| import Mathlib |
| import OAI.Probability.SKBarriers.Parisi.QuantileClamp |
| import OAI.Probability.SKBarriers.Interpolation.GuerraUpper |
|
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| open scoped Topology |
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| def finiteParisiValues (β : ℝ) : Set ℝ := |
| {p | ∃ k : ℕ, ∃ Q : Fin (k+1) → ℝ, Monotone Q ∧ |
| (∀ j, Q j ∈ Set.Icc 0 1) ∧ p = extendedQuantileParisi k β Q} |
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| def finiteParisiInf (β : ℝ) : ℝ := sInf (finiteParisiValues β) |
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| def quenchedPressure (β : ℝ) (N : ℕ) : ℝ := |
| (∫ J, logPartition β J ∂disorderLaw N)/(N:ℝ) |
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| theorem finiteParisiValues_nonempty (β : ℝ) : (finiteParisiValues β).Nonempty := by |
| refine ⟨extendedQuantileParisi 0 β (fun _ => 0),0,fun _ => 0,?_,?_,rfl⟩ |
| · exact monotone_const |
| · intro j; exact ⟨le_rfl,zero_le_one⟩ |
|
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| theorem quenched_le_finiteParisiValue {N : ℕ} (hN : 0 < N) (β : ℝ) |
| {p : ℝ} (hp : p ∈ finiteParisiValues β) : quenchedPressure β N ≤ p := by |
| obtain ⟨k,Q,hm,hQ,rfl⟩ := hp |
| exact quenched_le_extendedQuantileParisi hN β Q |
| ((cumulativeGapMap_nonneg_iff k Q).mpr ⟨(hQ 0).1,hm⟩) |
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| theorem finiteParisiValues_bddBelow (β : ℝ) : BddBelow (finiteParisiValues β) := |
| ⟨quenchedPressure β 1,fun _ hp => quenched_le_finiteParisiValue (by norm_num) β hp⟩ |
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| theorem finiteParisiInf_le {k : ℕ} (β : ℝ) (Q : Fin (k+1) → ℝ) |
| (hm : Monotone Q) (hQ : ∀ j, Q j ∈ Set.Icc 0 1) : |
| finiteParisiInf β ≤ extendedQuantileParisi k β Q := |
| csInf_le (finiteParisiValues_bddBelow β) ⟨k,Q,hm,hQ,rfl⟩ |
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| theorem quenched_le_finiteParisiInf {N : ℕ} (hN : 0 < N) (β : ℝ) : |
| quenchedPressure β N ≤ finiteParisiInf β := |
| le_csInf (finiteParisiValues_nonempty β) (fun _ hp => quenched_le_finiteParisiValue hN β hp) |
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| def finiteComparisonError (β : ℝ) (k : ℕ) (η : ℝ) (N : ℕ) : ℝ := |
| (β^2/4)*(20*((k:ℝ)/Real.sqrt ((N:ℝ)*(β^2*η))+1/Real.sqrt (k:ℝ))+ |
| 2*(((k+1:ℕ):ℝ)*η)*(1+((k+1:ℕ):ℝ)*η)+4*(((k+1:ℕ):ℝ)*η)) |
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| def finiteComparisonErrorLimit (β : ℝ) (k : ℕ) (η : ℝ) : ℝ := |
| (β^2/4)*(20*(1/Real.sqrt (k:ℝ))+ |
| 2*(((k+1:ℕ):ℝ)*η)*(1+((k+1:ℕ):ℝ)*η)+4*(((k+1:ℕ):ℝ)*η)) |
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| theorem finiteParisiInf_sub_error_le {N k : ℕ} (hN : 0 < N) (hk : 0 < k) |
| {β η : ℝ} (hβ : 0 < β) (hη : 0 < η) |
| (hsmall : (k:ℝ)/Real.sqrt ((N:ℝ)*(β^2*η)) ≤ 1) : |
| finiteParisiInf β-finiteComparisonError β k η N ≤ quenchedPressure β N := by |
| obtain ⟨Q,hm,hQ,H⟩ := exists_admissible_scalar_lower_bound hN hk hβ hη hsmall |
| exact (sub_le_sub_right (finiteParisiInf_le β Q hm hQ) _).trans H |
|
|
| theorem finiteComparisonRatio_tendsto (k : ℕ) {β η : ℝ} (hβ : 0 < β) (hη : 0 < η) : |
| Tendsto (fun N : ℕ => (k:ℝ)/Real.sqrt ((N:ℝ)*(β^2*η))) atTop (𝓝 0) := by |
| exact (Real.tendsto_sqrt_atTop.comp |
| (Filter.Tendsto.atTop_mul_const (by positivity : 0 < β^2*η) |
