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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 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 | import Mathlib
import OAI.Probability.SKBarriers.Dynamics.ContinuousCrossing
namespace OAI
section
noncomputable section
open scoped BigOperators Topology
open Classical MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
def discreteMixedMass {n : ℕ} (β : ℝ) (J : Disorder n) (k : ℕ) : ℝ :=
∑x : Config n,if discreteDistance β J x k≤(1/4:ℝ) then gibbs β J x else 0
def continuousMixedMass {n : ℕ} (β : ℝ) (J : Disorder n) (t : ℝ) : ℝ :=
∑x : Config n,if continuousDistance β J x t≤(1/4:ℝ) then gibbs β J x else 0
theorem discreteMixedMass_le_sign {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) (k : ℕ)
(v : Config n → Config n) : discreteMixedMass β J k ≤
4*∑x,gibbs β J x*kernelEvent β J (fun y => overlap (v x) y≤0) k x := by
unfold discreteMixedMass
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro x _
by_cases h : discreteDistance β J x k≤(1/4:ℝ)
· simp only [ite_eq_left h]
have H := discrete_sign_mass_of_tv hn β J x (v x) k h
change (1/4:ℝ)≤kernelEvent β J (fun y => overlap (v x) y≤0) k x at H
have := mul_le_mul_of_nonneg_left H (gibbs_pos β J x).le
linarith only [this]
· simp only [ite_eq_right h]
exact mul_nonneg (by norm_num) (mul_nonneg (gibbs_pos β J x).le (kernelEvent_nonneg _ _ _ _ _))
theorem continuousMixedMass_le_sign {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) {t : ℝ} (ht : 0≤t)
(v : Config n → Config n) : continuousMixedMass β J t ≤
4*∑x,gibbs β J x*∑y,if overlap (v x) y≤0 then continuousKernel β J t x y else 0 := by
unfold continuousMixedMass
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro x _
by_cases h : continuousDistance β J x t≤(1/4:ℝ)
· simp only [ite_eq_left h]
have H := continuous_sign_mass_of_tv hn β J x (v x) ht h
have := mul_le_mul_of_nonneg_left H (gibbs_pos β J x).le
linarith only [this]
· simp only [ite_eq_right h]
apply mul_nonneg (by norm_num)
apply mul_nonneg (gibbs_pos β J x).le
apply Finset.sum_nonneg
intro y _
split_ifs
· exact continuousKernel_nonneg β J ht x y
· exact le_rfl
theorem averaged_sign_clock {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) {t : ℝ} (ht : 0≤t)
(v : Config n → Config n) :
(∑x,gibbs β J x*∑y,if overlap (v x) y≤0 then continuousKernel β J t x y else 0)=
∑' k : ℕ,clockWeight ((n:ℝ)*t) k*(∑x,gibbs β J x*kernelEvent β J (fun y => overlap (v x) y≤0) k x) := by
simp_rw [continuous_event_eq_clock hn β J _ ht]
simp_rw [← tsum_mul_left]
rw [← Summable.tsum_finsetSum (fun x (_ : x∈(Finset.univ : Finset (Config n))) =>
(clock_kernelEvent_summable hn β J (fun y => overlap (v x) y≤0) (by positivity) x).mul_left (gibbs β J x))]
congr 1
funext k
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro x _
ring
theorem averaged_kernelEvent_nonneg {n : ℕ} (β : ℝ) (J : Disorder n) (A : Config n → Config n → Prop) (k : ℕ) :
0≤∑x,gibbs β J x*kernelEvent β J (A x) k x := by
