMass_le_sign hn β J ⌊Real.exp L⌋₊ v).trans ?_ |
| apply mul_le_mul_of_nonneg_left _ (by norm_num) |
| apply h |
| exact (Nat.floor_mono (Real.exp_le_exp.mpr (by linarith only [hL]))).trans (Nat.floor_le_ceil _) |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
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| end OAI |
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