| import OAI.Probability.Ising.RandomCluster |
| import OAI.Probability.Ising.Colorings |
|
|
| namespace OAI |
|
|
| open scoped BigOperators Classical |
|
|
| namespace BufferedIsing |
|
|
| / |
| is supplied by summing compatible fair cluster colors. -/ |
| noncomputable def fkEdgeWeight {n m : ℕ} (G : FiniteGraph n m) |
| (F : Finset (Fin n)) (ω : Finset (Fin m)) : ℝ := |
| ∏ e ∈ keptEdges G F, |
| if e ∈ ω then 1 - Real.exp (-2 * G.coupling e) else Real.exp (-2 * G.coupling e) |
|
|
| noncomputable def jointColorWeight {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) (ω : Finset (Fin m)) (σ : KeptVertex F → Bool) : ℝ := |
| if Respects (comparisonSetoid G I J F ω) σ then fkEdgeWeight G F ω else 0 |
|
|
| noncomputable def comparisonSpinWeight {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) (σ : KeptVertex F → Bool) : ℝ := |
| ∑ ω ∈ (keptEdges G F).powerset, jointColorWeight G I J F ω σ |
|
|
| lemma sum_jointColorWeight {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) (ω : Finset (Fin m)) : |
| (∑ σ, jointColorWeight G I J F ω σ) = fkWeight G I J F ω := by |
| calc |
| _ = fkEdgeWeight G F ω * ∑ σ : KeptVertex F → Bool, |
| if Respects (comparisonSetoid G I J F ω) σ then 1 else 0 := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro σ _ |
| simp only [jointColorWeight] |
| split_ifs <;> simp |
| _ = _ := by |
| rw [sum_respects_one] |
| exact mul_comm _ _ |
|
|
| lemma sum_jointColorWeight_spin_product {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) (ω : Finset (Fin m)) (i j : KeptVertex F) : |
| (∑ σ, jointColorWeight G I J F ω σ * spin (σ i) * spin (σ j)) = |
| if (comparisonSetoid G I J F ω).r i j then fkWeight G I J F ω else 0 := by |
| calc |
| _ = fkEdgeWeight G F ω * ∑ σ : KeptVertex F → Bool, |
| if Respects (comparisonSetoid G I J F ω) σ then spin (σ i) * spin (σ j) |
| else 0 := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro σ _ |
| simp only [jointColorWeight] |
| split_ifs <;> ring |
| _ = _ := by |
| rw [sum_respects_spin_product] |
| split_ifs |
| · exact mul_comm _ _ |
| · exact mul_zero _ |
|
|
| lemma sum_comparisonSpinWeight {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) : |
| (∑ σ, comparisonSpinWeight G I J F σ) = |
| ∑ ω ∈ (keptEdges G F).powerset, fkWeight G I J F ω := by |
| unfold comparisonSpinWeight |
| rw [Finset.sum_comm] |
| simp only [sum_jointColorWeight] |
|
|
| lemma sum_comparisonSpinWeight_spin_product {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) (i j : KeptVertex F) : |
| (∑ σ, comparisonSpinWeight G I J F σ * spin (σ i) * spin (σ j)) = |
| ∑ ω ∈ (keptEdges G F).powerset, |
| if (comparisonSetoid G I J F ω).r i j then fkWeight G I J F ω else 0 := by |
| unfold comparisonSpinWeight |
| simp only [Finset.sum_mul] |
| rw [Finset.sum_comm] |
| simp only [sum_jointColorWeight_spin_product] |
|
|
| / |
| lemma qZero_eq_color_correlation {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) (i j : KeptVertex F) : |
| qZero G I J F i j = |
| (∑ σ, comparisonSpinWeight G I J F σ * spin (σ i) * spin (σ j)) / |
| ∑ σ, comparisonSpinWeight G I J F σ := by |
| rw [sum_comparisonSpinWeight_spin_product, sum_comparisonSpinWeight] |
| rfl |
|
|
| lemma sum_subsets_product {E : Type*} [DecidableEq E] (s : Finset E) |
| (f g : E → ℝ) : |
| (∑ ω ∈ s.powerset, ∏ e ∈ s, if e ∈ ω then f e else g e) = |
| ∏ e ∈ s, (f e + g e) := by |
| rw [Finset.prod_add] |
| apply Finset.sum_congr rfl |
