| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
| open scoped BigOperators InnerProductSpace |
|
|
| namespace ClassicalON |
|
|
| abbrev Site := ℤ × ℤ |
|
|
| def PositiveNeighbor (x y : Site) : Prop := |
| (y.1 = x.1 + 1 ∧ y.2 = x.2) ∨ (y.1 = x.1 ∧ y.2 = x.2 + 1) |
|
|
| structure LatticeGraph where |
| vertices : Finset Site |
| edges : Finset (vertices × vertices) |
| nearest : ∀ e ∈ edges, PositiveNeighbor e.1.val e.2.val |
|
|
| abbrev Spin (n : ℕ) := ↥(Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1) |
| abbrev Configuration (n : ℕ) (G : LatticeGraph) := G.vertices → Spin n |
|
|
| def sphereProbability (n : ℕ) : Measure (Spin n) := |
| let surface := (volume : Measure (EuclideanSpace ℝ (Fin n))).toSphere |
| (surface Set.univ)⁻¹ • surface |
|
|
| def referenceLaw (n : ℕ) (G : LatticeGraph) : Measure (Configuration n G) := |
| Measure.pi (fun _ => sphereProbability n) |
|
|
| def interaction (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ) |
| (σ : Configuration n G) : ℝ := |
| ∑ e : G.edges, b e * ⟪(σ e.val.1).val, (σ e.val.2).val⟫_ℝ |
|
|
| def partition (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ) : ℝ := |
| ∫ σ, Real.exp (interaction n G b σ) ∂referenceLaw n G |
|
|
| def correlation (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ) |
| (x y : G.vertices) : ℝ := |
| (∫ σ, ⟪(σ x).val, (σ y).val⟫_ℝ * Real.exp (interaction n G b σ) |
| ∂referenceLaw n G) / partition n G b |
|
|
| def siteDistance (x y : Site) : ℝ := |
| Real.sqrt (((x.1 : ℝ) - (y.1 : ℝ)) ^ 2 + ((x.2 : ℝ) - (y.2 : ℝ)) ^ 2) |
|
|
| def ExponentialDecay : Prop := |
| ∀ (n : ℕ), 3 ≤ n → ∀ (β : ℝ), 0 < β → |
| ∃ A m : ℝ, 0 < m ∧ ∀ (G : LatticeGraph) (b : G.edges → ℝ), |
| (∀ e, 0 ≤ b e ∧ b e ≤ β) → ∀ x y : G.vertices, |
| 0 ≤ correlation n G b x y ∧ |
| correlation n G b x y ≤ A * Real.exp (-m * siteDistance x.val y.val) |
|
|
| theorem main : ExponentialDecay := by |
| sorry |
|
|
| end ClassicalON |
|
|
| end |
|
|
| end OAI |
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|