| (tendsto_natCast_atTop_atTop (R := ℝ)))).const_div_atTop (k:ℝ) |
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| theorem finiteComparisonError_tendsto (k : ℕ) {β η : ℝ} (hβ : 0 < β) (hη : 0 < η) : |
| Tendsto (finiteComparisonError β k η) atTop (𝓝 (finiteComparisonErrorLimit β k η)) := by |
| unfold finiteComparisonError |
| have H := finiteComparisonRatio_tendsto k hβ hη |
| simpa only [finiteComparisonErrorLimit,zero_add] using |
| ((((H.add_const (1/Real.sqrt (k:ℝ))).const_mul 20).add_const |
| (2*(((k+1:ℕ):ℝ)*η)*(1+((k+1:ℕ):ℝ)*η))).add_const |
| (4*(((k+1:ℕ):ℝ)*η))).const_mul (β^2/4) |
|
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| theorem finiteComparisonErrorLimit_continuous (β : ℝ) (k : ℕ) : |
| Continuous (finiteComparisonErrorLimit β k) := by |
| unfold finiteComparisonErrorLimit |
| fun_prop |
|
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| theorem eventually_lt_quenched_of_error {k : ℕ} (hk : 0 < k) |
| {β η a : ℝ} (hβ : 0 < β) (hη : 0 < η) |
| (ha : a < finiteParisiInf β-finiteComparisonErrorLimit β k η) : |
| ∀ᶠ N : ℕ in atTop, a < quenchedPressure β N := by |
| have he := (finiteComparisonError_tendsto k hβ hη).eventually_lt_const |
| (show finiteComparisonErrorLimit β k η < finiteParisiInf β-a by linarith) |
| have hr := (finiteComparisonRatio_tendsto k hβ hη).eventually_lt_const (by norm_num : (0:ℝ)<1) |
| filter_upwards [he,hr,eventually_gt_atTop (0:ℕ)] with N hE hR hN |
| have H := finiteParisiInf_sub_error_le hN hk hβ hη hR.le |
| linarith |
|
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| theorem eventually_lt_quenched_of_mesh {k : ℕ} (hk : 0 < k) {β a : ℝ} (hβ : 0 < β) |
| (ha : a < finiteParisiInf β-(β^2/4)*(20/Real.sqrt (k:ℝ))) : |
| ∀ᶠ N : ℕ in atTop, a < quenchedPressure β N := by |
| have H := (finiteComparisonErrorLimit_continuous β k).continuousAt.tendsto.comp |
| (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)) |
| have he : finiteComparisonErrorLimit β k 0 = (β^2/4)*(20/Real.sqrt (k:ℝ)) := by |
| simp only [finiteComparisonErrorLimit,mul_zero,add_zero,zero_mul,mul_one_div] |
| rw [he] at H |
| obtain ⟨n,hn⟩ := (H.eventually (eventually_lt_nhds |
| (show (β^2/4)*(20/Real.sqrt (k:ℝ)) < finiteParisiInf β-a by linarith))).exists |
| apply eventually_lt_quenched_of_error hk hβ (show 0 < 1/((n:ℝ)+1) by positivity) |
| dsimp only [Function.comp_def] at hn |
| linarith |
|
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| theorem quenchedPressure_tendsto_finiteParisiInf {β : ℝ} (hβ : 0 < β) : |
| Tendsto (quenchedPressure β) atTop (𝓝 (finiteParisiInf β)) := by |
| apply tendsto_order.mpr |
| constructor |
| · intro a ha |
| have H : Tendsto (fun k : ℕ => (β^2/4)*(20/Real.sqrt (k:ℝ))) atTop (𝓝 0) := by |
| simpa only [mul_zero,Function.comp_def] using ((Real.tendsto_sqrt_atTop.comp |
| (tendsto_natCast_atTop_atTop (R := ℝ))).const_div_atTop 20).const_mul (β^2/4) |
| have HE := H.eventually (eventually_lt_nhds (sub_pos.mpr ha)) |
| obtain ⟨k,hk,hE⟩ := ((eventually_gt_atTop (0:ℕ)).and HE).exists |
| exact eventually_lt_quenched_of_mesh hk hβ (by linarith) |
| · intro a ha |
| filter_upwards [eventually_gt_atTop (0:ℕ)] with N hN |
| exact (quenched_le_finiteParisiInf hN β).trans_lt ha |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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