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d5b562f | 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 | 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
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