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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 | import Mathlib
import OAI.Probability.SKBarriers.SpinGlass.Basic
import OAI.Probability.SKBarriers.Dynamics.HeatBathChoice
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
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
end SK.Analytic
end
end
end
end OAI
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