| import Mathlib |
| import OAI.Probability.SKBarriers.Dynamics.HeatBathStationary |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace SK.Analytic |
| namespace FiniteMarkov |
| variable {X : Type*} |
|
|
| theorem sum_cons [Fintype X] (m : ℕ) (f : (Fin (m+1) → X) → ℝ) : |
| (∑ s,f s)=∑ x : X,∑ t : Fin m → X,f (Fin.cons x t) := by |
| classical |
| calc |
| _ = ∑ z : X × (Fin m → X),f (Fin.cons z.1 z.2) := by |
| apply Fintype.sum_equiv (Fin.consEquiv (fun _ : Fin (m+1) => X)).symm |
| intro s |
| simp only [Fin.consEquiv_symm_apply,Fin.cons_self_tail] |
| _ = _ := Fintype.sum_prod_type _ |
|
|
| def walkWeight (K : X → X → ℝ) : (m : ℕ) → X → (Fin m → X) → ℝ |
| | 0,_,_ => 1 |
| | m+1,x,s => K x (s 0)*walkWeight K m (s 0) (Fin.tail s) |
|
|
| @[simp] theorem walkWeight_zero (K : X → X → ℝ) (x : X) (s : Fin 0 → X) : |
| walkWeight K 0 x s=1 := rfl |
|
|
| @[simp] theorem walkWeight_cons (K : X → X → ℝ) (m : ℕ) (x y : X) (s : Fin m → X) : |
| walkWeight K (m+1) x (Fin.cons y s)=K x y*walkWeight K m y s := rfl |
|
|
| theorem walkWeight_nonneg (K : X → X → ℝ) (hK : ∀ x y,0≤K x y) (m : ℕ) (x : X) (s : Fin m → X) : |
| 0≤walkWeight K m x s := by |
| induction m generalizing x with |
| | zero => exact zero_le_one |
| | succ m ih => exact mul_nonneg (hK _ _) (ih _ _) |
|
|
| theorem walkWeight_sum [Fintype X] (K : X → X → ℝ) (hK : ∀ x,∑ y,K x y=1) (m : ℕ) (x : X) : |
| ∑ s : Fin m → X,walkWeight K m x s=1 := by |
| classical |
| induction m generalizing x with |
| | zero => simp |
| | succ m ih => |
| rw [sum_cons] |
| simp only [walkWeight_cons,← Finset.mul_sum,ih,mul_one,hK] |
|
|
| def pathWeight (p : X → ℝ) (K : X → X → ℝ) (m : ℕ) (s : Fin (m+1) → X) : ℝ := |
| p (s 0)*walkWeight K m (s 0) (Fin.tail s) |
|
|
| theorem pathWeight_nonneg (p : X → ℝ) (K : X → X → ℝ) |
| (hp : ∀ x,0≤p x) (hK : ∀ x y,0≤K x y) (m : ℕ) (s : Fin (m+1) → X) : |
| 0≤pathWeight p K m s := mul_nonneg (hp _) (walkWeight_nonneg K hK _ _ _) |
|
|
| theorem pathWeight_sum [Fintype X] (p : X → ℝ) (K : X → X → ℝ) |
| (hp : ∑ x,p x=1) (hK : ∀ x,∑ y,K x y=1) (m : ℕ) : |
| ∑ s : Fin (m+1) → X,pathWeight p K m s=1 := by |
| rw [sum_cons] |
| simp only [pathWeight,Fin.cons_zero,Fin.tail_cons,← Finset.mul_sum,walkWeight_sum K hK,mul_one,hp] |
|
|
| theorem pathWeight_first [Fintype X] (p : X → ℝ) (K : X → X → ℝ) |
| (hK : ∀ x,∑ y,K x y=1) (m : ℕ) (f : X → ℝ) : |
| (∑ s : Fin (m+1) → X,pathWeight p K m s*f (s 0))=∑ x,p x*f x := by |
| rw [sum_cons] |
| simp only [pathWeight,Fin.cons_zero,Fin.tail_cons,mul_right_comm (p _) _ (f _),← Finset.mul_sum, |
| walkWeight_sum K hK,mul_one] |
|
|
| theorem pathWeight_marginal [Fintype X] (p : X → ℝ) (K : X → X → ℝ) |
| (hK : ∀ x,∑ y,K x y=1) (hstat : ∀ y,∑ x,p x*K x y=p y) |
| (m : ℕ) (k : Fin (m+1)) (f : X → ℝ) : |
| (∑ s : Fin (m+1) → X,pathWeight p K m s*f (s k))=∑ y,p y*f y := by |
| classical |
| induction m with |
| | zero => |
| have hk : k=0 := by exact Fin.ext (by omega) |
| subst k |
| exact pathWeight_first p K hK 0 f |
| | succ m ih => |
| refine Fin.cases (pathWeight_first p K hK (m+1) f) (fun k => ?_) k |
