| import OAI.Probability.Ising.GibbsMeasure |
| import OAI.Probability.Ising.Interpolation |
|
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| namespace OAI |
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| open scoped BigOperators Classical |
|
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| namespace BufferedIsing.DSS |
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| noncomputable section |
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| variable {V : Type*} [Fintype V] |
|
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| / |
| def insertSpin (v : V) (b : Bool) (η : {u : V // u ≠ v} → Bool) : V → Bool := |
| (Equiv.funSplitAt v Bool).symm (b, η) |
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|
| omit [Fintype V] in |
| @[simp] lemma insertSpin_self (v : V) (b : Bool) (η : {u : V // u ≠ v} → Bool) : |
| insertSpin v b η v = b := by simp [insertSpin, Equiv.funSplitAt, Equiv.piSplitAt] |
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|
| omit [Fintype V] in |
| @[simp] lemma insertSpin_other (v : V) (b : Bool) (η : {u : V // u ≠ v} → Bool) |
| (u : {u : V // u ≠ v}) : insertSpin v b η u = η u := by |
| simp [insertSpin, Equiv.funSplitAt, Equiv.piSplitAt, u.property] |
|
|
| lemma sum_insert (v : V) (f : (V → Bool) → ℝ) : |
| (∑ σ, f σ) = |
| (∑ η, f (insertSpin v true η)) + ∑ η, f (insertSpin v false η) := by |
| rw [← (Equiv.funSplitAt v Bool).symm.sum_comp f, Fintype.sum_prod_type] |
| simp only [insertSpin, Fintype.sum_bool] |
|
|
| lemma sum_remove (v : V) (f : V → ℝ) : |
| (∑ u, f u) = f v + ∑ u : {u : V // u ≠ v}, f u := by |
| rw [← Finset.sum_erase_add _ _ (Finset.mem_univ v)] |
| have he : (∑ u ∈ Finset.univ.erase v, f u) = ∑ u : {u : V // u ≠ v}, f u := by |
| exact Finset.sum_subtype _ (fun _ => by simp) _ |
| rw [he, add_comm] |
|
|
| / |
| def removedField (J : V → V → ℝ) (h : V → ℝ) (v : V) (b : Bool) |
| (u : {u : V // u ≠ v}) : ℝ := h u + (J v u + J u v) * sgn b |
|
|
| def removedCoupling (J : V → V → ℝ) (v : V) |
| (u w : {u : V // u ≠ v}) : ℝ := J u w |
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|
| lemma hamiltonian_insert (J : V → V → ℝ) (h : V → ℝ) (v : V) |
| (b : Bool) (η : {u : V // u ≠ v} → Bool) : |
| hamiltonian J h (insertSpin v b η) = J v v + h v * sgn b + |
| hamiltonian (removedCoupling J v) (removedField J h v b) η := by |
| unfold hamiltonian |
| rw [sum_remove v (fun u => ∑ w, J u w * sgn (insertSpin v b η u) * sgn (insertSpin v b η w))] |
| rw [sum_remove v (fun u => h u * sgn (insertSpin v b η u))] |
| simp_rw [sum_remove v (fun w => J _ w * sgn (insertSpin v b η _) * sgn (insertSpin v b η w))] |
| simp only [insertSpin_self, insertSpin_other, mul_assoc, sgn_sq, mul_one, |
| Finset.sum_add_distrib, removedCoupling, removedField, add_mul, Finset.sum_add_distrib] |
| have he : (∑ u : {u : V // u ≠ v}, J v u * (sgn b * sgn (η u))) + |
| (∑ u : {u : V // u ≠ v}, J u v * (sgn (η u) * sgn b)) = |
| ∑ u : {u : V // u ≠ v}, (J v u + J u v) * sgn b * sgn (η u) := by |
| rw [← Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro u _ |
| ring |
| simp only [add_mul, Finset.sum_add_distrib, mul_assoc] at he |
