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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 | import Mathlib
import OAI.Probability.SKBarriers.Coverage.CoverageTheorem
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
open scoped Topology
theorem stretchedLogScale_tendsto :
Tendsto (fun n : ℕ => (n:ℝ)^kappa) atTop atTop :=
(tendsto_rpow_atTop (by norm_num [kappa])).comp tendsto_natCast_atTop_atTop
theorem stretchedLogScale_sublinear :
Tendsto (fun n : ℕ => (n:ℝ)^kappa/(n:ℝ)) atTop (𝓝 0) := by
have H := (tendsto_rpow_neg_atTop (by norm_num [kappa] : 0 < 1-kappa)).comp
(tendsto_natCast_atTop_atTop (R := ℝ))
apply H.congr'
filter_upwards [eventually_gt_atTop (0:ℕ)] with n hn
have hn' : (0:ℝ) < n := by exact_mod_cast hn
change (n:ℝ)^(-(1-kappa)) = (n:ℝ)^kappa/(n:ℝ)
rw [show -(1-kappa) = kappa-1 by ring,Real.rpow_sub hn',Real.rpow_one]
theorem stretched_uniform_coverage {β : ℝ} (hβ : 1 < β) :
∃ q : ℝ, 0 < q ∧ q < 1 ∧ ∀ D : ℝ, 0 < D → ∃ D₃ : ℝ, 0 < D₃ ∧
∃ C : (n : ℕ) → Set (Disorder n), (∀ n, MeasurableSet (C n)) ∧
Tendsto (fun n => (disorderLaw n).real (C n)) atTop (𝓝 1) ∧
∀ n J, J ∈ C n → ∀ S : Finset (Config n),
finiteMass (gibbs β J) (S.filter (fun x =>
finiteMass (gibbs β J) (S.filter (fun y => q ≤ |overlap x y|)) <
Real.exp (-D₃*((n:ℝ)^kappa)))) < Real.exp (-D*((n:ℝ)^kappa)) :=
uniform_coverage hβ (fun n => (n:ℝ)^kappa) stretchedLogScale_tendsto stretchedLogScale_sublinear
end SK.Analytic
end
end
end
end OAI
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