| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.ReplicaMarginal |
| import OAI.Probability.SKBarriers.Replicas.PairGibbsGap |
| import OAI.Probability.SKBarriers.Replicas.ReplicaEventSplit |
| import OAI.Probability.SKBarriers.Replicas.TripleContinuumGap |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def signedLockingSet (n : ℕ) (h b : ℝ) : Finset (ReplicaConfig n 3) := by |
| classical |
| exact Finset.univ.filter (fun s => h≤tripleQ s ∧ |
| |overlap (s 0) (s 1)|≤b ∧ |overlap (s 0) (s 2)|≤b ∧ |
| 2*(n:ℝ)^(-(1:ℝ)/100)< |overlap (s 0) (s 1)-overlap (s 0) (s 2)|) |
| |
| def orientedLockingSet (n : ℕ) (β : ℝ) (α : ℝ → ℝ) (h b τ : ℝ) : Finset (ReplicaConfig n 3) := by |
| classical |
| exact Finset.univ.filter (fun s => tripleQ s∈Icc h 1 ∧ tripleR s∈Icc (0:ℝ) b ∧ |
| |tripleDelta s|≤b ∧ (n:ℝ)^(-(1:ℝ)/100)≤|tripleDelta s| ∧ |
| |tripleQ s-scalarCDFOverlap β α (tripleQ s)|≤τ) |
| |
| theorem signedLocking_split {n : ℕ} (β h b τ : ℝ) (hh : 0≤h) (α : ℝ → ℝ) |
| {s : ReplicaConfig n 3} (hs : s∈signedLockingSet n h b) : |
| s∈orientedLockingSet n β α h b τ ∨ replicaFlip 0 s∈orientedLockingSet n β α h b τ ∨ |
| Fin.tail s∈pairMismatchSet n β α τ := by |
| classical |
| obtain ⟨hq,hx,hy,hd⟩ := (Finset.mem_filter.mp hs).2 |
| have hb := tripleRDelta_abs_le s hx hy |
| have hq1 := (abs_le.mp (abs_overlap_le_one (s 1) (s 2))).2 |
| have hδ : (n:ℝ)^(-(1:ℝ)/100)≤|tripleDelta s| := by |
| dsimp only [tripleDelta] |
| rw [abs_div,abs_of_pos (by norm_num : (0:ℝ)<2)] |
| linarith only [hd] |
| by_cases hg : |tripleQ s-scalarCDFOverlap β α (tripleQ s)|≤τ |
| · by_cases hr : 0≤tripleR s |
| · left |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,⟨⟨hq,hq1⟩,⟨hr,(abs_le.mp hb.1).2⟩,hb.2,hδ,hg⟩⟩ |
| · right; left |
| apply Finset.mem_filter.mpr |
| refine ⟨Finset.mem_univ _,?_⟩ |
| simp only [tripleQ_flip,tripleR_flip,tripleDelta_flip,abs_neg] |
| exact ⟨⟨hq,hq1⟩,⟨by linarith only [hr],by linarith only [(abs_le.mp hb.1).1]⟩,hb.2,hδ,hg⟩ |
| · right; right |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,⟨hh.trans hq,(not_le.mp hg).le⟩⟩ |
|
|
| theorem signed_fixed_scale_locking {β h : ℝ} (hβ : 0<β) (hh : 0<h) (hh1 : h≤1) : |
| ∃ b : ℝ,0<b ∧ ∀ᶠ n : ℕ in atTop, |
| (∫ J,replicaGibbsMass β J (signedLockingSet n h b) ∂disorderLaw n)≤ |
| Real.exp (-(n:ℝ)^((9:ℝ)/10)) := by |
| classical |
| obtain ⟨μ,hμ,hmin,τ,r₀,δ₀,c,hτ,hr₀,hδ₀,hc,hgap⟩ := exists_tripleConstrainedPressure_gap hβ hh hh1 |
| let α := cdf (μ : Measure ℝ) |
| have ha (z : ℝ) : α z∈Icc (0:ℝ) 1 := ⟨cdf_nonneg _ _,cdf_le_one _ _⟩ |
| have ha1 : α 1=1 := supported_probability_cdf_one μ hμ |
| let b := min r₀ δ₀ |
| have hb : 0<b := lt_min hr₀ hδ₀ |
| have HP := eventual_pairMismatch_polynomial_bound (A:=τ) (a:=0) (t:=(9:ℝ)/10) (ε:=1/2) |
| hβ hτ le_rfl (by norm_num) (by norm_num) (by norm_num) α ha ha1 hmin |
| have HT := eventual_replicaGibbsMass_polynomial_gap (c:=c) (a:=(1:ℝ)/50) (t:=(9:ℝ)/10) (ε:=1/4) |
| hβ 3 3 (by decide) hc (by norm_num) (by norm_num) (by norm_num) (by norm_num) |
| refine ⟨b,hb,?_⟩ |
| filter_upwards [HP,HT,eventually_gt_atTop (0:ℕ)] with n hnP hnT hn0 |
| have hp : (0:ℝ)<n := by exact_mod_cast hn0 |
| let T := orientedLockingSet n β α h b τ |
| have HT' : (∫ J,replicaGibbsMass β J T ∂disorderLaw n)≤(1/4)*Real.exp (-(n:ℝ)^((9:ℝ)/10)) := by |
| apply hnT (Fin 3 → Fin (n+1)) (by simp) tripleReplicaCode (fun s t H => tripleReplicaCode_gram hn0 H) |
| intro s hs |
| obtain ⟨hq,hr,hδ,hδlower,hdev⟩ := (Finset.mem_filter.mp hs).2 |
| have hfeas := replicaGram_feasible s |
| rw [replicaGram_triple_overlap hn0] at hfeas ⊢ |
| have HG := hgap n hn0 (tripleR s) (tripleDelta s) (tripleQ s) |
| ⟨hr.1,hr.2.trans (min_le_left _ _)⟩ (hδ.trans (min_le_right _ _)) hq hdev hfeas |
| have HD := pow_le_pow_left₀ (Real.rpow_nonneg hp.le _) hδlower 2 |
| rw [sq_abs] at HD |
| have HE : ((n:ℝ)^(-(1:ℝ)/100))^2=(n:ℝ)^(-(1:ℝ)/50) := by |
| rw [← Real.rpow_natCast ((n:ℝ)^(-(1:ℝ)/100)) 2,← Real.rpow_mul hp.le] |
| norm_num |
| rw [HE] at HD |
| have HDC := mul_le_mul_of_nonneg_left HD hc.le |
| norm_num only [Nat.cast_ofNat] |
| linarith only [HG,HDC] |
| have Hsplit := replicaGibbsMass_integral_or_or_flip (0:Fin 3) β (signedLockingSet n h b) T |
| (Finset.univ.filter (fun s : ReplicaConfig n 3 => Fin.tail s∈pairMismatchSet n β α τ)) |
| (fun s hs => by |
| rcases signedLocking_split β h b τ hh.le α hs with H|H|H |
| · exact Or.inl H |
| · exact Or.inr (Or.inl H) |
| · exact Or.inr (Or.inr (Finset.mem_filter.mpr ⟨Finset.mem_univ _,H⟩))) |
| simp_rw [replicaGibbsMass_tail] at Hsplit |
| simp only [neg_zero,Real.rpow_zero,mul_one] at hnP |
| linarith only [Hsplit,HT',hnP] |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
