| import Mathlib |
| import OAI.Probability.SKGap.Stability.ImplicitComparison |
| import OAI.Probability.SKGap.Stability.StableLiteralComparison |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Matrix.Norms.Frobenius |
| namespace SKGapCutoff.Recipe |
| open SKGap.Stein Primary Static |
| universe u |
|
|
| theorem buffered_root_comparison_weak {j K B Aroot ε c r₀ R ρ : ℝ} |
| (hj : 0≤j) (hK : 0≤K) (hB : 0≤B) (hA : 1≤Aroot) (hc : 0<c) |
| (hr₀ : 0<r₀) (hρ : 0<ρ) (hε : 0≤ε) (hAR : Real.exp (R/2)≤Aroot) |
| (hbuffer : (K+4*j)*(2*ρ)<r₀) |
| (hR : (1+2*j)*(1+(1+(K+3*j)/c)*(K+4*j))*(2*ρ)≤R) |
| (hsmall : (3*Real.exp (R/2)+2)*((1+2*j)*(1+(1+(K+3*j)/c)*(K+4*j))*(2*ρ))≤ε) |
| (himplicit : (1+2*j)*(1+(1+(K+3*j)/c)*(K+4*j))*(2*ρ)≤ |
| c/(2*(|j| *Real.exp (R/2)*(3*Real.exp (R/2)+16)+1))) |
| (k : ℕ) (f : KernelExpr) : |
| ∃A≥1,∀W C : ℝ,0≤W→0≤C→∀(Ω : Type u) (n : Ω→ℕ) |
| (J : ∀b,Interaction (n b)) (h e : ∀b,Fin (n b)→ℝ) (P : ∀b,Observables (n b)), |
| (∀b,0<n b)→(∀b,(J b).IsSymm)→(∀b,vectorNorm (e b)≤1)→ |
| (∀b,StartedMatrixEvent j K (Real.exp (R/2)) A (c/2) B W C (k+3) (2+2*(k+1)) (J b))→ |
| (∀b,¬SKGap.rootBad j Aroot ε c r₀ (J b) (h b))→ |
| (∀b,4*(residualDerivativeBudget j K B k+residualDerivativeBudget j K B (k+1))≤ρ*Real.sqrt (n b:ℝ))→ |
| (∀b x,0≤P b x)→(∀b,∑x,P b x=1)→ |
| (∀b x i,conditionalMean (P b) x i=mag j (J b) (h b) 1 x i)→ |
| ∃r : ∀b,Fin (n b)→ℝ,(∀b,SKGap.tapField j (J b) (h b) (r b)=0 ∧ |
| ∀v,SKGap.tapField j (J b) (h b) v=0→v=r b) ∧ |
| ∃w y : ∀b,VectorFields (n b),∃c₀ : ∀b,Observables (n b), |
| (∀b x,residualCutoff ρ j (J b) (h b) k x≠0→ |
| LiteralEquations (J b) j f (fld j (J b) (h b) (k+1)) (mag j (J b) (h b) (k+1)) |
| (w b) (y b) (fun v=>j*(1-onsager j (J b) (h b) (k+1) v-SKGap.overlap (r b))) |
| (c₀ b) (r b) (e b) x) ∧ |
| LocalUniformWeak (fun b=>{x | residualCutoff ρ j (J b) (h b) k x≠0}) P |
| (fun b=>literalTerminal j (J b) (h b) k (w b) (c₀ b)) ∧ |
| LocalUniformWeak (fun b=>{x | residualCutoff ρ j (J b) (h b) k x≠0}) P |
| (fun b x=>j*c₀ b x*(∑i,residual j (J b) (h b) (k+1) x i* |
| (Real.tanh (fld j (J b) (h b) (k+1) x i)-Real.tanh (r b i)))) := by |
| let η:=(1+2*j)*(1+(1+(K+3*j)/c)*(K+4*j))*(2*ρ) |
| have hη : 0≤η:=by dsimp [η];positivity |
| obtain ⟨A,hA',hh⟩:=stable_literal_comparison_weak hj hK hB hc hη himplicit k f |
| refine ⟨A,hA',?_⟩ |
| intro W C hW hC Ω n J h e P hn hJ he hevent hbad hdim hP hp hm |
| have hact (b) (v : Fin (n b)→ℝ) : vectorNorm ((J b).mulVec v)≤K*vectorNorm v := |
| (vectorNorm_matrix_mul _ v).trans (mul_le_mul_of_nonneg_right (hevent b).1 (vectorNorm_nonneg v)) |
| choose r hr hu hs using fun b=>stable_primary_buffer (hn b) hj hK hA hc hr₀ |
