| import OAI.Probability.FreeIsing.Estimates.ComponentPairCount |
| import OAI.Probability.FreeIsing.Estimates.PairCorrelation |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory |
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| namespace Problem367 |
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| theorem componentWordDistance_eq_reducedWordDistance {α : Type*} [DecidableEq α] |
| (u v : List α) : componentWordDistance u v = reducedWordDistance u v := by |
| induction u generalizing v with |
| | nil => rfl |
| | cons a u ih => |
| cases v with |
| | nil => rfl |
| | cons b v => |
| by_cases hab : a = b <;> simp [componentWordDistance, reducedWordDistance, hab, ih] |
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| def componentVertex {d n : ℕ} {c : Fin d} (u : ReducedTail c n) : TreeVertex d := |
| ⟨c :: u.1, u.2.2⟩ |
|
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| theorem componentVertex_injective {d n : ℕ} {c : Fin d} : |
| Function.Injective (componentVertex : ReducedTail c n → TreeVertex d) := by |
| intro u v h |
| apply Subtype.ext |
| exact (List.cons.inj (congrArg Subtype.val h)).2 |
|
|
| theorem HasFreeIsingLaw.component_pair_correlation {d : ℕ} {theta : ℝ} |
| {mu : Measure (TreeVertex d → IsingSpin)} |
| (hmu : HasFreeIsingLaw theta mu) (hlo : -1 ≤ theta) (hhi : theta ≤ 1) |
| {c : Fin d} {k l : ℕ} (u : ReducedTail c k) (v : ReducedTail c l) : |
| (∫ x, spinSign (x (componentVertex u)) * spinSign (x (componentVertex v)) ∂mu) = |
| theta ^ componentWordDistance u.1 v.1 := by |
| rw [hmu.pair_correlation hlo hhi] |
| simp [componentVertex, componentWordDistance_eq_reducedWordDistance] |
|
|
| theorem HasFreeIsingLaw.component_level_covariance_critical {d : ℕ} {theta : ℝ} |
| {mu : Measure (TreeVertex d → IsingSpin)} |
| (hmu : HasFreeIsingLaw theta mu) (hlo : -1 ≤ theta) (hhi : theta ≤ 1) |
| (c : Fin d) (hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1) (k l : ℕ) : |
| (∑ u : ReducedTail c k, ∑ v : ReducedTail c l, theta ^ k * theta ^ l * |
| (∫ x, spinSign (x (componentVertex u)) * spinSign (x (componentVertex v)) ∂mu)) = |
| 1 + (min k l : ℕ) * (1 - theta ^ 2) := by |
| simp_rw [hmu.component_pair_correlation hlo hhi] |
| exact sum_component_weighted_correlation_critical theta c hcritical k l |
|
|
| theorem HasFreeIsingLaw.component_covariance_sum_critical_le {d : ℕ} {theta : ℝ} |
| {mu : Measure (TreeVertex d → IsingSpin)} |
| (hmu : HasFreeIsingLaw theta mu) (hlo : -1 ≤ theta) (hhi : theta ≤ 1) |
| (c : Fin d) (hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1) (n : ℕ) : |
| (∑ k ∈ Finset.range (n + 1), ∑ l ∈ Finset.range (n + 1), |
| ∑ u : ReducedTail c k, ∑ v : ReducedTail c l, theta ^ k * theta ^ l * |
| (∫ x, spinSign (x (componentVertex u)) * spinSign (x (componentVertex v)) ∂mu)) ≤ |
| (n + 1 : ℝ) ^ 3 := by |
| simp_rw [hmu.component_pair_correlation hlo hhi] |
| exact sum_component_weighted_correlation_critical_le theta c hcritical n |
|
|
| theorem HasFreeIsingLaw.component_root_correlation {d : ℕ} {theta : ℝ} |
| {mu : Measure (TreeVertex d → IsingSpin)} |
| (hmu : HasFreeIsingLaw theta mu) (hlo : -1 ≤ theta) (hhi : theta ≤ 1) |
