| import Mathlib |
| import OAI.Probability.SKGap.Brownian.GramPathRobust |
| import OAI.Probability.SKGap.Matrix.WordSimultaneousPrediction |
| import OAI.Probability.SKGap.Stability.WordStablePrediction |
| import OAI.Probability.SKGap.Stability.QuenchedStableFields |
| import OAI.Probability.SKGap.Terminal.PairInteraction |
| import OAI.Probability.SKGap.Entropy.QuenchedFieldAverageExponential |
| import OAI.Probability.SKGap.Localization.CurieWeiss |
| import OAI.Probability.SKGap.Gaussian.IndependentBoundedAverageTailLaw |
| import OAI.Probability.SKGap.Matrix.DoublePlantedGOEArray |
| import OAI.Probability.SKGap.Localization.DoublePlantedLawTransfer |
| import OAI.Probability.SKGap.Terminal.PairSpinFrequency |
| import OAI.Probability.SKGap.Matrix.GapHalfSmallOpNorm |
| import OAI.Probability.SKGap.Localization.OriginalEventRegeneration |
| import OAI.Probability.SKGap.Localization.PoissonBinomial |
| import OAI.Probability.SKGap.Brownian.DiscreteUpperContinuous |
| import OAI.Probability.SKGap.Brownian.WorstContinuousPolynomialUpper |
| import OAI.Probability.SKGap.Localization.ScalarSlowConeStep |
| import OAI.Probability.SKGap.Terminal.SpinDifferenceEnergyReplace |
| import OAI.Probability.SKGap.Localization.TanhContDiff |
| import OAI.Probability.SKGap.Localization.JumpBilinear |
| import OAI.Probability.SKGap.Matrix.StaticCovarianceMatrix |
| import OAI.Probability.SKGap.Localization.Prediction |
| import OAI.Probability.SKGap.Matrix.RecipeNodeTrace |
| import OAI.Probability.SKGap.Matrix.StartedNodeTrace |
| import OAI.Probability.SKGap.Stability.LocalStartedImplicit |
| import OAI.Probability.SKGap.Localization.StartedSmall |
| import OAI.Probability.SKGap.Terminal.TerminalChannel |
| import OAI.Probability.SKGap.Localization.ModelLocalization |
|
|
| namespace OAI |
|
|
| namespace SKGap.FullMain |
| open MeasureTheory ProbabilityTheory Filter Real Set |
| open scoped Topology BigOperators ENNReal |
| open GaussianStep GaussianHistory ObservationBridge SKGapCutoff.Recipe SKGapCutoff.Static |
| noncomputable section |
|
|
| lemma whp_of_measurable_full {β : ℝ} (Ω : ∀n,Set (Disorder n)) |
| (hΩ : ∀n,MeasurableSet (Ω n)) |
| (hfull : Tendsto (fun n=>(disorderLaw β n).real (Ω n)) atTop (𝓝 1)) : |
| Whp β (fun n g=>g∈Ω n) := by |
| apply (whp_iff_real _ _).mpr |
| have H:=(tendsto_const_nhds (x:=(1:ℝ))).sub hfull |
| simpa only [sub_self,show ∀n,{g:Disorder n | ¬g∈Ω n}=(Ω n)ᶜ from fun _=>rfl, probReal_compl_eq_one_sub (hΩ _)] using H |
|
|
| lemma weighted_accuracy {B C₀ r W : ℝ} (hB : 0≤B) (hC₀ : 0≤C₀) (hr : 0<r) (hW : 0<W) : |
| ∃ρ:ℝ,0<ρ ∧ B*ρ<r ∧ 4*C₀*ρ≤W := by |
| let ρ:=min (r/(2*(B+1))) (W/(2*(4*C₀+1))) |
| have hρ : 0<ρ:=lt_min (by positivity) (by positivity) |
| have hleft0 : ρ ≤ r/(2*(B+1)) := min_le_left _ _ |
| have hleft := (le_div_iff₀ (show 0 < 2*(B+1) by positivity)).mp hleft0 |
| have hright0 : ρ ≤ W/(2*(4*C₀+1)) := min_le_right _ _ |
| have hright := (le_div_iff₀ (show 0 < 2*(4*C₀+1) by positivity)).mp hright0 |
