| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.MatrixReplicaBridge |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem restrictedPressure_mono {n d : ℕ} (hn : 0 < n) (hd : 0 < d) (β : ℝ) |
| {S T : Finset (ReplicaConfig n d)} (hS : S.Nonempty) (hST : S ⊆ T) : |
| restrictedPressure β S ≤ restrictedPressure β T := by |
| apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg n) |
| apply integral_mono (restrictedLogPartition_integrable hn hd β hS) |
| (restrictedLogPartition_integrable hn hd β (hS.mono hST)) |
| intro J |
| apply Real.log_le_log (restrictedPartition_pos β hS J) |
| exact Finset.sum_le_sum_of_subset_of_nonneg hST (fun s _ _ => (Real.exp_pos _).le) |
|
|
| theorem replicaGibbsMass_integral_code_union {n d : ℕ} (hn : 0 < n) (hd : 0 < d) |
| (β : ℝ) {C : Type} [Fintype C] (code : ReplicaConfig n d → C) |
| (hcode : ∀ s t,code s=code t → replicaGram s=replicaGram t) |
| (S : Finset (ReplicaConfig n d)) (η E : ℝ) (hE : 0 ≤ E) |
| (hgap : ∀ s∈S,matrixConstrainedPressure n d β (replicaGram s) ≤ (d:ℝ)*finiteParisiInf β-η) |
| (hprob : ∀ T : Finset (ReplicaConfig n d),T.Nonempty → |
| restrictedPressure β T ≤ (d:ℝ)*finiteParisiInf β-η → |
| (∫ J,replicaGibbsMass β J T ∂disorderLaw n) ≤ E) : |
| (∫ J,replicaGibbsMass β J S ∂disorderLaw n) ≤ (Fintype.card C:ℝ)*E := by |
| classical |
| let T (c : C) := S.filter (fun s => code s=c) |
| have he (J : Disorder n) : replicaGibbsMass β J S=∑ c,replicaGibbsMass β J (T c) := by |
| exact (Finset.sum_fiberwise S code (fun s => ∏ a,gibbs β J (s a))).symm |
| simp_rw [he] |
| rw [integral_finsetSum _ (fun c _ => replicaGibbsMass_integrable β (T c))] |
| calc |
| _ ≤ ∑ _c : C,E := by |
| apply Finset.sum_le_sum |
| intro c _ |
| by_cases hc : (T c).Nonempty |
| · obtain ⟨s,hs⟩ := hc |
| have hs' := Finset.mem_filter.mp hs |
| have hsub : T c ⊆ matrixReplicaSet n d (replicaGram s) := by |
| intro t ht |
| apply (mem_matrixReplicaSet _ _).mpr |
| exact hcode t s ((Finset.mem_filter.mp ht).2.trans hs'.2.symm) |
| apply hprob (T c) ⟨s,hs⟩ |
| exact (restrictedPressure_mono hn hd β ⟨s,hs⟩ hsub).trans |
| (by rw [restrictedPressure_matrixReplicaSet]; exact hgap s hs'.1) |
| · have ht : T c=∅ := Finset.not_nonempty_iff_eq_empty.mp hc |
| simpa only [ht,replicaGibbsMass,Finset.sum_empty,integral_zero] using hE |
| _ = _ := by simp |
|
|
| theorem eventual_replicaGibbsMass_code_gap {β : ℝ} (hβ : 0 < β) (d : ℕ) (hd : 0 < d) : |
| ∃ K : ℝ,0 < K ∧ ∀ᶠ n : ℕ in atTop, ∀ (C : Type) [Fintype C] |
| (code : ReplicaConfig n d → C), |
| (∀ s t,code s=code t → replicaGram s=replicaGram t) → |
| ∀ (S : Finset (ReplicaConfig n d)) (η : ℝ), 0 ≤ η → |
| 2*(d:ℝ)*K*(n:ℝ)^(-(1:ℝ)/24) ≤ η → |
| (∀ s∈S,matrixConstrainedPressure n d β (replicaGram s) ≤ (d:ℝ)*finiteParisiInf β-η) → |
| (∫ J,replicaGibbsMass β J S ∂disorderLaw n) ≤ (Fintype.card C:ℝ)* |
| (Real.exp (-(n:ℝ)*η/4)+2*Real.exp (-(n:ℝ)*η^2/(16*Real.pi^2*β^2*(d:ℝ)^2))) := by |
| obtain ⟨K,hK,H⟩ := eventual_replicaGibbsMass_pressure_gap hβ d hd |
| refine ⟨K,hK,?_⟩ |
| filter_upwards [H,eventually_gt_atTop (0:ℕ)] with n hn hn0 |
| intro C _ code hcode S η hη herr hgap |
| exact replicaGibbsMass_integral_code_union hn0 hd β code hcode S η _ (by positivity) |
| hgap (fun T hT hP => hn T η hT hη herr hP) |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
