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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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | 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
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