| import OAI.MathematicalPhysics.CriticalSK.Main | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators Topology NNReal ENNReal | |
| open MeasureTheory ProbabilityTheory Filter | |
| namespace CriticalSK | |
| def continuousMixingAt {n : ℕ} (W : Disorder n) (ε : ℝ) : ℝ := | |
| sInf {t : ℝ | 0 ≤ t ∧ ∀ x, continuousDistance W t x ≤ ε} | |
| def discreteMixingAt {n : ℕ} (W : Disorder n) (ε : ℝ) : ℕ := | |
| sInf {k : ℕ | ∀ x, discreteDistance W k x ≤ ε} | |
| def entropy {n : ℕ} (W : Disorder n) (g : Spin n → ℝ) : ℝ := | |
| mean W (fun x => g x * Real.log (g x)) - mean W g * Real.log (mean W g) | |
| def relaxationTime {n : ℕ} (W : Disorder n) : ℝ := | |
| sSup {r : ℝ | ∃ f : Spin n → ℝ, 0 < variance W f ∧ | |
| r = variance W f / dirichlet W f} | |
| def logSobolevConstant {n : ℕ} (W : Disorder n) : ℝ := | |
| sSup {r : ℝ | ∃ f : Spin n → ℝ, 0 < entropy W (fun x => f x ^ 2) ∧ | |
| r = entropy W (fun x => f x ^ 2) / dirichlet W f} | |
| def ExponentInProbability (X : (n : ℕ) → Disorder n → ℝ) (a : ℝ) : Prop := | |
| ∀ δ : ℝ, 0 < δ → | |
| Tendsto (fun n : ℕ => (disorderLaw n).real | |
| {W | (n : ℝ) ^ (a - δ) ≤ X n W ∧ X n W ≤ (n : ℝ) ^ (a + δ)}) | |
| atTop (𝓝 1) | |
| end CriticalSK | |
| end | |
| end OAI | |