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c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Arrays.ReplicaContractions
namespace OAI
/-! Two-replica Gibbs calculus used by the off-diagonal Ward equation. -/
noncomputable section
open IsingPerceptron MeasureTheory
open scoped BigOperators
namespace InvariantIsing
variable {S : Type*} [Fintype S]
omit [Fintype S] in
lemma GibbsReference_pair {w : S → ℝ} (hw : GibbsReference w) :
GibbsReference (fun x : S × S => w x.1 * w x.2) := by
refine ⟨fun x => mul_nonneg (hw.1 x.1) (hw.1 x.2), ?_⟩
obtain ⟨σ, hσ⟩ := hw.2
exact ⟨(σ, σ), mul_pos hσ hσ⟩
lemma finitePartition_pair (w H : S → ℝ) :
finitePartition (fun x : S × S => w x.1 * w x.2)
(fun x => H x.1 + H x.2) = finitePartition w H * finitePartition w H := by
simp only [finitePartition, Fintype.sum_prod_type, Real.exp_add, Finset.sum_mul_sum]
apply Finset.sum_congr rfl
intro σ _
apply Finset.sum_congr rfl
intro τ _
ring
lemma finiteGibbs_pair (w H : S → ℝ) (x : S × S) :
finiteGibbs (fun x : S × S => w x.1 * w x.2) (fun x => H x.1 + H x.2) x =
finiteGibbs w H x.1 * finiteGibbs w H x.2 := by
simp only [finiteGibbs, finitePartition_pair, Real.exp_add, div_eq_mul_inv, mul_inv_rev]
ring
lemma gibbsAverage_pair (w H : S → ℝ) (A : S → S → ℝ) :
gibbsAverage (fun x : S × S => w x.1 * w x.2) (fun x => H x.1 + H x.2)
(fun x => A x.1 x.2) = gibbsPairAverage w H A := by
simp only [gibbsAverage, Fintype.sum_prod_type, finiteGibbs_pair, gibbsPairAverage]
lemma gibbsPairAverage_add (w H : S → ℝ) (A B : S → S → ℝ) :
gibbsPairAverage w H (fun σ τ => A σ τ + B σ τ) =
gibbsPairAverage w H A + gibbsPairAverage w H B := by
simp only [gibbsPairAverage, mul_add, Finset.sum_add_distrib]
lemma gibbsPairAverage_fst {w : S → ℝ} (hw : GibbsReference w) (H A : S → ℝ) :
gibbsPairAverage w H (fun σ _ => A σ) = gibbsAverage w H A := by
calc
_ = ∑ σ, finiteGibbs w H σ * A σ * ∑ τ, finiteGibbs w H τ := by
simp only [gibbsPairAverage, Finset.mul_sum]
apply Finset.sum_congr rfl
intro σ _
apply Finset.sum_congr rfl
intro τ _
ring
_ = _ := by simp only [finiteGibbs_sum hw H, mul_one, gibbsAverage]
lemma gibbsPairAverage_swap (w H : S → ℝ) (A : S → S → ℝ) :
gibbsPairAverage w H (fun σ τ => A τ σ) = gibbsPairAverage w H A := by
unfold gibbsPairAverage
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro σ _
apply Finset.sum_congr rfl
intro τ _
ring
lemma gibbsPairAverage_snd {w : S → ℝ} (hw : GibbsReference w) (H A : S → ℝ) :
gibbsPairAverage w H (fun _ τ => A τ) = gibbsAverage w H A := by
rw [← gibbsPairAverage_fst hw H A]
exact gibbsPairAverage_swap w H (fun σ _ => A σ)
lemma gibbsAverage_pair_sum {w : S → ℝ} (hw : GibbsReference w) (H A : S → ℝ) :
gibbsAverage (fun x : S × S => w x.1 * w x.2) (fun x => H x.1 + H x.2)
(fun x => A x.1 + A x.2) = 2 * gibbsAverage w H A := by
rw [gibbsAverage_pair w H (fun σ τ => A σ + A τ), gibbsPairAverage_add,
gibbsPairAverage_fst hw, gibbsPairAverage_snd hw]
ring
/-- The two-replica derivative has one Hamiltonian insertion in each existing
replica and two Gibbs normalization subtractions. -/
lemma hasDerivAt_gibbsPairAverage {w : S → ℝ} (hw : GibbsReference w)
{H : ℝ → S → ℝ} {A : ℝ → S → S → ℝ}
{dH : S → ℝ} {dA : S → S → ℝ} {t : ℝ}
(hH : ∀ σ, HasDerivAt (fun s => H s σ) (dH σ) t)
(hA : ∀ σ τ, HasDerivAt (fun s => A s σ τ) (dA σ τ) t) :
HasDerivAt (fun s => gibbsPairAverage w (H s) (A s))
(gibbsPairAverage w (H t) dA +
gibbsPairAverage w (H t) (fun σ τ => A t σ τ * (dH σ + dH τ)) -
2 * gibbsPairAverage w (H t) (A t) * gibbsAverage w (H t) dH) t := by
have hd := hasDerivAt_gibbsAverage (GibbsReference_pair hw)
(fun x : S × S => (hH x.1).add (hH x.2)) (fun x : S × S => hA x.1 x.2)
simp only [Pi.add_apply, gibbsAverage_pair] at hd
rw [gibbsAverage_pair w (H t) (fun σ τ => A t σ τ * (dH σ + dH τ)),
gibbsAverage_pair w (H t) (fun σ τ => dH σ + dH τ)] at hd
rw [gibbsPairAverage_add, gibbsPairAverage_fst hw, gibbsPairAverage_snd hw] at hd
convert hd using 1
ring
end InvariantIsing
end
end OAI
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