| linarith |
|
|
| lemma mass_insert (J : V → V → ℝ) (h : V → ℝ) (v : V) |
| (b : Bool) (η : {u : V // u ≠ v} → Bool) : |
| mass J h (insertSpin v b η) = Real.exp (J v v + h v * sgn b) * |
| mass (removedCoupling J v) (removedField J h v b) η := by |
| simp only [mass, hamiltonian_insert, Real.exp_add] |
|
|
| / |
| def pinMean (J : V → V → ℝ) (h : V → ℝ) (v : V) (b : Bool) (o : V) : ℝ := |
| average (removedCoupling J v) (removedField J h v b) (fun η => sgn (insertSpin v b η o)) |
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|
| lemma pinMean_self (J : V → V → ℝ) (h : V → ℝ) (v : V) (b : Bool) : |
| pinMean J h v b v = sgn b := by |
| simp only [pinMean, insertSpin_self, average_const] |
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|
| lemma pinMean_other (J : V → V → ℝ) (h : V → ℝ) (v : V) (b : Bool) |
| (o : V) (hov : o ≠ v) : |
| pinMean J h v b o = mean (removedCoupling J v) (removedField J h v b) ⟨o, hov⟩ := by |
| unfold pinMean mean |
| congr 1 |
| funext η |
| exact congrArg sgn (insertSpin_other v b η ⟨o, hov⟩) |
|
|
| lemma pinMean_mono (J : V → V → ℝ) (hJ : ∀ u v, 0 ≤ J u v) |
| (h : V → ℝ) (v o : V) : pinMean J h v false o ≤ pinMean J h v true o := by |
| by_cases hov : o = v |
| · subst o |
| simp [pinMean_self, sgn] |
| · rw [pinMean_other _ _ _ _ _ hov, pinMean_other _ _ _ _ _ hov] |
| apply mean_mono (removedCoupling J v) (fun u w => hJ u.val w.val) |
| intro u |
| dsimp [removedField, sgn] |
| nlinarith [hJ v u, hJ u v] |
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| lemma average_mul_part (J : V → V → ℝ) (h : V → ℝ) (f : (V → Bool) → ℝ) : |
| average J h f * part J h = ∑ σ, mass J h σ * f σ := by |
| exact div_mul_cancel₀ _ (part_pos J h).ne' |
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| def pinPart (J : V → V → ℝ) (h : V → ℝ) (v : V) (b : Bool) : ℝ := |
| Real.exp (J v v + h v * sgn b) * part (removedCoupling J v) (removedField J h v b) |
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|
| lemma pinPart_pos (J : V → V → ℝ) (h : V → ℝ) (v : V) (b : Bool) : |
| 0 < pinPart J h v b := mul_pos (Real.exp_pos _) (part_pos _ _) |
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|
| lemma part_split (J : V → V → ℝ) (h : V → ℝ) (v : V) : |
| part J h = pinPart J h v true + pinPart J h v false := by |
| rw [part, sum_insert v] |
| simp_rw [mass_insert, ← Finset.mul_sum] |
| simp only [pinPart, part] |
| congr! |
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|
| lemma mean_split (J : V → V → ℝ) (h : V → ℝ) (v o : V) : |
| mean J h o = (pinPart J h v true * pinMean J h v true o + |
| pinPart J h v false * pinMean J h v false o) / part J h := by |
| unfold mean average |
| rw [sum_insert v] |
| congr 1 |
| have H (b : Bool) : (∑ η, mass J h (insertSpin v b η) * sgn (insertSpin v b η o)) = |
| pinPart J h v b * pinMean J h v b o := by |
| simp_rw [mass_insert, mul_assoc, ← Finset.mul_sum] |
| rw [pinPart, pinMean, mul_assoc, mul_comm (part _ _), average_mul_part] |
| congr! |
| rw [H, H] |
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|
| lemma mean_self_split (J : V → V → ℝ) (h : V → ℝ) (v : V) : |