| refine ⟨ρ,hρ,?_,?_⟩ |
| · nlinarith [mul_nonneg hB hρ.le] |
| · nlinarith [mul_nonneg hC₀ hρ.le] |
|
|
| lemma exponential_pair_eventually {a d : ℝ} (ha : 0<a) (hd : 0<d) : |
| ∀ᶠ n:ℕ in atTop,exp (-a*(n:ℝ))+exp (-d*(n:ℝ))≤exp (-(min a d/2)*(n:ℝ)) := by |
| have hm : 0 < min a d := lt_min ha hd |
| filter_upwards [constant_exp_absorb 2 hm] with n hn |
| calc |
| _≤exp (-(min a d)*(n:ℝ))+exp (-(min a d)*(n:ℝ)) := |
| add_le_add (exp_le_exp.mpr (by |
| have H := mul_le_mul_of_nonneg_right (min_le_left a d) (Nat.cast_nonneg n : (0:ℝ) ≤ n) |
| nlinarith)) |
| (exp_le_exp.mpr (by |
| have H := mul_le_mul_of_nonneg_right (min_le_right a d) (Nat.cast_nonneg n : (0:ℝ) ≤ n) |
| nlinarith)) |
| _=2*exp (-(min a d)*(n:ℝ)):=by ring |
| _≤_:=hn |
|
|
| theorem quenched_poincare {β : ℝ} (hβ : 0<β) (hβ1 : β<1) : |
| ∃C:ℝ,0<C ∧ Whp β (fun n g=>∀f:Spin n→ℝ,variance g 0 f≤C*dirichlet g 0 f) := by |
| obtain ⟨T,aT,hT,haT,Ω,hΩ,hΩfull,hterminal⟩:=terminal_gap_high_probability hβ |
| obtain ⟨Ar,K,ε,c,r₀,a,hAr,hK,hs,hε,hc,hr₀,ha,hroot⟩:=quenched_grid_root_stability hβ hβ1 hT |
| obtain ⟨D₀,hD₀,hcov⟩:=quenched_posterior_covariance hβ hβ1 hAr hs hε hc hr₀ |
| let b:=min a aT/2 |
| have hb : 0<b:=half_pos (lt_min ha haT) |
| obtain ⟨W,hW,hnum⟩:=localization_numeric_choice (D₀^2) 2 T a b hT hb |
| let P:=physicalNormBound (β^2) |
| let C₀:=(1+(P+3*β^2)/c)*(P+4*β^2) |
| have hP : 0≤P:=(physicalNormBound_pos (β^2)).le |
| have hC₀ : 0≤C₀:=by dsimp [C₀]; positivity |
| obtain ⟨ρ,hρ,hbuffer,hlead⟩:=weighted_accuracy (B:=P+4*β^2) (by positivity) hC₀ hr₀ hW |
| obtain ⟨Cmean,hCmean,hmean⟩:=disorder_stable_square_mean hβ hβ1 hc hr₀ ρ hρ hbuffer |
| let C:=2*(D₀^2*T+3)/exp (-2*D₀^2*T) |
| have hC : 0<C:=by dsimp [C]; positivity |
| refine ⟨C,hC,?_⟩ |
| have hevents:=(((whp_of_measurable_full Ω hΩ hΩfull).and hroot).and hcov).and |
| (show Whp β (fun n g=>StableSquareMeanProperty (β^2) c r₀ ρ C₀ Cmean (coupling g)) from hmean) |
| apply hevents.mono |
| filter_upwards [hnum (2*Cmean),exponential_pair_eventually ha haT] with n hn hr |
| intro g hg |
| rcases hg with ⟨⟨⟨hgΩ,hgr⟩,hgc⟩,hgm⟩ |
| rcases hn with ⟨hn,hmesh,hmeshD,hloss,hcost⟩ |
| have hcost' : localizationCost n (2*Cmean) a b (D₀^2) 2 T≤D₀^2*T+3 := by |
| convert hcost using 1; ring |
| exact poincare_from_model_data g hn (β^2) Ar K ε c r₀ ρ C₀ Cmean D₀ W a aT b T |
| hAr.le hD₀ hW.le hCmean ha hb hT hlead hgr.1 hgr.2 (hterminal n g hgΩ) hgc hgm hr |
| hmesh hmeshD hloss hcost' |
|
|
| end |
| end SKGap.FullMain |
|
|
| namespace SKGap |
| open MeasureTheory ProbabilityTheory Filter Real Set |
| open scoped Topology |
|
|
| theorem sk_main (β : ℝ) (hβ0 : 0 < β) (hβ1 : β < 1) : MainStatement β := by |
| obtain ⟨C,hC,hwhp⟩:=FullMain.quenched_poincare hβ0 hβ1 |
| have hP : Tendsto (fun n:ℕ=>disorderLaw β n (poincareEvent n C)) atTop (𝓝 1):= |
| FullMain.whp_full_probability hwhp |
| exact ⟨C,hC,hP,gap_probability_of_poincare β C hC hP⟩ |
|
|
| end SKGap |
|
|
| end OAI |
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