| import OAI.Probability.DilutedSpin.CavitySplit |
| import OAI.Probability.DilutedSpin.NormalizationSymmetry |
|
|
| namespace OAI |
|
|
| section |
| namespace DilutedSpinGlass |
| open Filter |
| open scoped Topology NNReal BigOperators |
|
|
| lemma oneNewRate_tendsto (α : ℝ≥0) (q : ℕ) : |
| Tendsto (fun N => (oneNewRate α q N:ℝ)) atTop (𝓝 ((α:ℝ)*(q+1))) := by |
| have hh := ((tendsto_natCast_div_add_atTop (1:ℝ)).pow q).const_mul ((α:ℝ)*(q+1)) |
| simpa only [oneNewRate,NNReal.coe_mul,NNReal.coe_add,NNReal.coe_natCast,NNReal.coe_one, |
| NNReal.coe_pow,NNReal.coe_div,one_pow,mul_one] using hh |
|
|
| lemma reservoir_deficit_identity (α : ℝ≥0) (q N : ℕ) : |
| (α:ℝ)*N-(reservoirRate α q N:ℝ) = |
| (α:ℝ)*((N:ℝ)/(N+1))*(∑ i∈Finset.range q,((N:ℝ)/(N+1))^i) := by |
| have hh := geom_sum_mul ((N:ℝ)/(N+1)) q |
| have hd : (N:ℝ)+1 ≠ 0 := by positivity |
| change (α:ℝ)*N-(α:ℝ)*N*((N:ℝ)/(N+1))^q = _ |
| have hx : ((N:ℝ)/(N+1)-1) = -1/(N+1) := by field_simp; ring |
| rw [hx] at hh |
| have he : 1-((N:ℝ)/(N+1))^q = (∑ i∈Finset.range q,((N:ℝ)/(N+1))^i)/(N+1) := by |
| simp only [div_eq_mul_inv,neg_mul,one_mul,mul_neg] at hh ⊢ |
| linarith |
| rw [show (α:ℝ)*N-(α:ℝ)*N*((N:ℝ)/(N+1))^q = (α:ℝ)*N*(1-((N:ℝ)/(N+1))^q) by ring,he] |
| ring |
|
|
| lemma reservoir_deficit_tendsto (α : ℝ≥0) (q : ℕ) : |
| Tendsto (fun N => (α:ℝ)*N-(reservoirRate α q N:ℝ)) atTop (𝓝 ((α:ℝ)*q)) := by |
| simp_rw [reservoir_deficit_identity] |
| have hh := ((tendsto_natCast_div_add_atTop (1:ℝ)).const_mul (α:ℝ)).mul |
| (tendsto_finsetSum (Finset.range q) (fun i _ => (tendsto_natCast_div_add_atTop (1:ℝ)).pow i)) |
| simpa only [one_pow,Finset.sum_const,Finset.card_range,nsmul_eq_mul,mul_one,mul_one] using hh |
|
|
| open Classical in |
| lemma cavity_bad_count (q N : ℕ) : |
| (∑ x : Fin (q+1) → Fin (N+1),if CavityGood x then (0:ℝ) else 1) = |
| (N+1:ℝ)^(q+1)-(N:ℝ)^(q+1)-(q+1:ℝ)*(N:ℝ)^q := by |
| have hg := sum_cavityKeep (q := q) (N := N) (fun _ => (1:ℝ)) |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fun,Fintype.card_fin, |
| Fintype.card_prod,nsmul_eq_mul,mul_one,Nat.cast_pow,Nat.cast_add,Nat.cast_one,Nat.cast_mul] at hg |
| have he : (∑ x : Fin (q+1) → Fin (N+1),if CavityGood x then (0:ℝ) else 1) = |
| (∑ _x : Fin (q+1) → Fin (N+1),(1:ℝ))- |
| (∑ x : Fin (q+1) → Fin (N+1),if CavityGood x then (1:ℝ) else 0) := by |
| rw [← Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro x _ |
| split_ifs <;> norm_num |
| rw [he,hg] |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fun,Fintype.card_fin, |
| nsmul_eq_mul,mul_one,Nat.cast_pow,Nat.cast_add,Nat.cast_one] |
| ring |
|
|
| noncomputable def cavityBadRate (α : ℝ≥0) (q N : ℕ) : ℝ := |
| (α:ℝ)*(N+1)-(reservoirRate α q N:ℝ)-(oneNewRate α q N:ℝ) |
|
|
| open Classical in |
| lemma cavityBadRate_count (α : ℝ≥0) (q N : ℕ) : |
| cavityBadRate α q N = (α:ℝ)*(N+1)* |
| (∑ x : Fin (q+1) → Fin (N+1),if CavityGood x then (0:ℝ) else 1)/(N+1)^(q+1) := by |
| rw [cavity_bad_count] |
| simp only [cavityBadRate,oneNewRate,reservoirRate,NNReal.coe_mul,NNReal.coe_add, |
| NNReal.coe_natCast,NNReal.coe_one,NNReal.coe_pow,NNReal.coe_div,div_pow,pow_succ] |
| have hd : (N:ℝ)+1≠0 := by positivity |
| field_simp |