| mean J h v = (pinPart J h v true - pinPart J h v false) / |
| (pinPart J h v true + pinPart J h v false) := by |
| rw [mean_split J h v v, pinMean_self, pinMean_self, part_split J h v] |
| simp [sgn, sub_eq_add_neg] |
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|
| lemma mean_affine (J : V → V → ℝ) (h : V → ℝ) (v o : V) : |
| mean J h o = (pinMean J h v true o + pinMean J h v false o) / 2 + |
| mean J h v * (pinMean J h v true o - pinMean J h v false o) / 2 := by |
| rw [mean_split J h v o, mean_self_split J h v, part_split J h v] |
| have hz : pinPart J h v true + pinPart J h v false ≠ 0 := |
| (add_pos (pinPart_pos _ _ _ _) (pinPart_pos _ _ _ _)).ne' |
| field_simp |
| ring |
|
|
| / |
| def fieldAt (v : V) (t : ℝ) : V → ℝ := fun u => if u = v then t else 0 |
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|
| omit [Fintype V] in |
| @[simp] lemma fieldAt_self (v : V) (t : ℝ) : fieldAt v t v = t := by simp [fieldAt] |
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| omit [Fintype V] in |
| @[simp] lemma fieldAt_other (v : V) (t : ℝ) (u : V) (hu : u ≠ v) : |
| fieldAt v t u = 0 := by simp [fieldAt, hu] |
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|
| omit [Fintype V] in |
| lemma removedField_shift (J : V → V → ℝ) (h : V → ℝ) (v : V) (b : Bool) (t : ℝ) : |
| removedField J (h + fieldAt v t) v b = removedField J h v b := by |
| ext u |
| simp [removedField, u.property] |
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|
| lemma pinMean_shift (J : V → V → ℝ) (h : V → ℝ) (v : V) (b : Bool) (o : V) (t : ℝ) : |
| pinMean J (h + fieldAt v t) v b o = pinMean J h v b o := by |
| simp only [pinMean, removedField_shift] |
|
|
| lemma pinPart_shift (J : V → V → ℝ) (h : V → ℝ) (v : V) (b : Bool) (t : ℝ) : |
| pinPart J (h + fieldAt v t) v b = pinPart J h v b * Real.exp (t * sgn b) := by |
| simp only [pinPart, removedField_shift, Pi.add_apply, fieldAt_self, add_mul, |
| ← add_assoc, Real.exp_add] |
| ring |
|
|
| lemma mean_field_update (J : V → V → ℝ) (h : V → ℝ) (v : V) (t : ℝ) : |
| mean J (h + fieldAt v t) v = (mean J h v + Real.tanh t) / |
| (1 + mean J h v * Real.tanh t) := by |
| rw [mean_self_split J (h + fieldAt v t) v, mean_self_split J h v] |
| simp only [pinPart_shift, sgn, ↓reduceIte, Bool.false_eq_true, mul_one, mul_neg_one] |
| rw [Real.tanh_eq] |
| have hp := pinPart_pos J h v true |
| have hm := pinPart_pos J h v false |
| have he := Real.exp_pos t |
| have hf := Real.exp_pos (-t) |
| have hsum : pinPart J h v true + pinPart J h v false ≠ 0 := (add_pos hp hm).ne' |
| have hexp : Real.exp t + Real.exp (-t) ≠ 0 := (add_pos he hf).ne' |
| field_simp |
| ring_nf |
| field_simp |
| ring |
|
|
| lemma hamiltonian_flip (J : V → V → ℝ) (h : V → ℝ) (σ : V → Bool) : |
| hamiltonian J (-h) (fun u => !(σ u)) = hamiltonian J h σ := by |
| simp [hamiltonian, sgn_neg] |
|
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| lemma mass_flip (J : V → V → ℝ) (h : V → ℝ) (σ : V → Bool) : |
| mass J (-h) (fun u => !(σ u)) = mass J h σ := by |
| simp only [mass, hamiltonian_flip] |
|
|
| def flipEquiv : (V → Bool) ≃ (V → Bool) where |
| toFun σ u := !(σ u) |