| intro ω hω |
| have hsub := Finset.mem_powerset.mp hω |
| rw [Finset.prod_ite] |
| have hf : s.filter (fun e => e ∈ ω) = ω := by |
| ext e |
| simp only [Finset.mem_filter] |
| exact ⟨And.right, fun he => ⟨hsub he, he⟩⟩ |
| have hg : s.filter (fun e => e ∉ ω) = s \ ω := by |
| ext e |
| simp |
| rw [hf, hg] |
|
|
| / |
| vertices are immaterial: every incident edge is absent from keptEdges. -/ |
| def extendKept {n : ℕ} (F : Finset (Fin n)) (σ : KeptVertex F → Bool) : Spins n := |
| fun v => if hv : v ∉ F then σ ⟨v, hv⟩ else false |
|
|
| @[simp] lemma extendKept_apply {n : ℕ} (F : Finset (Fin n)) |
| (σ : KeptVertex F → Bool) (v : KeptVertex F) : |
| extendKept F σ v.val = σ v := by |
| simp [extendKept, v.property] |
|
|
| def BondCompatible {n m : ℕ} (G : FiniteGraph n m) |
| (F : Finset (Fin n)) (σ : KeptVertex F → Bool) (ω : Finset (Fin m)) : Prop := |
| ∀ e ∈ ω, extendKept F σ (G.left e) = extendKept F σ (G.right e) |
|
|
| lemma respects_comparison_iff {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) (ω : Finset (Fin m)) (hω : ω ⊆ keptEdges G F) |
| (σ : KeptVertex F → Bool) : |
| Respects (comparisonSetoid G I J F ω) σ ↔ |
| Respects (comparisonSetoid G I J F ∅) σ ∧ BondCompatible G F σ ω := by |
| simp only [comparisonSetoid, respects_eqvGen] |
| constructor |
| · intro H |
| constructor |
| · intro u v huv |
| rcases huv with hi | hj | ⟨e, he, _⟩ |
| · exact H u v (Or.inl hi) |
| · exact H u v (Or.inr (Or.inl hj)) |
| · exact (Finset.notMem_empty e he).elim |
| · intro e he |
| have hk := (Finset.mem_filter.mp (hω he)).2 |
| let u : KeptVertex F := ⟨G.left e, hk.1⟩ |
| let v : KeptVertex F := ⟨G.right e, hk.2⟩ |
| have He := H u v (Or.inr (Or.inr ⟨e, he, Or.inl ⟨rfl, rfl⟩⟩)) |
| simpa only [← extendKept_apply F σ u, ← extendKept_apply F σ v] using He |
| · rintro ⟨H, Hω⟩ u v huv |
| rcases huv with hi | hj | ⟨e, he, huv | hvu⟩ |
| · exact H u v (Or.inl hi) |
| · exact H u v (Or.inr (Or.inl hj)) |
| · have He := Hω e he |
| rw [huv.1, huv.2] at He |
| simpa only [extendKept_apply] using He |
| · have He := Hω e he |
| rw [hvu.1, hvu.2] at He |
| simpa only [extendKept_apply] using He.symm |
|
|
| lemma prod_conditioned_subset {E : Type*} [DecidableEq E] (s ω : Finset E) |
| (hω : ω ⊆ s) (P : E → Prop) [DecidablePred P] (f g : E → ℝ) : |
| (if ∀ e ∈ ω, P e then ∏ e ∈ s, if e ∈ ω then f e else g e else 0) = |
| ∏ e ∈ s, if e ∈ ω then (if P e then f e else 0) else g e := by |
| by_cases H : ∀ e ∈ ω, P e |
| · rw [ite_eq_left H] |
| apply Finset.prod_congr rfl |
| intro e _ |
| by_cases he : e ∈ ω |
| · simp only [he, H e he, ite_true] |
| · simp only [he, ite_false] |
| · rw [ite_eq_right H] |
| push Not at H |
| obtain ⟨e, he, hP⟩ := H |
| symm |
| apply Finset.prod_eq_zero (hω he) |
| simp only [he, hP, ite_true, ite_false] |
|
|
| lemma jointColorWeight_expansion {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) (ω : Finset (Fin m)) (hω : ω ⊆ keptEdges G F) |
| (σ : KeptVertex F → Bool) : |
| jointColorWeight G I J F ω σ = |
| if Respects (comparisonSetoid G I J F ∅) σ then |
| ∏ e ∈ keptEdges G F, if e ∈ ω then |
| (if extendKept F σ (G.left e) = extendKept F σ (G.right e) |
| then 1 - Real.exp (-2 * G.coupling e) else 0) |
| else Real.exp (-2 * G.coupling e) |
| else 0 := by |
| unfold jointColorWeight |
| rw [respects_comparison_iff G I J F ω hω σ] |
| by_cases H : Respects (comparisonSetoid G I J F ∅) σ |
