File size: 2,730 Bytes
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namespace OAI
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SKGap
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
def spinValue (b : Bool) : ℝ := if b then 1 else -1
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
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)
def weight {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (x : Spin n) : ℝ :=
Real.exp (hamiltonian g h x)
def partition {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) : ℝ :=
∑ x, weight g h x
def mass {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (x : Spin n) : ℝ :=
weight g h x / partition g h
def expectation {n : ℕ} (g : Disorder n) (h : Fin n → ℝ) (f : Spin n → ℝ) : ℝ :=
∑ x, mass g h x * f x
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)
def flip {n : ℕ} (i : Fin n) (x : Spin n) : Spin n :=
Function.update x i (!(x i))
/-- Conditional expectation under the actual Gibbs law given all other spins. -/
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))
/-- Unscaled form: each site has a rate-one clock. -/
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)
def disorderLaw (β : ℝ) (n : ℕ) : Measure (Disorder n) :=
Measure.pi (fun _ : Edge n => gaussianReal 0 (Real.toNNReal (β ^ 2 / n)))
def poincareEvent (n : ℕ) (C : ℝ) : Set (Disorder n) :=
{g | ∀ f : Spin n → ℝ, variance g 0 f ≤ C * dirichlet g 0 f}
/-- Rayleigh quotient definition of the uniform-site discrete heat-bath gap. -/
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)}
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)
end SKGap
end
end OAI
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