| (mul_nonneg (by norm_num) hρ.le) hε hAR hbuffer hR hsmall (J b) (hJ b) (h b) (hact b) (hbad b) |
| let E:=fun b=>{x | residualCutoff ρ j (J b) (h b) k x≠0} |
| have hstable : ∀b x,x∈flipNeighborhood (flipNeighborhood (E b))→ |
| LiteralStableAt (J b) j (fld j (J b) (h b) (k+1)) (mag j (J b) (h b) (k+1)) |
| (fun v=>j*(1-onsager j (J b) (h b) (k+1) v-SKGap.overlap (r b))) (r b) R η c x := by |
| intro b x hx |
| let BB:=residualDerivativeBudget j K B k+residualDerivativeBudget j K B (k+1) |
| have hBB : 0≤BB:=add_nonneg (residualDerivativeBudget_nonneg hK hB _) |
| (residualDerivativeBudget_nonneg hK hB _) |
| have hD₀ (v : Spin (n b)) : SKGap.opNorm (derivativeMatrix (residual j (J b) (h b) k) v)≤BB := |
| (residual_derivative_bound (hn b) hK hB (J b) (h b) (hevent b).1 k |
| (fun v l hl=>(hevent b).2.1 (h b) v l (by omega)) v).trans |
| (le_add_of_nonneg_right (residualDerivativeBudget_nonneg hK hB _)) |
| have hD₁ (v : Spin (n b)) : SKGap.opNorm (derivativeMatrix (residual j (J b) (h b) (k+1)) v)≤BB := |
| (residual_derivative_bound (hn b) hK hB (J b) (h b) (hevent b).1 (k+1) |
| (fun v l hl=>(hevent b).2.1 (h b) v l (by omega)) v).trans |
| (le_add_of_nonneg_left (residualDerivativeBudget_nonneg hK hB _)) |
| have H0:=vector_neighborhood_twice_buffer (residual j (J b) (h b) k) (E b) hBB hD₀ (hdim b) |
| (fun v hv=>(residualCutoff_support (hn b) hρ (J b) (h b) k v hv).1) |
| have H1:=vector_neighborhood_twice_buffer (residual j (J b) (h b) (k+1)) (E b) hBB hD₁ (hdim b) |
| (fun v hv=>(residualCutoff_support (hn b) hρ (J b) (h b) k v hv).2) |
| rcases hx with hx|⟨i,hx⟩ |
| · exact hs b k x (H0 x hx).1 (H1 x hx).1 |
| · have h0:=(H0 (flip x i) hx).2 i |
| have h1:=(H1 (flip x i) hx).2 i |
| simp only [flip_flip] at h0 h1 |
| exact hs b k x h0 h1 |
| obtain ⟨w,y,c₀,heq,hw⟩:=hh W C hW hC Ω n J h r e E P hn hJ he hevent hstable hP hp hm |
| exact ⟨r,fun b=>⟨hr b,hu b⟩,w,y,c₀,heq,hw⟩ |
|
|
| end SKGapCutoff.Recipe |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace SKGapCutoff.Recipe |
| open Primary Static SKGap.Stein Matrix |
| variable {n : ℕ} |
|
|
| theorem literal_comparison_identity (hn : 0<n) (j : ℝ) (J : Interaction n) |
| (hJ : J.IsSymm) (h r e : Fin n→ℝ) (k : ℕ) (f : KernelExpr) |
| (w y : VectorFields n) (c : Observables n) (x : Spin n) |
| (hr : SKGap.tapField j J h r=0) |
| (heq : LiteralEquations J j f (fld j J h (k+1)) (mag j J h (k+1)) w y |
| (fun v=>j*(1-onsager j J h (k+1) v-SKGap.overlap r)) c r e x) : |
| literalTerminal j J h k w c x= |
| (∑i,((fld j J h (k+1) x i-r i- |
| j*(1-onsager j J h (k+1) x-SKGap.overlap r)*Real.tanh (fld j J h (k+1) x i))* |
| f.eval (fld j J h (k+1) x i) (r i) (j*(1-onsager j J h (k+1) x-SKGap.overlap r))*e i- |
