| import Mathlib |
| import OAI.Probability.SKGap.Entropy.WeightedMean |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace SKGapCutoff.Recipe |
| open Primary Static |
| variable {n : ℕ} |
|
|
| def squareMass (P : Observables n) (f : Observables n) (x : Spin n) : ℝ := |
| P x*(f x)^2/(∑y,P y*(f y)^2) |
|
|
| def squareMean (P : Observables n) (f : Observables n) (i : Fin n) : ℝ := |
| ∑x,squareMass P f x*spin x i |
|
|
| lemma squareMass_probability (P f : Observables n) (hP : ∀x,0≤P x) |
| (hN : 0<∑x,P x*(f x)^2) : |
| (∀x,0 ≤ squareMass P f x) ∧ ∑x,squareMass P f x=1 := by |
| refine ⟨fun x=>div_nonneg (mul_nonneg (hP x) (sq_nonneg _)) hN.le,?_⟩ |
| simp only [squareMass,←Finset.sum_div,div_self (ne_of_gt hN)] |
|
|
| lemma squareMean_centered (P f : Observables n) (t : Fin n→ℝ) |
| (hN : 0<∑x,P x*(f x)^2) : |
| squareMean P f-t=fun i=>(∑x,P x*(f x)^2*(spin x i-t i))/(∑x,P x*(f x)^2) := by |
| ext i |
| simp only [squareMean,squareMass,Pi.sub_apply,div_mul_eq_mul_div,←Finset.sum_div, |
| mul_sub,Finset.sum_sub_distrib,←Finset.sum_mul,sub_div] |
| rw [mul_div_cancel_left₀ _ (ne_of_gt hN)] |
|
|
| lemma normalized_square_bound (hn : 0<n) (P f : Observables n) (t : Fin n→ℝ) |
| (hN : 0<∑x,P x*(f x)^2) (L A : ℝ) |
| (H : vectorNorm (fun i=>∑x,P x*(f x)^2*(spin x i-t i))≤ |
| L*Real.sqrt (n:ℝ)*(∑x,P x*(f x)^2)+A*starSquared P f) : |
| vectorNorm (squareMean P f-t)/Real.sqrt (n:ℝ)≤ |
| L+A/Real.sqrt (n:ℝ)*(1+varianceEnergy P f/(∑x,P x*(f x)^2)) := by |
| have hs : 0<Real.sqrt (n:ℝ):=Real.sqrt_pos.mpr (Nat.cast_pos.mpr hn) |
| rw [squareMean_centered P f t hN] |
| have he : vectorNorm (fun i=>(∑x,P x*(f x)^2*(spin x i-t i))/(∑x,P x*(f x)^2))= |
| vectorNorm (fun i=>∑x,P x*(f x)^2*(spin x i-t i))/(∑x,P x*(f x)^2) := by |
| simp only [div_eq_mul_inv,mul_comm _ (∑x,P x*(f x)^2)⁻¹] |
| change SKGap.vectorNorm _ = _ |
| rw [SKGap.vectorNorm_smul,abs_of_pos (inv_pos.mpr hN)] |
| rfl |
| rw [he] |
| calc |
| _ ≤ (L*Real.sqrt (n:ℝ)*(∑x,P x*(f x)^2)+A*starSquared P f)/(∑x,P x*(f x)^2)/Real.sqrt (n:ℝ) := |
| div_le_div_of_nonneg_right (div_le_div_of_nonneg_right H hN.le) hs.le |
| _ = _ := by unfold starSquared;field_simp [ne_of_gt hN,ne_of_gt hs] |
|
|
| universe u |
| variable {Ω : Type u} {N : Ω→ℕ} {j R B W C ρ : ℝ} |
| variable {J : ∀a,Interaction (N a)} {h : ∀a,Fin (N a)→ℝ} |
|
|
| theorem gibbs_normalized_square_mean (m : ℕ) {c r₀ ε : ℝ} |
| (hj : 0≤j) (hR : 0≤R) (hc : 0<c) (hr₀ : 0<r₀) (hρ : 0<ρ) (hε : 0<ε) |
| (hbuffer : (R+4*j)*ρ<r₀) (hm : 2<(m:ℝ)*ρ^2/4) (he : 2*(2*m:ℕ)*ε≤1) |
| (hB : 0≤B) (hW : 0≤W) (hC : 0≤C) |
| (hn : ∀a,0<N a) (hJ : ∀a,(J a).IsSymm) (hdiag : ∀a i,J a i i=0) |
| (hevent : ∀a,RecipeMatrixEvent j R (residualCoefficientBudget j 2 (2*m) 0) B W C (2*m+1) (2*(2*m)) (J a)) |
| (hH : ∀a y,vectorNorm (SKGap.tapField j (J a) (h a) y)≤r₀*Real.sqrt (N a:ℝ)→∀v, |
| c*SKGap.vectorSqNorm v≤SKGap.quadraticForm (SKGap.fieldHessian j (J a) y (SKGap.spinVariance y)) v) : |
| ∃r : ∀a,Fin (N a)→ℝ, |
| (∀a,SKGap.tapField j (J a) (h a) (r a)=0 ∧ |
| ∀y,SKGap.tapField j (J a) (h a) y=0→y=r a) ∧ |
