| import Mathlib |
| import OAI.Probability.SKGap.Localization.CubeApprox |
| import OAI.Probability.SKGap.Localization.LocalVolumeChange |
| import OAI.Probability.SKGap.Matrix.ConditionalMiddleNorm |
| import OAI.Probability.SKGap.Gaussian.GaussianReductionPointwise |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal |
| attribute [local fun_prop] continuous_radialDensity |
|
|
| def fieldMollifier {n : ℕ} (j σ : ℝ) (J : Matrix (Fin n) (Fin n) ℝ) |
| (h y : Fin n → ℝ) : ℝ := |
| radialDensity σ (tapField j J h y)*exp ((n:ℝ)*j*(1-overlap y)^2/2) |
|
|
| def fieldIntegral {n : ℕ} (j σ : ℝ) (J : Matrix (Fin n) (Fin n) ℝ) |
| (h : Fin n → ℝ) (E : Set (Fin n → ℝ)) : ℝ≥0∞ := |
| ∫⁻ y in E, ENNReal.ofReal (fieldMollifier j σ J h y) |
|
|
| lemma continuous_fieldMollifier {n : ℕ} (j σ : ℝ) : |
| Continuous (fun p : Matrix (Fin n) (Fin n) ℝ × ((Fin n → ℝ) × (Fin n → ℝ))=> |
| fieldMollifier j σ p.1 p.2.1 p.2.2) := by |
| unfold fieldMollifier tapField onsager overlap magnetization |
| exact ((continuous_radialDensity σ).comp (by fun_prop)).mul (by fun_prop) |
|
|
| def conditionalFieldWeight {n : ℕ} [NeZero n] (j t σ : ℝ) |
| (E : Set (((Fin n×Fin n) → ℝ) × (Fin n → ℝ))) (y : Fin n → ℝ) : ℝ≥0∞ := |
| conditionalGoeLaw (j/(n:ℝ)) (t+σ^2) (magnetization y) (empiricalObservation j t σ y) |
| {W | (W,y) ∈ E} |
|
|
| lemma field_noise_convolution {n : ℕ} [NeZero n] {j t σ : ℝ} (ht : 0 ≤ t) (hσ : 0 < σ) |
| (y : Fin n → ℝ) (f : ((Fin n×Fin n) → ℝ) → ℝ≥0∞) (g : MatrixCoordinates (Fin n) → ℝ) : |
| (∫⁻ x, ENNReal.ofReal (fieldMollifier j σ |
| (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) (fun i=>t+x i) y)* |
| f (fun p=>goeMatrix (j/(n:ℝ)) g p.1 p.2) |
| ∂Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal))= |
| ENNReal.ofReal (exp ((n:ℝ)*j*(1-overlap y)^2/2))* |
| (ENNReal.ofReal (radialDensity (sqrt (t+σ^2)) |
| (empiricalObservation j t σ y-goeMatrix (j/(n:ℝ)) g*ᵥmagnetization y))* |
| f (fun p=>goeMatrix (j/(n:ℝ)) g p.1 p.2)) := by |
| simp only [fieldMollifier,tapField_planted_observation j t σ, |
| ENNReal.ofReal_mul (radialDensity_pos hσ _).le] |
| rw [lintegral_mul_const _ (by fun_prop),lintegral_mul_const _ (by fun_prop),gaussian_radial_convolution ht hσ] |
| ring |
|
|
| theorem field_reduction_fixed {n : ℕ} [NeZero n] {j t σ : ℝ} |
| (hj : 0 ≤ j) (ht : 0 ≤ t) (hσ : 0 < σ) |
| (E : Set (((Fin n×Fin n) → ℝ) × (Fin n → ℝ))) (hE : MeasurableSet E) (y : Fin n → ℝ) : |
| (∫⁻ g, ∫⁻ x, ENNReal.ofReal (fieldMollifier j σ |
| (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) (fun i=>t+x i) y)* |
| E.indicator (fun _=>1) ((fun p=>goeMatrix (j/(n:ℝ)) g p.1 p.2),y) |
| ∂Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal) |
| ∂gaussianCoordinates (MatrixCoordinates (Fin n)))= |
| ENNReal.ofReal (sqrt (scalarS j (scalarEmpirical y t σ)/ |
| (scalarS j (scalarEmpirical y t σ)+j*scalarQMoment (empiricalLaw y)))*scalarIntegrand j t σ y)* |
| conditionalFieldWeight j t σ E y := by |
| let f : ((Fin n×Fin n) → ℝ) → ℝ≥0∞ := fun W=>E.indicator (fun _=>1) (W,y) |
| have hf : Measurable f := measurable_const.indicator (hE.preimage (by fun_prop)) |
| change (∫⁻ g, ∫⁻ x, ENNReal.ofReal (fieldMollifier j σ _ _ y)*f _ ∂_ ∂_)=_ |
| simp_rw [field_noise_convolution ht hσ y f] |
| have hgf : Measurable (fun g : MatrixCoordinates (Fin n) → ℝ => f (fun p=>goeMatrix (j/(n:ℝ)) g p.1 p.2)) := hf.comp (by unfold goeMatrix; fun_prop) |
| have hgr : Measurable (fun g : MatrixCoordinates (Fin n) → ℝ => ENNReal.ofReal |
| (radialDensity (sqrt (t+σ^2)) (empiricalObservation j t σ y-goeMatrix (j/(n:ℝ)) g*ᵥmagnetization y))) := by |
| apply Measurable.ennreal_ofReal |
