| import Mathlib |
| import OAI.Probability.SKBarriers.Dynamics.ForbiddenDynamics |
| import OAI.Probability.SKBarriers.Dynamics.IdentityMain |
|
|
| namespace OAI |
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|
| section |
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| noncomputable section |
| open scoped BigOperators Topology |
| open Classical MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def forbiddenEndpointError (n : ℕ) : ℝ := Real.exp (-3*levelLogScale n)+48*Real.exp (-8*levelLogScale n) |
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| theorem forbiddenEndpointError_tendsto : Tendsto forbiddenEndpointError atTop (𝓝 0) := by |
| have H := (stretched_exp_decay (by norm_num : (0:ℝ)<3)).add ((stretched_exp_decay (by norm_num : (0:ℝ)<8)).const_mul 48) |
| change Tendsto (fun n => Real.exp (-3*levelLogScale n)+48*Real.exp (-8*levelLogScale n)) atTop (𝓝 0) |
| simpa only [mul_zero,add_zero] using H |
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| theorem forbidden_main_of_static_gap (β : ℝ) {q a₀ a₁ : ℝ} |
| (ha₀ : 0<a₀) (ha : a₀<a₁) (haq : a₁<q) |
| (hcov : ∀D : ℝ,0<D → ∃c : ℝ,0<c ∧ ∃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 (-c*levelLogScale n)))<Real.exp (-D*levelLogScale n)) |
| (hstatic : ∀ᶠ n : ℕ in atTop,(∫J,replicaGibbsMass β J (pairStripSet n a₀ a₁) ∂disorderLaw n)≤Real.exp (-(n:ℝ)^((4:ℝ)/5))) : |
| Tendsto (continuousBadMass β) atTop (𝓝 1) ∧ Tendsto (discreteBadMass β) atTop (𝓝 1) := by |
| obtain ⟨c,hc,C,hC,hC1,hcovC⟩ := hcov 3 (by norm_num) |
| let V (n : ℕ) : Set (Disorder n) := {J | replicaGibbsMass β J (pairStripSet n a₀ a₁)≤Real.exp (-(c+10)*levelLogScale n)} |
| let D (n : ℕ) : Set (Disorder n) := C n ∩ V n |
| have hV (n : ℕ) : MeasurableSet (V n) := measurableSet_le (continuous_replicaGibbsMass β _).measurable measurable_const |
| have hV1 : Tendsto (fun n => (disorderLaw n).real (V n)) atTop (𝓝 1) := by |
| apply high_probability_small_replica β _ (by norm_num [kappa] : kappa<(4:ℝ)/5) 1 (c+10) |
| simpa only [one_mul] using hstatic |
| have hD (n : ℕ) : MeasurableSet (D n) := (hC n).inter (hV n) |
| have hD1 : Tendsto (fun n => (disorderLaw n).real (D n)) atTop (𝓝 1) := high_probability_inter _ _ hV hC1 hV1 |
| apply main_of_quenched_mixed_bounds β D hD hD1 forbiddenEndpointError |
| (fun n => by unfold forbiddenEndpointError; positivity) forbiddenEndpointError_tendsto |
| have Hgap : Tendsto (fun n : ℕ => 2/(n:ℝ)) atTop (𝓝 0) := by |
| simpa only [div_eq_mul_inv,mul_zero,Function.comp_apply] using (tendsto_inv_atTop_zero.comp (tendsto_natCast_atTop_atTop (R:=ℝ))).const_mul 2 |
| filter_upwards [eventually_gt_atTop (0:ℕ),Hgap.eventually_lt_const (sub_pos.mpr ha), |
| site_clock_tail_tendsto.eventually_le_const (by norm_num : (0:ℝ)<1/8)] with n hn hgap htail |
| intro J hJ |
| have hcovJ := gibbs_coverage_predicate β J q _ _ (hcovC n J hJ.1) (fun _ => True) |
| simp only [true_and] at hcovJ |
| have H := mixedMass_le_pairStrip hn β J ha₀ haq (Real.exp_pos _) hgap |
| (show 0≤levelLogScale n by unfold levelLogScale; positivity) htail hcovJ |
| have Hnum := pairStrip_numerical_error (show 0≤levelLogScale n by unfold levelLogScale; positivity) hJ.2 |
| exact ⟨H.1.trans Hnum,H.2.trans Hnum⟩ |
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| end SK.Analytic |
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| end |
| end |
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| end OAI |
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