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16a4018 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | 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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