| rw [sum_cons] |
| simp only [pathWeight,Fin.cons_zero,Fin.tail_cons,Fin.cons_succ,walkWeight] |
| simp_rw [sum_cons m] |
| simp only [Fin.cons_zero,Fin.tail_cons] |
| have he (x y : X) (s : Fin m → X) : |
| p x*(K x y*walkWeight K m y s)*f ((Fin.cons y s : Fin (m+1) → X) k)= |
| (p x*K x y)*(walkWeight K m y s*f ((Fin.cons y s : Fin (m+1) → X) k)) := by ring |
| simp_rw [he,← Finset.mul_sum] |
| rw [Finset.sum_comm] |
| simp_rw [← Finset.sum_mul,hstat] |
| have HH := ih k |
| rw [sum_cons] at HH |
| simpa only [pathWeight,Fin.cons_zero,Fin.tail_cons,mul_assoc,← Finset.mul_sum] using HH |
|
|
| theorem walkWeight_pos_step (K : X → X → ℝ) (hK : ∀ x y,0≤K x y) |
| (m : ℕ) (x : X) (s : Fin m → X) (h : 0<walkWeight K m x s) : |
| ∀ i : Fin m,0<K ((Fin.cons x s : Fin (m+1) → X) i.castSucc) (s i) := by |
| induction m generalizing x with |
| | zero => intro i; exact Fin.elim0 i |
| | succ m ih => |
| have hfirst : 0<K x (s 0) := lt_of_le_of_ne (hK _ _) (by intro he; simp [walkWeight,← he] at h) |
| have htail : 0<walkWeight K m (s 0) (Fin.tail s) := (mul_pos_iff_of_pos_left hfirst).mp h |
| intro i |
| refine Fin.cases (by simpa using hfirst) (fun i => ?_) i |
| simpa only [Fin.castSucc_succ,Fin.cons_succ,Fin.cons_self_tail,Fin.tail] using ih (s 0) (Fin.tail s) htail i |
|
|
| theorem pathWeight_pos_step (p : X → ℝ) (K : X → X → ℝ) |
| (hp : ∀ x,0≤p x) (hK : ∀ x y,0≤K x y) |
| (m : ℕ) (s : Fin (m+1) → X) (h : 0<pathWeight p K m s) : |
| ∀ i : Fin m,0<K (s i.castSucc) (s i.succ) := by |
| have hh : 0<walkWeight K m (s 0) (Fin.tail s) := |
| pos_of_mul_pos_right h (hp (s 0)) |
| intro i |
| simpa only [Fin.cons_self_tail,Fin.tail] using walkWeight_pos_step K hK m (s 0) (Fin.tail s) hh i |
|
|
| end FiniteMarkov |
|
|
| def stationaryPathWeight {n : ℕ} (β : ℝ) (J : Disorder n) (m : ℕ) : (Fin (m+1) → Config n) → ℝ := |
| FiniteMarkov.pathWeight (gibbs β J) (heatBath β J) m |
|
|
| theorem stationaryPathWeight_sum {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) (m : ℕ) : |
| ∑ s,stationaryPathWeight β J m s=1 := |
| FiniteMarkov.pathWeight_sum _ _ (gibbs_sum β J) (heatBath_sum hn β J) m |
|
|
| theorem stationaryPathWeight_marginal {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) |
| (m : ℕ) (k : Fin (m+1)) (f : Config n → ℝ) : |
| (∑ s,stationaryPathWeight β J m s*f (s k))=∑ y,gibbs β J y*f y := |
| FiniteMarkov.pathWeight_marginal _ _ (heatBath_sum hn β J) (heatBath_stationary hn β J) m k f |
|
|
| theorem stationaryPathWeight_pos_jump {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) |
| (m : ℕ) (s : Fin (m+1) → Config n) (hs : 0<stationaryPathWeight β J m s) |
| (v : Config n) (i : Fin m) : |overlap v (s i.succ)-overlap v (s i.castSucc)|≤2/(n:ℝ) := |
| heatBath_overlap_step hn β J v _ _ |
| (FiniteMarkov.pathWeight_pos_step _ _ (fun x => (gibbs_pos β J x).le) (heatBath_nonneg β J) m s hs i) |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
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| end OAI |
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