| invFun σ u := !(σ u) |
| left_inv σ := by ext u; simp |
| right_inv σ := by ext u; simp |
|
|
| lemma part_neg (J : V → V → ℝ) (h : V → ℝ) : part J (-h) = part J h := by |
| rw [part, ← flipEquiv.sum_comp] |
| simp only [flipEquiv, Equiv.coe_fn_mk, mass_flip] |
| rfl |
|
|
| lemma mean_neg (J : V → V → ℝ) (h : V → ℝ) (v : V) : |
| mean J (-h) v = -mean J h v := by |
| rw [mean, average, part_neg, ← flipEquiv.sum_comp] |
| simp only [flipEquiv, Equiv.coe_fn_mk, mass_flip, sgn_neg, mul_neg, |
| Finset.sum_neg_distrib, neg_div] |
| rfl |
|
|
| lemma mean_zero (J : V → V → ℝ) (v : V) : mean J 0 v = 0 := by |
| have H := mean_neg J 0 v |
| simp only [neg_zero] at H |
| linarith |
|
|
| lemma mean_nonneg (J : V → V → ℝ) (hJ : ∀ u v, 0 ≤ J u v) |
| (h : V → ℝ) (hh : 0 ≤ h) (v : V) : 0 ≤ mean J h v := by |
| simpa only [mean_zero] using mean_mono J hJ 0 h hh v |
|
|
| omit [Fintype V] in |
| @[simp] lemma fieldAt_zero (v : V) : fieldAt v 0 = 0 := by |
| ext u |
| simp [fieldAt] |
|
|
| lemma mean_mixture_identity (J : V → V → ℝ) (h : V → ℝ) (v o : V) |
| (t s α : ℝ) (b : Bool) : |
| mean J (h + fieldAt v t) o - |
| (α * mean J (h + fieldAt v s) o + (1 - α) * pinMean J h v b o) = |
| (mean J (h + fieldAt v t) v - |
| (α * mean J (h + fieldAt v s) v + (1 - α) * sgn b)) * |
| (pinMean J h v true o - pinMean J h v false o) / 2 := by |
| rw [mean_affine J (h + fieldAt v t) v o, mean_affine J (h + fieldAt v s) v o] |
| simp only [pinMean_shift] |
| cases b <;> norm_num [sgn] <;> ring |
|
|
| lemma mean_mixture_le (J : V → V → ℝ) (hJ : ∀ u v, 0 ≤ J u v) |
| (h : V → ℝ) (v o : V) (t s α : ℝ) (b : Bool) |
| (H : mean J (h + fieldAt v t) v ≤ |
| α * mean J (h + fieldAt v s) v + (1 - α) * sgn b) : |
| mean J (h + fieldAt v t) o ≤ |
| α * mean J (h + fieldAt v s) o + (1 - α) * pinMean J h v b o := by |
| have hh := mean_mixture_identity J h v o t s α b |
| have hd := sub_nonneg.mpr (pinMean_mono J hJ h v o) |
| have hp := mul_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr H) hd |
| linarith |
|
|
| lemma mean_mixture_ge (J : V → V → ℝ) (hJ : ∀ u v, 0 ≤ J u v) |
| (h : V → ℝ) (v o : V) (t s α : ℝ) (b : Bool) |
| (H : α * mean J (h + fieldAt v s) v + (1 - α) * sgn b ≤ |
| mean J (h + fieldAt v t) v) : |
| α * mean J (h + fieldAt v s) o + (1 - α) * pinMean J h v b o ≤ |
| mean J (h + fieldAt v t) o := by |
| have hh := mean_mixture_identity J h v o t s α b |
| have hd := sub_nonneg.mpr (pinMean_mono J hJ h v o) |
| have hp := mul_nonneg (sub_nonneg.mpr H) hd |
| linarith |
|
|
| lemma mean_mixture_eq (J : V → V → ℝ) (h : V → ℝ) (v o : V) (t s α : ℝ) (b : Bool) |
| (H : mean J (h + fieldAt v t) v = |
| α * mean J (h + fieldAt v s) v + (1 - α) * sgn b) : |
| mean J (h + fieldAt v t) o = |
| α * mean J (h + fieldAt v s) o + (1 - α) * pinMean J h v b o := by |
| have hh := mean_mixture_identity J h v o t s α b |
| rw [H, sub_self, zero_mul, zero_div] at hh |
| exact sub_eq_zero.mp hh |
|
|
| end |
|
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| end BufferedIsing.DSS |
|
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| end OAI |
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|