apply Finset.sum_nonneg
intro x _
exact mul_nonneg (gibbs_pos β J x).le (kernelEvent_nonneg _ _ _ _ _)
theorem averaged_kernelEvent_le_one {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) (A : Config n → Config n → Prop) (k : ℕ) :
(∑x,gibbs β J x*kernelEvent β J (A x) k x)≤1 := by
rw [← gibbs_sum β J]
apply Finset.sum_le_sum
intro x _
exact mul_le_of_le_one_right (gibbs_pos β J x).le (kernelEvent_le_one hn β J _ _ _)
theorem averaged_sign_clock_bound {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) {t : ℝ} (ht : 0≤t)
(v : Config n → Config n) (m : ℕ) {E : ℝ} (hE : 0≤E)
(h : ∀k ≤ m,(∑x,gibbs β J x*kernelEvent β J (fun y => overlap (v x) y≤0) k x)≤E) :
(∑x,gibbs β J x*∑y,if overlap (v x) y≤0 then continuousKernel β J t x y else 0)≤
E+∑' k : ℕ,if m<k then clockWeight ((n:ℝ)*t) k else 0 := by
rw [averaged_sign_clock hn β J ht v]
have ha : 0≤(n:ℝ)*t := by positivity
have hs : Summable (fun k : ℕ => clockWeight ((n:ℝ)*t) k*
(∑x,gibbs β J x*kernelEvent β J (fun y => overlap (v x) y≤0) k x)) := by
apply Summable.of_nonneg_of_le _ _ (clockWeight_hasSum ((n:ℝ)*t)).summable
· intro k; exact mul_nonneg (clockWeight_nonneg ha k) (averaged_kernelEvent_nonneg _ _ _ _)
· intro k; exact mul_le_of_le_one_right (clockWeight_nonneg ha k) (averaged_kernelEvent_le_one hn _ _ _ _)
have HB := hs.tsum_le_tsum (g := fun k => clockWeight ((n:ℝ)*t) k*E+(if m<k then clockWeight ((n:ℝ)*t) k else 0))
(fun k => by
by_cases hk : m<k
· simp only [ite_eq_left hk]
have H := mul_le_of_le_one_right (clockWeight_nonneg ha k) (averaged_kernelEvent_le_one hn β J (fun x y => overlap (v x) y≤0) k)
linarith only [H,mul_nonneg (clockWeight_nonneg ha k) hE]
· simp only [ite_eq_right hk,add_zero]
exact mul_le_mul_of_nonneg_left (h k (le_of_not_gt hk)) (clockWeight_nonneg ha k))
(((clockWeight_hasSum ((n:ℝ)*t)).summable.mul_right E).add (clockWeight_tail_summable ha m))
simpa only [Summable.tsum_add ((clockWeight_hasSum ((n:ℝ)*t)).summable.mul_right E) (clockWeight_tail_summable ha m),tsum_mul_right,clockWeight_sum,one_mul] using HB
theorem mixedMass_bounds_of_crossing {n : ℕ} (hn : 0<n) (β : ℝ) (J : Disorder n) (L E : ℝ)
(hL : 0≤L) (hE : 0≤E) (v : Config n → Config n)
(h : ∀k≤⌈Real.exp (2*L)⌉₊,(∑x,gibbs β J x*kernelEvent β J (fun y => overlap (v x) y≤0) k x)≤E) :
continuousMixedMass β J (Real.exp L)≤4*(E+(n:ℝ)*Real.exp (-L)) ∧
discreteMixedMass β J ⌊Real.exp L⌋₊≤4*E := by
constructor
· refine (continuousMixedMass_le_sign hn β J (Real.exp_pos L).le v).trans ?_
apply mul_le_mul_of_nonneg_left _ (by norm_num)
exact (averaged_sign_clock_bound hn β J (Real.exp_pos L).le v _ hE h).trans
(add_le_add le_rfl (attempt_horizon_tail n L))
· refine (discreteMixedMass_le_sign hn β J ⌊Real.exp L⌋₊ v).trans ?_
apply mul_le_mul_of_nonneg_left _ (by norm_num)
apply h
exact (Nat.floor_mono (Real.exp_le_exp.mpr (by linarith only [hL]))).trans (Nat.floor_le_ceil _)
end SK.Analytic
end
end
end OAI
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