| 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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|
