| import Mathlib |
| import OAI.Probability.SKBarriers.Interpolation.RestrictedConcentration |
| import OAI.Probability.SKBarriers.SpinGlass.FiniteProduct |
| import OAI.Probability.SKBarriers.SpinGlass.FiniteIndicators |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open Classical |
| namespace SK.Analytic |
|
|
| namespace FiniteLaw |
| variable {X I : Type*} [Fintype X] [Fintype I] [DecidableEq I] |
| theorem iid_expect_involutions (P : FiniteLaw X) (T : I → X → X) |
| (hT : ∀ i, Function.Involutive (T i)) (hP : ∀ i x, P.weight (T i x)=P.weight x) |
| (f : (I → X) → ℝ) : |
| (P.iid I).expect (fun z => f (fun i => T i (z i)))=(P.iid I).expect f := by |
| let e : (I → X) ≃ (I → X) := |
| ⟨fun z i => T i (z i),fun z i => T i (z i),fun z => funext (fun i => hT i (z i)),fun z => funext (fun i => hT i (z i))⟩ |
| unfold expect iid pi |
| apply Fintype.sum_equiv e |
| intro z |
| dsimp only [e,Equiv.coe_fn_mk] |
| simp only [hP] |
|
|
| theorem iid_two_then_one (P : FiniteLaw X) (a b : I) (hab : a≠b) (f : (Fin 3 → X) → ℝ) : |
| (P.iid I).expect (fun z => P.expect (fun x => f ![z a,z b,x]))=(P.iid (Fin 3)).expect f := by |
| classical |
| have hi : Function.Injective (![a,b] : Fin 2 → I) := by |
| intro i j H |
| fin_cases i <;> fin_cases j <;> simp_all |
| have H := P.iid_expect_injective (![a,b] : Fin 2 → I) hi |
| (fun z => P.expect (fun x => f ![z 0,z 1,x])) |
| calc |
| _ = (P.iid (Fin 2)).expect (fun z => P.expect (fun x => f ![z 0,z 1,x])) := H |
| _ = _ := by |
| rw [iid_expect_fin_succ P 2] |
| congr 1 |
| funext z |
| congr 1 |
| funext x |
| congr 1 |
| funext i |
| fin_cases i <;> simp [Fin.snoc] |
| end FiniteLaw |
|
|
| def orient {n : ℕ} (ε : Bool) (x : Config n) : Config n := if ε then flip x else x |
|
|
| @[simp] theorem orient_orient {n : ℕ} (ε : Bool) (x : Config n) : orient ε (orient ε x)=x := by |
| cases ε <;> simp [orient] |
| @[simp] theorem gibbs_orient {n : ℕ} (β : ℝ) (J : Disorder n) (ε : Bool) (x : Config n) : |
| gibbs β J (orient ε x)=gibbs β J x := by cases ε <;> simp [orient] |
| @[simp] theorem overlap_orient_abs {n : ℕ} (ε : Bool) (x y : Config n) : |
| |overlap (orient ε x) y|=|overlap x y| := by |
| cases ε |
| · simp [orient] |
| · change |overlap (flip x) y| = |overlap x y| |
| rw [overlap_comm,overlap_flip_right,abs_neg,overlap_comm] |
|
|
| theorem iid_gibbs_replica {n d : ℕ} (β : ℝ) (J : Disorder n) (S : Finset (ReplicaConfig n d)) : |
| ((gibbsFiniteLaw β J).iid (Fin d)).prob (fun z => z∈S)=replicaGibbsMass β J S := by |
| classical |
| unfold FiniteLaw.prob FiniteLaw.expect FiniteLaw.iid FiniteLaw.pi gibbsFiniteLaw replicaGibbsMass |
| simp only [mul_ite,mul_one,mul_zero,← Finset.sum_filter,Finset.filter_mem_eq_inter,Finset.univ_inter] |
|
|
| theorem iid_gibbs_oriented_replica {n d : ℕ} (β : ℝ) (J : Disorder n) |
| (ε : Fin d → Bool) (S : Finset (ReplicaConfig n d)) : |
| ((gibbsFiniteLaw β J).iid (Fin d)).prob (fun z => (fun i => orient (ε i) (z i))∈S)=replicaGibbsMass β J S := by |
| have H := (gibbsFiniteLaw β J).iid_expect_involutions (fun i => orient (ε i)) (fun i => orient_orient (ε i)) |
| (fun i z => gibbs_orient β J (ε i) z) (fun z => if z∈S then 1 else 0) |
| simp only [FiniteLaw.expect_indicator_eq_prob] at H |
| exact H.trans (iid_gibbs_replica β J S) |
|
|
| theorem bank_triple_oriented {n : ℕ} {I : Type*} [Fintype I] [DecidableEq I] |
| (β : ℝ) (J : Disorder n) (a b c : I) (hab : a≠b) (hac : a≠c) (hbc : b≠c) |
| (ε₀ ε₁ ε₂ : Bool) (S : Finset (ReplicaConfig n 3)) : |
| ((gibbsFiniteLaw β J).iid I).prob (fun z => (![orient ε₀ (z a),orient ε₁ (z b),orient ε₂ (z c)] : ReplicaConfig n 3)∈S)= |
| replicaGibbsMass β J S := by |
| classical |
| have hi : Function.Injective (![a,b,c] : Fin 3 → I) := by |
| intro i j H |
| fin_cases i <;> fin_cases j <;> simp_all |
| have H := (gibbsFiniteLaw β J).iid_prob_injective (![a,b,c] : Fin 3 → I) hi |
| (fun z => (fun i => orient (![ε₀,ε₁,ε₂] i) (z i))∈S) |
| have he (z : I → Config n) : (fun i => orient (![ε₀,ε₁,ε₂] i) ((z ∘ ![a,b,c]) i))= |
| (![orient ε₀ (z a),orient ε₁ (z b),orient ε₂ (z c)] : ReplicaConfig n 3) := by |
| funext i; fin_cases i <;> rfl |
| simp only [he] at H |
| exact H.trans (iid_gibbs_oriented_replica β J _ S) |
|
|
| theorem bank_pair_vertex_oriented {n : ℕ} {I : Type*} [Fintype I] [DecidableEq I] |
| (β : ℝ) (J : Disorder n) (a b : I) (hab : a≠b) |
| (ε₀ ε₁ ε₂ : Bool) (S : Finset (ReplicaConfig n 3)) : |
| ((gibbsFiniteLaw β J).iid I).expect (fun z => (gibbsFiniteLaw β J).prob (fun x => |
| (![orient ε₀ (z a),orient ε₁ (z b),orient ε₂ x] : ReplicaConfig n 3)∈S))= |
| replicaGibbsMass β J S := by |
| have H := (gibbsFiniteLaw β J).iid_two_then_one a b hab |
| (fun z => if (fun i => orient (![ε₀,ε₁,ε₂] i) (z i))∈S then 1 else 0) |
| have he (z : I → Config n) (x : Config n) : |
| (fun i => orient (![ε₀,ε₁,ε₂] i) ((![z a,z b,x] : ReplicaConfig n 3) i))= |
| (![orient ε₀ (z a),orient ε₁ (z b),orient ε₂ x] : ReplicaConfig n 3) := by |
| funext i; fin_cases i <;> rfl |
| simp only [he,FiniteLaw.expect_indicator_eq_prob] at H |
| exact H.trans (iid_gibbs_oriented_replica β J _ S) |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