| j*(1-onsager j J h (k+1) x-SKGap.overlap r)* |
| f.dz.eval (fld j J h (k+1) x i) (r i) (j*(1-onsager j J h (k+1) x-SKGap.overlap r))*e i))+ |
| j*(1-SKGap.overlap r-siteMean (fun v i=>phi (fld j J h (k+1) v i) (r i) |
| (j*(1-onsager j J h (k+1) v-SKGap.overlap r))) x)* |
| (∑i,(Real.tanh (fld j J h (k+1) x i)-Real.tanh (r i))*w x i)- |
| j*c x*(∑i,residual j J h (k+1) x i*(Real.tanh (fld j J h (k+1) x i)-Real.tanh (r i))) := by |
| let z:=fld j J h (k+1) |
| let m:=mag j J h (k+1) |
| let a:=fun v=>j*(1-onsager j J h (k+1) v-SKGap.overlap r) |
| have hn0 : (n:ℝ)≠0:=ne_of_gt (Nat.cast_pos.mpr hn) |
| have ht := (literalEquations_recipe J j 1 f z m w y a c r e one_ne_zero x).mp heq |
| have hw := fun i=>congrFun ht.1 i |
| simp only [literalInitial_source J j 1 f z m y a c r e one_ne_zero x] at hw |
| have hy := ht.2.1.trans (literalInitial_field J j 1 f z m w y a c r e x ht.1 ht.2.2) |
| have hc : (n:ℝ)*c x=∑i,(f.dz.eval (z x i) (r i) (a x)*e i+ |
| phiZ (z x i) (r i) (a x)*(y x i-j*c x*Real.tanh (r i))) := by |
| calc |
| _ = (n:ℝ)*siteMean ((literalInitial J j 1 f z m a c r e).implicitPartial y) x := by rw [←ht.2.2] |
| _ = _ := ?_ |
| rw [siteMean,mul_div_cancel₀ _ hn0] |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact literalInitial_partial J j 1 f z m y a c r e one_ne_zero x i |
| have hroot (i) : r i-h i-J.mulVec (fun q=>Real.tanh (r q)) i+ |
| j*(1-SKGap.overlap r)*Real.tanh (r i)=0 := by |
| have hh:=congrFun hr i |
| simp only [SKGap.tapField,SKGap.onsager,SKGap.magnetization,Pi.zero_apply] at hh |
| simpa only [mulVec,dotProduct] using (by linear_combination hh : |
| r i-h i-(∑q,J i q*Real.tanh (r q))+j*(1-SKGap.overlap r)*Real.tanh (r i)=0) |
| have hb : (∑i,(Real.tanh (z x i))^2)=(n:ℝ)*(1-onsager j J h (k+2) x) := by |
| simp only [onsager,siteMean,mag_succ,z,Finset.sum_sub_distrib,Finset.sum_const, |
| Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one] |
| field_simp |
| ring |
| have hq : (∑i,(Real.tanh (r i))^2)=(n:ℝ)*SKGap.overlap r := by |
| simp only [SKGap.overlap,SKGap.magnetization,mul_div_cancel₀ _ hn0] |
| have hh:=implicit_comparison J hJ (z x) r h (m x) (w x) (y x) |
| (fun i=>f.eval (z x i) (r i) (a x)*e i) |
| (fun i=>f.dz.eval (z x i) (r i) (a x)*e i) |
| j (onsager j J h (k+2) x) (onsager j J h (k+1) x) (SKGap.overlap r) |
| (siteMean (fun v i=>phi (z v i) (r i) (a v)) x) (c x) |
| hroot hb hq hw (fun i=>congrFun hy i) hc |
| have hmag : mag j J h (k+2) x=(fun i=>Real.tanh (z x i)):=funext fun i=>mag_succ .. |
| simpa only [literalTerminal,Primary.residual,hmag,mag_succ,z,m,a,mul_assoc] using hh |
|
|
| end SKGapCutoff.Recipe |
|
|
| end |
| end |
|
|
| end OAI |
|
|
