| import Mathlib |
| import OAI.Probability.SKBarriers.Dynamics.DynamicsMeasurable |
| import OAI.Probability.SKBarriers.Coverage.HighProbability |
| import OAI.Probability.SKBarriers.Coverage.CoverageSchedule |
| import OAI.Probability.SKBarriers.Dynamics.IdentityDynamics |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Topology |
| open Classical MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem site_clock_tail_tendsto : |
| Tendsto (fun n : ℕ => (n:ℝ)*Real.exp (-levelLogScale n)) atTop (𝓝 0) := by |
| simpa only [pow_one,neg_one_mul,levelLogScale] using |
| polynomial_rpow_exp_decay (by norm_num [kappa] : 0<kappa) zero_lt_one 1 |
|
|
| theorem bankEndpointError_nonneg (n : ℕ) : 0≤bankEndpointError n := by |
| unfold bankEndpointError |
| positivity |
|
|
| theorem bankEndpointError_tendsto : Tendsto bankEndpointError atTop (𝓝 0) := by |
| have H := (((stretched_exp_decay (by norm_num : (0:ℝ)<2)).const_mul 9).add site_clock_tail_tendsto).const_mul 4 |
| change Tendsto (fun n => 4*(9*Real.exp (-2*levelLogScale n)+(n:ℝ)*Real.exp (-levelLogScale n))) atTop (𝓝 0) |
| simpa only [mul_zero,add_zero] using H |
|
|
| theorem identity_main_of_static_locking (β : ℝ) (μ : ProbabilityMeasure ℝ) {t b b' q : ℝ} |
| (ht : 0<t) (hb : 0<b) (hpack : 3*t<b^2/2) (hcut : 3*t<b) (hcut' : 2*t<b') (hh : 2*t<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 (lockingUnion n μ t (levelRho n) q b b') ∂disorderLaw n)≤3*Real.exp (-(n:ℝ)^((4:ℝ)/5))) : |
| Tendsto (continuousBadMass β) atTop (𝓝 1) ∧ Tendsto (discreteBadMass β) atTop (𝓝 1) := by |
| let H : ℕ := ⌈2/b^2⌉₊ |
| let B : ℕ := H+1 |
| have hB : 0<B := Nat.succ_pos H |
| obtain ⟨c,hc,C,hC,hC1,hcovC⟩ := exists_coverage_schedule β q hcov |
| let A := 3*bankPrefix c B+20 |
| let V (n : ℕ) : Set (Disorder n) := {J | replicaGibbsMass β J (lockingUnion n μ t (levelRho n) q b b')≤Real.exp (-A*levelLogScale n)} |
| let D (n : ℕ) : Set (Disorder n) := finiteGoodEvent C B 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) := |
| high_probability_small_replica β _ (by norm_num [kappa] : kappa<(4:ℝ)/5) 3 A hstatic |
| have hD (n : ℕ) : MeasurableSet (D n) := (finiteGoodEvent_measurable C hC B n).inter (hV n) |
| have hD1 : Tendsto (fun n => (disorderLaw n).real (D n)) atTop (𝓝 1) := |
| high_probability_inter _ _ hV (finiteGoodEvent_tendsto C hC hC1 B) hV1 |
| apply main_of_quenched_mixed_bounds β D hD hD1 bankEndpointError bankEndpointError_nonneg bankEndpointError_tendsto |
| have hsmall : Tendsto (fun n => 3*Real.exp (-4*levelLogScale n)) atTop (𝓝 0) := by |
| simpa only [mul_zero] using (stretched_exp_decay (by norm_num : (0:ℝ)<4)).const_mul 3 |
| filter_upwards [eventually_gt_atTop (0:ℕ),stretchedLogScale_tendsto.eventually_ge_atTop 5, |
| eventually_exp_dominates_linear (bankPrefix c B+5),eventually_exp_dominates_linear (t/8), |
| hsmall.eventually_lt_const zero_lt_one, |
| greedyScale_eventual hB (show H<B by omega) ht hb le_rfl hpack hcut hcut' hh, |
| greedy_count_eventual hB ht] with n hn hL hdom hp hsmalln G hcount |
| intro J hJ |
| have hcovJ (j : ℕ) (hj : j<B) := hcovC j n J ((finiteGoodEvent_mem C B n J).mp hJ.1 j hj) |
| exact identity_quenched_bound hn hB β J μ t b b' q c (fun j => (hc j).le) hcovJ hJ.2 hL |
| ((greedy_level_count_bound hB ht n).trans hp) hdom hsmalln G hcount |
| |
| end SK.Analytic |
| |
| end |
| end |
| |
| end OAI |
| |
