| import Mathlib |
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| namespace OAI |
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| section |
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| section |
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| noncomputable section |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
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| namespace SK |
| abbrev Config (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 spin (b : Bool) : ℝ := if b then 1 else -1 |
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| def hamiltonian {n : ℕ} (J : Disorder n) (x : Config n) : ℝ := |
| (∑ e : Edge n, J e * spin (x e.val.1) * spin (x e.val.2)) / Real.sqrt n |
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| def partition {n : ℕ} (β : ℝ) (J : Disorder n) : ℝ := |
| ∑ x : Config n, Real.exp (β * hamiltonian J x) |
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| def gibbs {n : ℕ} (β : ℝ) (J : Disorder n) (x : Config n) : ℝ := |
| Real.exp (β * hamiltonian J x) / partition β J |
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| def coupling {n : ℕ} (J : Disorder n) (i j : Fin n) : ℝ := |
| if h : i < j then J ⟨(i, j), h⟩ else |
| if h : j < i then J ⟨(j, i), h⟩ else 0 |
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| def localField {n : ℕ} (J : Disorder n) (x : Config n) (i : Fin n) : ℝ := |
| (∑ j : Fin n, coupling J i j * spin (x j)) / Real.sqrt n |
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| def heatBath {n : ℕ} (β : ℝ) (J : Disorder n) (x y : Config n) : ℝ := |
| (∑ i : Fin n, if Function.update x i (y i) = y then |
| Real.exp (β * spin (y i) * localField J x i) / |
| (2 * Real.cosh (β * localField J x i)) else 0) / n |
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| def discreteKernel {n : ℕ} (β : ℝ) (J : Disorder n) : ℕ → Config n → Config n → ℝ |
| | 0, x, y => if x = y then 1 else 0 |
| | k + 1, x, y => ∑ z : Config n, heatBath β J x z * discreteKernel β J k z y |
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| def continuousKernel {n : ℕ} (β : ℝ) (J : Disorder n) (t : ℝ) |
| (x y : Config n) : ℝ := |
| ∑' k : ℕ, (Real.exp (-(n : ℝ) * t) * ((n : ℝ) * t) ^ k / |
| (Nat.factorial k : ℝ)) * discreteKernel β J k x y |
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| def discreteDistance {n : ℕ} (β : ℝ) (J : Disorder n) |
| (x : Config n) (k : ℕ) : ℝ := |
| (1 / 2 : ℝ) * ∑ y : Config n, |discreteKernel β J k x y - gibbs β J y| |
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| def continuousDistance {n : ℕ} (β : ℝ) (J : Disorder n) |
| (x : Config n) (t : ℝ) : ℝ := |
| (1 / 2 : ℝ) * ∑ y : Config n, |continuousKernel β J t x y - gibbs β J y| |
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| def disorderLaw (n : ℕ) : Measure (Disorder n) := |
| Measure.pi (fun _ : Edge n => gaussianReal 0 1) |
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| def kappa : ℝ := 1 / 10000 |
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| def timeScale (n : ℕ) : ℝ := Real.exp ((n : ℝ) ^ kappa) |
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| def continuousBadMass (β : ℝ) (n : ℕ) : ℝ := |
| ∫ J : Disorder n, (∑ x : Config n, if 1 / 4 < continuousDistance β J x (timeScale n) |
| then gibbs β J x else 0) ∂disorderLaw n |
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| def discreteBadMass (β : ℝ) (n : ℕ) : ℝ := |
| ∫ J : Disorder n, (∑ x : Config n, if 1 / 4 < discreteDistance β J x ⌊timeScale n⌋₊ |
| then gibbs β J x else 0) ∂disorderLaw n |
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| end SK |
| end |
| end |
| end |
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| end OAI |
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