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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 | import Mathlib
import OAI.Probability.SKBarriers.Replicas.PairGibbsGap
import OAI.Probability.SKBarriers.Parisi.CDFMinimizer
import OAI.Probability.SKBarriers.Coverage.FiniteCoverageMixing
namespace OAI
section
noncomputable section
open scoped BigOperators Topology
open Classical MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem exists_forbidden_interval {f : ℝ → ℝ} (hf : ContinuousOn f (Icc (0:ℝ) 1))
{l u : ℝ} (hl : 0≤l) (hu : u≤1) (h : ¬∀x∈Ioo l u,f x=x) :
∃a₀ a₁ d : ℝ,l<a₀ ∧ a₀<a₁ ∧ a₁<u ∧ 0<d ∧ ∀x∈Icc a₀ a₁,d≤|x-f x| := by
push Not at h
obtain ⟨z,hz,hfZ⟩ := h
have hz0 : 0<z := hl.trans_lt hz.1
have hz1 : z<1 := hz.2.trans_le hu
have hc : ContinuousAt (fun x => |x-f x|) z :=
(continuousAt_id.sub ((hf z ⟨hz0.le,hz1.le⟩).continuousAt (Icc_mem_nhds hz0 hz1))).abs
have hd : 0 < |z-f z| := abs_pos.mpr (sub_ne_zero.mpr (Ne.symm hfZ))
have hznhds : ∀ᶠ x in 𝓝 z,x∈Ioo l u := Ioo_mem_nhds hz.1 hz.2
have hh : ∀ᶠ x in 𝓝 z,x∈Ioo l u ∧ |z-f z|/2≤|x-f x| :=
hznhds.and (Filter.Tendsto.eventually_const_le (half_lt_self hd) hc)
obtain ⟨r,hr,hrball⟩ := Metric.mem_nhds_iff.mp hh
let a₀ := z-r/2
let a₁ := z+r/2
have hleft : a₀∈Metric.ball z r := by
rw [Metric.mem_ball,Real.dist_eq]
dsimp only [a₀]
rw [sub_sub_cancel_left,abs_neg,abs_of_pos (half_pos hr)]
exact half_lt_self hr
have hright : a₁∈Metric.ball z r := by
rw [Metric.mem_ball,Real.dist_eq]
dsimp only [a₁]
rw [add_sub_cancel_left,abs_of_pos (half_pos hr)]
exact half_lt_self hr
refine ⟨a₀,a₁,|z-f z|/2,(hrball hleft).1.1,?_,(hrball hright).1.2,half_pos hd,?_⟩
· dsimp only [a₀,a₁]; linarith only [hr]
· intro x hx
apply (hrball ?_).2
rw [Metric.mem_ball,Real.dist_eq]
have H : |x-z|≤r/2 := abs_le.mpr (by dsimp only [a₀,a₁] at hx; constructor <;> linarith only [hx.1,hx.2])
exact H.trans_lt (half_lt_self hr)
theorem pairStrip_annealed_bound {β : ℝ} (hβ : 0<β)
(μ : ProbabilityMeasure ℝ) (hμ : (μ : Measure ℝ) (Icc (0:ℝ) 1)=1)
(hmin : scalarCDFParisi β (cdf (μ : Measure ℝ))=finiteParisiInf β)
{a₀ a₁ d : ℝ} (ha₀ : 0≤a₀) (hd : 0<d)
(hgap : ∀x∈Icc a₀ a₁,d≤|x-scalarCDFOverlap β (cdf (μ : Measure ℝ)) x|) :
∀ᶠ n : ℕ in atTop,(∫J,replicaGibbsMass β J (pairStripSet n a₀ a₁) ∂disorderLaw n)≤Real.exp (-(n:ℝ)^((4:ℝ)/5)) := by
have H := eventual_pairMismatch_polynomial_bound (A:=d) (a:=0) (t:=(4:ℝ)/5) (ε:=1)
hβ hd le_rfl (by norm_num) (by norm_num) zero_lt_one
(cdf (μ : Measure ℝ)) (fun x => ⟨cdf_nonneg _ _,cdf_le_one _ _⟩) (supported_probability_cdf_one μ hμ) hmin
simp only [neg_zero,Real.rpow_zero,mul_one,one_mul] at H
filter_upwards [H] with n hn
refine (integral_mono (replicaGibbsMass_integrable β _) (replicaGibbsMass_integrable β _)
(fun J => replicaGibbsMass_mono β J ?_)).trans hn
intro s hs
have hh := (Finset.mem_filter.mp hs).2
exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,ha₀.trans hh.1,hgap _ hh⟩
end SK.Analytic
end
end
end OAI
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