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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 | 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
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