| import OAI.Probability.DilutedSpin.CompoundRate |
| import OAI.Probability.DilutedSpin.FunctionalStability |
| import OAI.Probability.DilutedSpin.UpperMoments |
|
|
| namespace OAI |
|
|
| section |
| namespace DilutedSpinGlass |
| open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal |
|
|
| lemma clippedModel_interaction_bounded {p : ℕ} (M : Model p) {K : ℝ} (hK : 0 ≤ K) : |
| ∀ᵐ z ∂(clippedModel M K).disorder.toMeasure, ‖z.1‖ ≤ K := by |
| change ∀ᵐ z ∂Measure.map (clipSample K) M.disorder.toMeasure, ‖z.1‖ ≤ K |
| apply (ae_map_iff (measurable_clipSample K).aemeasurable (measurableSet_le measurable_fst.norm measurable_const)).mpr |
| exact ae_of_all _ (fun z => (pi_norm_le_iff_of_nonneg hK).mpr (fun s => by |
| simpa only [Real.norm_eq_abs,clipSample] using clipReal_bound hK (z.1 s))) |
|
|
| lemma clippedModel_field_bounded {p : ℕ} (M : Model p) {K : ℝ} (hK : 0 ≤ K) : |
| ∀ᵐ h ∂(clippedModel M K).field.toMeasure, |h| ≤ K := by |
| change ∀ᵐ h ∂Measure.map (clipReal K) M.field.toMeasure, |h| ≤ K |
| apply (ae_map_iff (measurable_clipReal K).aemeasurable (measurableSet_le measurable_id.abs measurable_const)).mpr |
| exact ae_of_all _ (clipReal_bound hK) |
|
|
| lemma clippedModel_interaction_integrable {p : ℕ} (M : Model p) {K : ℝ} (hK : 0 ≤ K) : |
| Integrable (fun z : InteractionSample p => ‖z.1‖) (clippedModel M K).disorder.toMeasure := by |
| exact (integrable_const K).mono' (by fun_prop) ((clippedModel_interaction_bounded M hK).mono (fun z hz => by simpa only [norm_norm] using hz)) |
|
|
| lemma clippedModel_field_integrable {p : ℕ} (M : Model p) {K : ℝ} (hK : 0 ≤ K) : |
| Integrable (fun h : ℝ => |h|) (clippedModel M K).field.toMeasure := by |
| exact (integrable_const K).mono' (by fun_prop) ((clippedModel_field_bounded M hK).mono (fun z hz => by simpa only [Real.norm_eq_abs,abs_abs] using hz)) |
|
|
| lemma clippedModel_symmetric {p : ℕ} (M : Model p) (K : ℝ) |
| (hsym : ∀ e : Equiv.Perm (Fin p),IdentDistrib (fun z : InteractionSample p => z.1) |
| (fun z : InteractionSample p => fun s => z.1 (fun i => s (e i))) M.disorder.toMeasure M.disorder.toMeasure) |
| (e : Equiv.Perm (Fin p)) : |
| IdentDistrib (fun z : InteractionSample p => z.1) |
| (fun z : InteractionSample p => fun s => z.1 (fun i => s (e i))) |
| (clippedModel M K).disorder.toMeasure (clippedModel M K).disorder.toMeasure := by |
| constructor |
| · exact measurable_fst.aemeasurable |
| · exact (show Measurable (fun z : InteractionSample p => fun (s : Fin p → Spin) => z.1 (fun l => s (e l))) by fun_prop).aemeasurable |
| · change Measure.map _ (Measure.map (clipSample K) M.disorder.toMeasure)=Measure.map _ (Measure.map (clipSample K) M.disorder.toMeasure) |
| rw [Measure.map_map measurable_fst (measurable_clipSample K), |
| Measure.map_map (show Measurable (fun z : InteractionSample p => fun (s : Fin p → Spin) => z.1 (fun l => s (e l))) by fun_prop) (measurable_clipSample K)] |
| exact (interaction_clip_symmetric M K hsym e).map_eq |
|
|
| lemma clip_interaction_difference_integrable {p : ℕ} (M : Model p) {K : ℝ} (hK : 0 ≤ K) |
| (hi : Integrable (fun z : InteractionSample p => ‖z.1‖) M.disorder.toMeasure) : |
| Integrable (fun z => ‖(clipSample K z).1-z.1‖) M.disorder.toMeasure := by |
| apply (hi.const_mul 2).mono' ((measurable_fst.comp (measurable_clipSample K)).sub measurable_fst).norm.aestronglyMeasurable |
| exact ae_of_all _ (fun z => by |
| simp only [norm_norm] |
| change ‖(clipSample K z).1-z.1‖ ≤ 2*‖z.1‖ |
| rw [norm_sub_rev] |
