| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
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| namespace SKGap |
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| abbrev Spin (n : ℕ) := Fin n → Bool |
| abbrev Edge (n : ℕ) := {p : Fin n × Fin n // p.1 < p.2} |
| abbrev Disorder (n : ℕ) := Edge n → ℝ |
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| def spinValue (b : Bool) : ℝ := if b then 1 else -1 |
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| def coupling {n : ℕ} (g : Disorder n) (i k : Fin n) : ℝ := |
| if h : i < k then g ⟨(i, k), h⟩ else |
| if h : k < i then g ⟨(k, i), h⟩ else 0 |
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| def hamiltonian {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (x : Spin n) : ℝ := |
| (1 / 2 : ℝ) * ∑ i, ∑ k, spinValue (x i) * coupling g i k * spinValue (x k) + |
| ∑ i, h i * spinValue (x i) |
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| def weight {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (x : Spin n) : ℝ := |
| Real.exp (hamiltonian g h x) |
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| def partition {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) : ℝ := |
| ∑ x, weight g h x |
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| def mass {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (x : Spin n) : ℝ := |
| weight g h x / partition g h |
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| def expectation {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (f : Spin n → ℝ) : ℝ := |
| ∑ x, mass g h x * f x |
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| def variance {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (f : Spin n → ℝ) : ℝ := |
| expectation g h (fun x => (f x - expectation g h f) ^ 2) |
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| def flip {n : ℕ} (i : Fin n) (x : Spin n) : Spin n := |
| Function.update x i (!(x i)) |
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| / |
| def conditionalExpectation {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) |
| (i : Fin n) (f : Spin n → ℝ) (x : Spin n) : ℝ := |
| (weight g h x * f x + weight g h (flip i x) * f (flip i x)) / |
| (weight g h x + weight g h (flip i x)) |
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| / |
| def dirichlet {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (f : Spin n → ℝ) : ℝ := |
| ∑ i, expectation g h (fun x => (f x - conditionalExpectation g h i f x) ^ 2) |
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| def disorderLaw (β : ℝ) (n : ℕ) : Measure (Disorder n) := |
| Measure.pi (fun _ : Edge n => gaussianReal 0 (Real.toNNReal (β ^ 2 / n))) |
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| def poincareEvent (n : ℕ) (C : ℝ) : Set (Disorder n) := |
| {g | ∀ f : Spin n → ℝ, variance g 0 f ≤ C * dirichlet g 0 f} |
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| / |
| def discreteGap {n : ℕ} (g : Disorder n) : ℝ := |
| sInf {r : ℝ | ∃ f : Spin n → ℝ, 0 < variance g 0 f ∧ |
| r = dirichlet g 0 f / ((n : ℝ) * variance g 0 f)} |
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| def MainStatement (β : ℝ) : Prop := |
| ∃ C : ℝ, 0 < C ∧ |
| Tendsto (fun n : ℕ => disorderLaw β n (poincareEvent n C)) atTop (nhds 1) ∧ |
| Tendsto (fun n : ℕ => disorderLaw β n {g | 1 / (C * (n : ℝ)) ≤ discreteGap g}) |
| atTop (nhds 1) |
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| end SKGap |
| end |
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| end OAI |
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