|
|
| lemma cavityBadRate_nonneg (α : ℝ≥0) (q N : ℕ) : 0≤cavityBadRate α q N := by |
| classical |
| rw [cavityBadRate_count] |
| positivity |
|
|
| lemma cavityBadRate_tendsto (α : ℝ≥0) (q : ℕ) : |
| Tendsto (cavityBadRate α q) atTop (𝓝 0) := by |
| have hh := ((tendsto_const_nhds (x := (α:ℝ))).add (reservoir_deficit_tendsto α q)).sub (oneNewRate_tendsto α q) |
| have he : (α:ℝ)+(α:ℝ)*q-(α:ℝ)*(q+1)=0 := by ring |
| rw [he] at hh |
| convert hh using 1 |
| funext N |
| dsimp [cavityBadRate] |
| ring |
|
|
| end DilutedSpinGlass |
|
|
| end |
|
|
| section |
| namespace DilutedSpinGlass |
| open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory |
|
|
| /-- Move the singled-out interaction coordinate to the distinguished last slot, |
| preserving the relative order of all other slots. -/ |
| def cavityCoordinateEquiv {q : ℕ} (j : Fin (q+1)) : Equiv.Perm (Fin (q+1)) where |
| toFun := j.insertNth (Fin.last q) (fun i => i.castSucc) |
| invFun := Fin.lastCases j (fun i => j.succAbove i) |
| left_inv x := by |
| refine Fin.succAboveCases j ?_ (fun i => ?_) x |
| · simp only [Fin.insertNth_apply_same,Fin.lastCases_last] |
| · simp only [Fin.insertNth_apply_succAbove,Fin.lastCases_castSucc] |
| right_inv x := by |
| refine Fin.lastCases ?_ (fun i => ?_) x |
| · simp only [Fin.lastCases_last,Fin.insertNth_apply_same] |
| · simp only [Fin.lastCases_castSucc,Fin.insertNth_apply_succAbove] |
|
|
| lemma appendSpin_castSucc {q : ℕ} (s : Fin q → Spin) (ε : Spin) (i : Fin q) : |
| appendSpin (p := q+1) s ε i.castSucc = s i := by |
| simp only [appendSpin,Nat.add_sub_cancel,Fin.val_castSucc,dite_eq_left i.isLt] |
| rfl |
|
|
| lemma appendSpin_last {q : ℕ} (s : Fin q → Spin) (ε : Spin) : |
| appendSpin (p := q+1) s ε (Fin.last q) = ε := by |
| simp [appendSpin] |
|
|
| lemma cavityCoordinate_append {q : ℕ} (j : Fin (q+1)) (s : Fin q → Spin) (ε : Spin) : |
| (fun l => appendSpin (p := q+1) s ε (cavityCoordinateEquiv j l)) = j.insertNth ε s := by |
| funext l |
| refine Fin.succAboveCases j ?_ (fun i => ?_) l |
| · simp only [cavityCoordinateEquiv,Equiv.coe_fn_mk,Fin.insertNth_apply_same,appendSpin_last] |
| · simp only [cavityCoordinateEquiv,Equiv.coe_fn_mk,Fin.insertNth_apply_succAbove,appendSpin_castSucc] |
|
|
| /-- iid coordinate symmetry is sufficient for all one-new-index classes to |
| have the exact same interaction as the last-coordinate cavity convention. -/ |
| lemma cavity_coordinate_identDistrib {q : ℕ} {X : Type} [MeasurableSpace X] |
| (μ : Measure X) (θ : X → Interaction (q+1)) |
| (hsym : ∀ e : Equiv.Perm (Fin (q+1)), IdentDistrib θ |
| (fun z s => θ z (fun l => s (e l))) μ μ) |
| (j : Fin (q+1)) : |
| IdentDistrib (fun z (v : (Fin q → Spin) × Spin) => θ z (appendSpin v.1 v.2)) |
| (fun z (v : (Fin q → Spin) × Spin) => θ z (j.insertNth v.2 v.1)) μ μ := by |
| have h := (hsym (cavityCoordinateEquiv j)).comp |
| (show Measurable (fun t : Interaction (q+1) => fun v : (Fin q → Spin) × Spin => t (appendSpin v.1 v.2)) by fun_prop) |
| simpa only [Function.comp_def,cavityCoordinate_append] using h |
|
|
| end DilutedSpinGlass |
|
|
| end |
|
|
| end OAI |
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|