| · simp only [H, true_and, ite_true] |
| have hP := prod_conditioned_subset (keptEdges G F) ω hω |
| (fun e => extendKept F σ (G.left e) = extendKept F σ (G.right e)) |
| (fun e => 1 - Real.exp (-2 * G.coupling e)) |
| (fun e => Real.exp (-2 * G.coupling e)) |
| unfold BondCompatible fkEdgeWeight |
| by_cases Hω : ∀ e ∈ ω, extendKept F σ (G.left e) = extendKept F σ (G.right e) |
| · rw [ite_eq_left Hω] at hP ⊢ |
| exact hP |
| · rw [ite_eq_right Hω] at hP ⊢ |
| exact hP |
| · simp only [H, false_and, ite_false] |
|
|
| lemma comparisonSpinWeight_product {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) (σ : KeptVertex F → Bool) : |
| comparisonSpinWeight G I J F σ = |
| if Respects (comparisonSetoid G I J F ∅) σ then |
| ∏ e ∈ keptEdges G F, |
| ((if extendKept F σ (G.left e) = extendKept F σ (G.right e) |
| then 1 - Real.exp (-2 * G.coupling e) else 0) + |
| Real.exp (-2 * G.coupling e)) |
| else 0 := by |
| unfold comparisonSpinWeight |
| by_cases H : Respects (comparisonSetoid G I J F ∅) σ |
| · simp only [H, ite_true] |
| calc |
| _ = ∑ ω ∈ (keptEdges G F).powerset, ∏ e ∈ keptEdges G F, |
| if e ∈ ω then |
| (if extendKept F σ (G.left e) = extendKept F σ (G.right e) |
| then 1 - Real.exp (-2 * G.coupling e) else 0) |
| else Real.exp (-2 * G.coupling e) := by |
| apply Finset.sum_congr rfl |
| intro ω hω |
| rw [jointColorWeight_expansion G I J F ω (Finset.mem_powerset.mp hω) σ] |
| simp only [H, ite_true] |
| _ = _ := sum_subsets_product _ _ _ |
| · simp only [H, ite_false] |
| apply Finset.sum_eq_zero |
| intro ω hω |
| rw [jointColorWeight_expansion G I J F ω (Finset.mem_powerset.mp hω) σ] |
| simp only [H, ite_false] |
|
|
| lemma es_one_edge_identity (K : ℝ) (s t : Bool) : |
| ((if s = t then 1 - Real.exp (-2 * K) else 0) + Real.exp (-2 * K)) = |
| Real.exp (-K) * Real.exp (K * spin s * spin t) := by |
| rw [← Real.exp_add] |
| cases s <;> cases t <;> simp [spin] <;> ring |
|
|
| noncomputable def deletedEnergy {n m : ℕ} (G : FiniteGraph n m) |
| (F : Finset (Fin n)) (σ : KeptVertex F → Bool) : ℝ := |
| ∑ e ∈ keptEdges G F, |
| G.coupling e * spin (extendKept F σ (G.left e)) * spin (extendKept F σ (G.right e)) |
|
|
| / |
| contracted by restricting each to constant spin assignments. -/ |
| noncomputable def contractedZeroWeight {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) (σ : KeptVertex F → Bool) : ℝ := |
| if Respects (comparisonSetoid G I J F ∅) σ then Real.exp (deletedEnergy G F σ) else 0 |
|
|
| lemma comparisonSpinWeight_eq_Ising {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) (σ : KeptVertex F → Bool) : |
| comparisonSpinWeight G I J F σ = |
| Real.exp (-(∑ e ∈ keptEdges G F, G.coupling e)) * contractedZeroWeight G I J F σ := by |
| rw [comparisonSpinWeight_product] |
| unfold contractedZeroWeight |
| split_ifs |
| · simp only [es_one_edge_identity, Finset.prod_mul_distrib, ← Real.exp_sum, |
| Finset.sum_neg_distrib] |
| rfl |
| · exact (mul_zero _).symm |
|
|
| / |
| lemma qZero_eq_zero_field_correlation {n m : ℕ} (G : FiniteGraph n m) |
| (I J F : Finset (Fin n)) (i j : KeptVertex F) : |
| qZero G I J F i j = |
| (∑ σ, contractedZeroWeight G I J F σ * spin (σ i) * spin (σ j)) / |
| ∑ σ, contractedZeroWeight G I J F σ := by |
| rw [qZero_eq_color_correlation] |
| simp only [comparisonSpinWeight_eq_Ising, mul_assoc, ← Finset.mul_sum] |
| exact mul_div_mul_left _ _ (Real.exp_ne_zero _) |
|
|
| end BufferedIsing |
|
|
| end OAI |
|
|