| {c : Fin d} {k : ℕ} (u : ReducedTail c k) : |
| (∫ x, spinSign (x ⟨[c], List.IsChain.singleton c⟩) * |
| spinSign (x (componentVertex u)) ∂mu) = theta ^ k := by |
| rw [hmu.pair_correlation hlo hhi] |
| simp only [componentVertex, reducedWordDistance_cons_same, reducedWordDistance_nil_left, u.2.1] |
|
|
| theorem HasFreeIsingLaw.component_level_signal_critical {d : ℕ} {theta : ℝ} |
| {mu : Measure (TreeVertex d → IsingSpin)} |
| (hmu : HasFreeIsingLaw theta mu) (hlo : -1 ≤ theta) (hhi : theta ≤ 1) |
| (c : Fin d) (hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1) (k : ℕ) : |
| (∑ u : ReducedTail c k, theta ^ k * |
| (∫ x, spinSign (x ⟨[c], List.IsChain.singleton c⟩) * |
| spinSign (x (componentVertex u)) ∂mu)) = 1 := by |
| simp_rw [hmu.component_root_correlation hlo hhi] |
| have hlen (u : ReducedTail c k) : u.1.length = k := u.2.1 |
| simpa only [hlen, pow_two] using sum_component_radial_sq_critical theta c hcritical k |
|
|
| theorem HasFreeIsingLaw.component_signal_critical {d : ℕ} {theta : ℝ} |
| {mu : Measure (TreeVertex d → IsingSpin)} |
| (hmu : HasFreeIsingLaw theta mu) (hlo : -1 ≤ theta) (hhi : theta ≤ 1) |
| (c : Fin d) (hcritical : ((d - 1 : ℕ) : ℝ) * theta ^ 2 = 1) (n : ℕ) : |
| (∑ k ∈ Finset.range (n + 1), ∑ u : ReducedTail c k, theta ^ k * |
| (∫ x, spinSign (x ⟨[c], List.IsChain.singleton c⟩) * |
| spinSign (x (componentVertex u)) ∂mu)) = (n + 1 : ℝ) := by |
| simp [hmu.component_level_signal_critical hlo hhi c hcritical] |
|
|
| def componentIndexedVertex {d : ℕ} {c : Fin d} |
| (u : Σ k : ℕ, ReducedTail c k) : TreeVertex d := componentVertex u.2 |
|
|
| theorem componentIndexedVertex_injective {d : ℕ} {c : Fin d} : |
| Function.Injective (componentIndexedVertex (c := c)) := by |
| rintro ⟨k, u⟩ ⟨l, v⟩ h |
| have hw : u.1 = v.1 := (List.cons.inj (congrArg Subtype.val h)).2 |
| have hkl : k = l := by rw [← u.2.1, ← v.2.1, hw] |
| subst l |
| exact Sigma.ext rfl (heq_of_eq (Subtype.ext hw)) |
|
|
| def componentIndexFinset {d : ℕ} (c : Fin d) (n : ℕ) : |
| Finset (Σ k : ℕ, ReducedTail c k) := |
| (Finset.range (n + 1)).sigma (fun _ => Finset.univ) |
|
|
| theorem sum_componentIndexFinset {M : Type*} [AddCommMonoid M] |
| {d : ℕ} (c : Fin d) (n : ℕ) (f : (Σ k : ℕ, ReducedTail c k) → M) : |
| (∑ s ∈ componentIndexFinset c n, f s) = |
| ∑ k ∈ Finset.range (n + 1), ∑ u : ReducedTail c k, f ⟨k, u⟩ := by |
| rw [componentIndexFinset, Finset.sum_sigma] |
|
|
| theorem sum_componentIndexFinset_pair {M : Type*} [AddCommMonoid M] |
| {d : ℕ} (c : Fin d) (n : ℕ) |
| (f : (Σ k : ℕ, ReducedTail c k) → (Σ k : ℕ, ReducedTail c k) → M) : |
| (∑ u ∈ componentIndexFinset c n, ∑ v ∈ componentIndexFinset c n, f u v) = |
| ∑ k ∈ Finset.range (n + 1), ∑ l ∈ Finset.range (n + 1), |
| ∑ u : ReducedTail c k, ∑ v : ReducedTail c l, f ⟨k, u⟩ ⟨l, v⟩ := by |
| simp_rw [sum_componentIndexFinset] |
| apply Finset.sum_congr rfl |
| intro k _ |
| rw [Finset.sum_comm] |
|
|
| end Problem367 |
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| end |
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| end OAI |
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