| exact clipSample_difference_bound hK z) |
|
|
| lemma clip_field_difference_integrable {p : ℕ} (M : Model p) {K : ℝ} (hK : 0 ≤ K) |
| (hi : Integrable (fun h : ℝ => |h|) M.field.toMeasure) : |
| Integrable (fun h => |clipReal K h-h|) M.field.toMeasure := by |
| apply (hi.const_mul 2).mono' ((measurable_clipReal K).sub measurable_id).abs.aestronglyMeasurable |
| exact ae_of_all _ (fun h => by |
| simp only [Real.norm_eq_abs,abs_abs] |
| change |clipReal K h-h| ≤ 2*|h| |
| have h1 := abs_sub (clipReal K h) h |
| have h2 := clipReal_abs_le hK h |
| linarith) |
| end DilutedSpinGlass |
|
|
| end |
|
|
| section |
| namespace DilutedSpinGlass |
| open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory |
| open scoped BigOperators |
|
|
| noncomputable def replicaInsertionMoment {p n : ℕ} (M : Model p) (R : ℝ) |
| (P : Fin p → FiniteLaw (Fin n → Spin)) : ℝ := |
| ∫ z : InteractionSample p, (-z.2.2.1)^n*(FiniteLaw.pi P).expect |
| (fun s => ∏ a : Fin n,∏ l : Fin p,factorCut R (z.2.2.2 l) (s l a)) ∂M.disorder.toMeasure |
|
|
| /-- The literal order-n replica coefficient of the three independent-label |
| insertions. The old spin-pattern law P and fresh-label spin-pattern law B |
| are arbitrary; their laws are not assumed to agree. -/ |
| lemma replicaInsertion_combination {p n : ℕ} (M : Model p) (hM : Admissible M) |
| (j : Fin p) (R : ℝ) (P B : FiniteLaw (Fin n → Spin)) : |
| replicaInsertionMoment M R (fun _ : Fin p => P) - |
| (p:ℝ)*replicaInsertionMoment M R (fun l : Fin p => if l=j then P else B) + |
| ((p-1:ℕ):ℝ)*replicaInsertionMoment M R (fun _ : Fin p => B) = |
| (∫ z : InteractionSample p,(-z.2.2.1)^n ∂M.disorder.toMeasure)* |
| ((∫ z : InteractionSample p,patternFeature R P (z.2.2.2 j) ∂M.disorder.toMeasure)^p - |
| (p:ℝ)*(∫ z : InteractionSample p,patternFeature R P (z.2.2.2 j) ∂M.disorder.toMeasure)* |
| (∫ z : InteractionSample p,patternFeature R B (z.2.2.2 j) ∂M.disorder.toMeasure)^(p-1) + |
| ((p-1:ℕ):ℝ)*(∫ z : InteractionSample p,patternFeature R B (z.2.2.2 j) ∂M.disorder.toMeasure)^p) := by |
| classical |
| unfold replicaInsertionMoment |
| rw [upper_pattern_moment M hM j,upper_pattern_moment M hM j,upper_pattern_moment M hM j] |
| simp only [Finset.prod_const,Finset.card_univ,Fintype.card_fin] |
| have he : (∏ l : Fin p, ∫ z : InteractionSample p, |
| patternFeature R (if l=j then P else B) (z.2.2.2 j) ∂M.disorder.toMeasure) = |
| (∫ z : InteractionSample p,patternFeature R P (z.2.2.2 j) ∂M.disorder.toMeasure)* |
| (∫ z : InteractionSample p,patternFeature R B (z.2.2.2 j) ∂M.disorder.toMeasure)^(p-1) := by |
| convert product_one_distinguished j |
| (∫ z : InteractionSample p,patternFeature R P (z.2.2.2 j) ∂M.disorder.toMeasure) |
| (∫ z : InteractionSample p,patternFeature R B (z.2.2.2 j) ∂M.disorder.toMeasure) using 1 |
| · apply Finset.prod_congr rfl |
| intro l _ |
| split_ifs <;> rfl |
| · simp |
| rw [he] |
| ring |
|
|
| lemma replicaInsertion_coefficient_nonpos {p n : ℕ} (M : Model p) (hM : Admissible M) |
| (hn : 1≤n) (j : Fin p) (R t : ℝ) (ht : 0≤t) (P B : FiniteLaw (Fin n → Spin)) : |
| -(t^n/(n:ℝ))*(replicaInsertionMoment M R (fun _ : Fin p => P) - |
| (p:ℝ)*replicaInsertionMoment M R (fun l : Fin p => if l=j then P else B) + |
| ((p-1:ℕ):ℝ)*replicaInsertionMoment M R (fun _ : Fin p => B)) ≤ 0 := by |
| rw [replicaInsertion_combination M hM j, ← mul_assoc] |
| exact admissible_upper_coefficient_nonpos M hM n hn t _ _ ht |
|
|
| end DilutedSpinGlass |
|
|
| end |
|
|
| end OAI |
|
|