| ∃A:ℝ,0≤A ∧ ∀a (f:Observables (N a)), |
| 0<(∑x,fieldGibbs (J a) (h a) x*(f x)^2)→ |
| vectorNorm (squareMean (fieldGibbs (J a) (h a)) f-SKGap.magnetization (r a))/Real.sqrt (N a:ℝ)≤ |
| ((1+(R+3*j)/c)*(R+4*j))*ρ+A/Real.sqrt (N a:ℝ)* |
| (1+varianceEnergy (fieldGibbs (J a) (h a)) f/(∑x,fieldGibbs (J a) (h a) x*(f x)^2)) := by |
| obtain ⟨r,hr,A,hA,hb⟩:=gibbs_square_mean (h:=h) m hj hR hc hr₀ hρ hε hbuffer hm he hB hW hC hn hJ hdiag hevent hH |
| exact ⟨r,hr,A,hA,fun a f hN=>normalized_square_bound (hn a) _ f _ hN _ A (hb a f)⟩ |
|
|
| end SKGapCutoff.Recipe |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace SKGapCutoff.Recipe |
| open Primary Static |
|
|
| structure StableRecipeInstance (j R B W C c r₀ : ℝ) (m : ℕ) where |
| n : ℕ |
| positive : 0<n |
| J : Interaction n |
| field : Fin n→ℝ |
| symmetric : J.IsSymm |
| diagonal : ∀i,J i i=0 |
| event : RecipeMatrixEvent j R (residualCoefficientBudget j 2 (2*m) 0) B W C (2*m+1) (2*(2*m)) J |
| stability : ∀y,vectorNorm (SKGap.tapField j J field y)≤r₀*Real.sqrt (n:ℝ)→∀v, |
| c*SKGap.vectorSqNorm v≤SKGap.quadraticForm (SKGap.fieldHessian j J y (SKGap.spinVariance y)) v |
|
|
| theorem uniform_stable_square_mean {j R B W C c r₀ ρ ε : ℝ} (m : ℕ) |
| (hj : 0≤j) (hR : 0≤R) (hc : 0<c) (hr₀ : 0<r₀) (hρ : 0<ρ) (hε : 0<ε) |
| (hbuffer : (R+4*j)*ρ<r₀) (hm : 2<(m:ℝ)*ρ^2/4) (he : 2*(2*m:ℕ)*ε≤1) |
| (hB : 0≤B) (hW : 0≤W) (hC : 0≤C) : |
| ∃A:ℝ,0≤A ∧ ∀{n:ℕ},0<n→∀(J:Interaction n) (h:Fin n→ℝ), |
| J.IsSymm→(∀i,J i i=0)→ |
| RecipeMatrixEvent j R (residualCoefficientBudget j 2 (2*m) 0) B W C (2*m+1) (2*(2*m)) J→ |
| (∀y,vectorNorm (SKGap.tapField j J h y)≤r₀*Real.sqrt (n:ℝ)→∀v, |
| c*SKGap.vectorSqNorm v≤SKGap.quadraticForm (SKGap.fieldHessian j J y (SKGap.spinVariance y)) v)→ |
| ∃r:Fin n→ℝ,SKGap.tapField j J h r=0 ∧ (∀y,SKGap.tapField j J h y=0→y=r) ∧ |
| ∀f:Observables n,0<(∑x,fieldGibbs J h x*(f x)^2)→ |
| vectorNorm (squareMean (fieldGibbs J h) f-SKGap.magnetization r)/Real.sqrt (n:ℝ)≤ |
| ((1+(R+3*j)/c)*(R+4*j))*ρ+A/Real.sqrt (n:ℝ)* |
| (1+varianceEnergy (fieldGibbs J h) f/(∑x,fieldGibbs J h x*(f x)^2)) := by |
| let Ω:=StableRecipeInstance j R B W C c r₀ m |
| obtain ⟨r,hr,A,hA,hb⟩:=gibbs_normalized_square_mean (Ω:=Ω) (N:=fun a=>a.n) |
| (J:=fun a=>a.J) (h:=fun a=>a.field) m hj hR hc hr₀ hρ hε hbuffer hm he hB hW hC |
| (fun a=>a.positive) (fun a=>a.symmetric) (fun a=>a.diagonal) (fun a=>a.event) (fun a=>a.stability) |
| refine ⟨A,hA,?_⟩ |
| intro n hn J h hJ hd hev hst |
| let a:Ω:=⟨n,hn,J,h,hJ,hd,hev,hst⟩ |
| exact ⟨r a,(hr a).1,(hr a).2,hb a⟩ |
|
|
| lemma choose_selection_depth {ρ : ℝ} (hρ : 0<ρ) : |
| ∃m:ℕ,∃ε:ℝ,0<ε ∧ 2<(m:ℝ)*ρ^2/4 ∧ 2*(2*m:ℕ)*ε≤1 := by |
| obtain ⟨m,hm⟩:=exists_nat_gt (8/ρ^2) |
| have hρ2 : 0<ρ^2:=sq_pos_of_pos hρ |
| have hm' : 2<(m:ℝ)*ρ^2/4 := by |
| have H:8<(m:ℝ)*ρ^2:=(div_lt_iff₀ hρ2).mp hm |
| linarith |
| refine ⟨m,1/(2*(2*m:ℕ)+1),by positivity,hm',?_⟩ |
| have hp : 0≤(2*(2*m:ℕ):ℝ):=by positivity |
| rw [mul_one_div] |
| exact (div_le_one (by positivity)).mpr (by linarith) |
|
|
| end SKGapCutoff.Recipe |
|
|
| end |
| end |
|
|
| end OAI |
|
|
