| import Mathlib |
| import OAI.Probability.SKBarriers.SpinGlass.Basic |
| import OAI.Probability.SKBarriers.Dynamics.HeatBathChoice |
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|
| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
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|
| theorem finite_event_le_tv {Ω : Type*} [Fintype Ω] (p q : Ω → ℝ) |
| (hp : ∑ x, p x = 1) (hq : ∑ x, q x = 1) (A : Ω → Prop) [DecidablePred A] : |
| (∑ x, if A x then q x else 0)-(∑ x, if A x then p x else 0) ≤ |
| (1/2:ℝ)*∑ x, |p x-q x| := by |
| have hsum : ∑ x, (q x-p x) = 0 := by rw [Finset.sum_sub_distrib,hq,hp,sub_self] |
| have ht (x : Ω) : 2*(if A x then q x-p x else 0)-(q x-p x) ≤ |q x-p x| := by |
| by_cases hx : A x |
| · simp only [ite_eq_left hx] |
| linarith [le_abs_self (q x-p x)] |
| · simp only [ite_eq_right hx,mul_zero,zero_sub] |
| exact neg_le_abs (q x-p x) |
| have H := Finset.sum_le_sum (fun x (_ : x ∈ (Finset.univ : Finset Ω)) => ht x) |
| have he : (∑ x, if A x then q x-p x else 0) = |
| (∑ x, if A x then q x else 0)-(∑ x, if A x then p x else 0) := by |
| rw [← Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro x _ |
| split_ifs <;> ring |
| simp only [Finset.sum_sub_distrib,← Finset.mul_sum,hsum,sub_zero,he] at H |
| have ha : (∑ x, |q x-p x|) = ∑ x, |p x-q x| := by |
| apply Finset.sum_congr rfl |
| intro x _ |
| exact abs_sub_comm _ _ |
| rw [ha] at H |
| linarith |
|
|
| theorem gibbs_nonpositive_overlap_mass {n : ℕ} (β : ℝ) (J : Disorder n) (v : Config n) : |
| (1/2:ℝ) ≤ ∑ y : Config n, if overlap v y ≤ 0 then gibbs β J y else 0 := by |
| classical |
| let e : Config n ≃ Config n := ⟨flip,flip,flip_flip,flip_flip⟩ |
| have he : (∑ y : Config n, if -overlap v y ≤ 0 then gibbs β J y else 0) = |
| ∑ y : Config n, if overlap v y ≤ 0 then gibbs β J y else 0 := by |
| calc |
| _ = ∑ y : Config n, if overlap v (e y) ≤ 0 then gibbs β J (e y) else 0 := by |
| change _ = ∑ y : Config n, if overlap v (flip y) ≤ 0 then gibbs β J (flip y) else 0 |
| simp only [overlap_flip_right,gibbs_flip] |
| _ = _ := Equiv.sum_comp e (fun y => if overlap v y ≤ 0 then gibbs β J y else 0) |
| have ht (y : Config n) : gibbs β J y ≤ |
| (if overlap v y ≤ 0 then gibbs β J y else 0)+ |
| (if -overlap v y ≤ 0 then gibbs β J y else 0) := by |
| by_cases h : overlap v y ≤ 0 |
| · simp only [ite_eq_left h] |
| split_ifs <;> linarith [gibbs_pos β J y] |
| · have h' : -overlap v y ≤ 0 := by linarith [lt_of_not_ge h] |
| simp only [ite_eq_right h,ite_eq_left h',zero_add,le_refl] |
| have H := Finset.sum_le_sum (fun y (_ : y ∈ (Finset.univ : Finset (Config n))) => ht y) |
| rw [gibbs_sum,Finset.sum_add_distrib,he] at H |
| linarith |
|
|
| theorem discrete_sign_mass_of_tv {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n) |
| (x v : Config n) (k : ℕ) (hTV : discreteDistance β J x k ≤ (1/4:ℝ)) : |
| (1/4:ℝ) ≤ ∑ y : Config n, if overlap v y ≤ 0 then discreteKernel β J k x y else 0 := by |
| have H := finite_event_le_tv (discreteKernel β J k x) (gibbs β J) |
| (discreteKernel_sum hn β J k x) (gibbs_sum β J) (fun y => overlap v y ≤ 0) |
| have HG := gibbs_nonpositive_overlap_mass β J v |
| change _ ≤ discreteDistance β J x k at H |
| linarith |
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| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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