| import Mathlib |
| import OAI.Probability.SKBarriers.Dynamics.HeatBathStationary |
| import OAI.Probability.SKBarriers.Dynamics.TVSignBridge |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def clockWeight (a : ℝ) (k : ℕ) : ℝ := Real.exp (-a)*a^k/(k.factorial:ℝ) |
|
|
| theorem clockWeight_nonneg {a : ℝ} (ha : 0 ≤ a) (k : ℕ) : 0 ≤ clockWeight a k := by |
| unfold clockWeight |
| positivity |
|
|
| theorem clockWeight_hasSum (a : ℝ) : HasSum (clockWeight a) 1 := by |
| have H := (NormedSpace.expSeries_div_hasSum_exp a).mul_left (Real.exp (-a)) |
| rw [← Real.exp_eq_exp_ℝ,← Real.exp_add,neg_add_cancel,Real.exp_zero] at H |
| unfold clockWeight |
| simpa only [mul_div_assoc] using H |
|
|
| theorem clockWeight_sum (a : ℝ) : ∑' k : ℕ, clockWeight a k = 1 := (clockWeight_hasSum a).tsum_eq |
|
|
| theorem uniformization_summable {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n) |
| {t : ℝ} (ht : 0 ≤ t) (x y : Config n) : |
| Summable (fun k : ℕ => clockWeight ((n:ℝ)*t) k*discreteKernel β J k x y) := by |
| refine Summable.of_nonneg_of_le (fun k => mul_nonneg (clockWeight_nonneg (by positivity) k) |
| (discreteKernel_nonneg β J k x y)) (fun k => ?_) (clockWeight_hasSum ((n:ℝ)*t)).summable |
| exact mul_le_of_le_one_right (clockWeight_nonneg (by positivity) k) (discreteKernel_le_one hn β J k x y) |
|
|
| theorem continuousKernel_eq_clock {n : ℕ} (β : ℝ) (J : Disorder n) (t : ℝ) |
| (x y : Config n) : continuousKernel β J t x y = |
| ∑' k : ℕ, clockWeight ((n:ℝ)*t) k*discreteKernel β J k x y := by |
| simp only [continuousKernel,clockWeight,neg_mul] |
|
|
| theorem continuousKernel_nonneg {n : ℕ} (β : ℝ) (J : Disorder n) {t : ℝ} (ht : 0 ≤ t) |
| (x y : Config n) : 0 ≤ continuousKernel β J t x y := by |
| rw [continuousKernel_eq_clock] |
| apply tsum_nonneg |
| intro k |
| apply mul_nonneg |
| · exact clockWeight_nonneg (by positivity) k |
| · exact discreteKernel_nonneg β J k x y |
|
|
| theorem continuousKernel_sum {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n) |
| {t : ℝ} (ht : 0 ≤ t) (x : Config n) : ∑ y : Config n, continuousKernel β J t x y = 1 := by |
| classical |
| simp_rw [continuousKernel_eq_clock] |
| rw [← Summable.tsum_finsetSum (fun y (_ : y ∈ (Finset.univ : Finset (Config n))) => |
| uniformization_summable hn β J ht x y)] |
| simp_rw [← Finset.mul_sum,discreteKernel_sum hn,mul_one] |
| exact clockWeight_sum _ |
|
|
| theorem continuousKernel_stationary {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n) |
| {t : ℝ} (ht : 0 ≤ t) (y : Config n) : |
| ∑ x : Config n, gibbs β J x*continuousKernel β J t x y = gibbs β J y := by |
| classical |
| simp_rw [continuousKernel_eq_clock] |
| simp_rw [← tsum_mul_left] |
| rw [← Summable.tsum_finsetSum (fun x (_ : x ∈ (Finset.univ : Finset (Config n))) => |
| (uniformization_summable hn β J ht x y).mul_left (gibbs β J x))] |
| have HH (k : ℕ) : (∑ x : Config n, gibbs β J x*(clockWeight ((n:ℝ)*t) k*discreteKernel β J k x y)) = |
| clockWeight ((n:ℝ)*t) k*gibbs β J y := by |
| simp_rw [mul_left_comm (gibbs β J _) (clockWeight ((n:ℝ)*t) k),← Finset.mul_sum, |
| discreteKernel_stationary hn] |
| simp_rw [HH] |
| rw [tsum_mul_right,clockWeight_sum,one_mul] |
|
|
| theorem continuous_sign_mass_of_tv {n : ℕ} (hn : 0 < n) (β : ℝ) (J : Disorder n) |
| (x v : Config n) {t : ℝ} (ht : 0 ≤ t) (hTV : continuousDistance β J x t ≤ (1/4:ℝ)) : |
| (1/4:ℝ) ≤ ∑ y : Config n, if overlap v y ≤ 0 then continuousKernel β J t x y else 0 := by |
| have H := finite_event_le_tv (continuousKernel β J t x) (gibbs β J) |
| (continuousKernel_sum hn β J ht x) (gibbs_sum β J) (fun y => overlap v y ≤ 0) |
| have HG := gibbs_nonpositive_overlap_mass β J v |
| change _ ≤ continuousDistance β J x t at H |
| linarith |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
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|
