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135052a | 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 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | 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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