| 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 |
|
|