| apply (continuous_radialDensity _).measurable.comp |
| unfold Matrix.mulVec dotProduct goeMatrix |
| fun_prop |
| erw [lintegral_const_mul _ (hgr.mul hgf),goe_pointwise_conditional_integral |
| (div_nonneg hj (Nat.cast_nonneg n)) (add_pos_of_nonneg_of_pos ht (sq_pos_of_pos hσ)) _ _ f hf] |
| have hfi : (∫⁻ W,f W ∂conditionalGoeLaw (j/(n:ℝ)) (t+σ^2) (magnetization y) |
| (empiricalObservation j t σ y))=conditionalFieldWeight j t σ E y := by |
| change (∫⁻ W, {W | (W,y)∈E}.indicator (fun _=>(1 : ℝ≥0∞)) W ∂_)=_ |
| have hset : MeasurableSet {W : (Fin n×Fin n) → ℝ | (W,y)∈E} := hE.preimage (by fun_prop) |
| rw [lintegral_indicator hset,lintegral_const] |
| simp [conditionalFieldWeight] |
| rw [hfi,← mul_assoc,← ENNReal.ofReal_mul (exp_pos _).le] |
| congr 1 |
| congr 1 |
| rw [mul_comm,scalar_density_cancellation hj ht hσ y] |
| end SKGap |
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity |
| open scoped BigOperators ENNReal |
|
|
| lemma fieldMollifier_eq_localAction {n : ℕ} (j σ : ℝ) (J : Matrix (Fin n) (Fin n) ℝ) |
| (h y : Field n) : fieldMollifier j σ J h y= |
| ((sqrt (2*Real.pi)*σ)⁻¹)^n*exp (localAction j σ J h y) := by |
| unfold fieldMollifier radialDensity localAction varianceAverage |
| rw [Fintype.card_fin,vectorNorm_sq,mul_assoc,← exp_add] |
| simp only [vectorSqNorm,dotProduct,pow_two] |
|
|
| lemma fieldIntegral_eq_localVolume {n : ℕ} {σ : ℝ} (hσ : 0 < σ) (j : ℝ) |
| (J : Matrix (Fin n) (Fin n) ℝ) (h : Field n) (E : Set (Field n)) : |
| fieldIntegral j σ J h E=localVolume j σ J h E := by |
| unfold fieldIntegral localVolume |
| simp_rw [fieldMollifier_eq_localAction,ENNReal.ofReal_mul (by positivity : 0 ≤ ((sqrt (2*Real.pi)*σ)⁻¹)^n)] |
| exact lintegral_const_mul _ (Real.measurable_exp.comp (localAction_continuous j σ J h).measurable |>.ennreal_ofReal) |
| end SKGap |
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal |
|
|
| lemma outer_add_rank_le_two {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (u v w z : ι → ℝ) : (vecMulVec u v+vecMulVec w z).rank ≤ 2 := by |
| let Q : Matrix ι (Fin 2) ℝ := fun i k=>![u i,w i] k |
| let R : Matrix (Fin 2) ι ℝ := fun k i=>![v i,z i] k |
| have he : vecMulVec u v+vecMulVec w z=Q*R := by |
| ext i k |
| change u i*v k+w i*z k=∑ l : Fin 2,Q i l*R l k |
| simp [Q,R,Fin.sum_univ_two] |
| rw [he] |
| exact (rank_mul_le_left Q R).trans (by simpa using rank_le_card_width Q) |
|
|
| lemma operator_outer_vector_le {n : ℕ} (u v : Field n) : |
| opNorm (vecMulVec u v) ≤ vectorNorm u*vectorNorm v := |
| operator_vecMulVec_le (WithLp.toLp 2 u) (WithLp.toLp 2 v) |
|
|
| lemma vectorNorm_const_one (n : ℕ) : vectorNorm (fun _ : Fin n=>(1:ℝ))=sqrt (n:ℝ) := by |
| apply (sq_eq_sq₀ (vectorNorm_nonneg _) (sqrt_nonneg _)).mp |
| rw [vectorNorm_sq,sq_sqrt (Nat.cast_nonneg _)] |
| simp [vectorSqNorm] |
|
|
| lemma spinVariance_norm_bound {n : ℕ} (y : Field n) : |
| vectorNorm (spinVariance y) ≤ sqrt (n:ℝ) := by |
| apply vectorNorm_le_of_sq_sum (sqrt_nonneg _) |
| rw [sq_sqrt (Nat.cast_nonneg _)] |
| calc |
| _ ≤ ∑ _i : Fin n,(1:ℝ) := Finset.sum_le_sum (fun i _=>by |
| have h0 := (spinVariance_pos y i).le |
| have h1 := spinVariance_le_one y i |
| nlinarith) |
| _ = _ := by simp |
|
|
| lemma weightedMagnetization_norm_bound {n : ℕ} (y : Field n) : |
| vectorNorm (fun i=>magnetization y i*spinVariance y i) ≤ sqrt (n:ℝ) := by |
| have hh := vectorNorm_diag_le zero_le_one (x := magnetization y) |
| (a := spinVariance y) (fun i=>(abs_of_pos (spinVariance_pos y i)).trans_le (spinVariance_le_one y i)) |
| simpa only [one_mul,mul_comm] using hh.trans (by simpa only [one_mul] using magnetization_norm_bound y) |
|
|
| def plantedJacobianPerturbation {n : ℕ} (j : ℝ) (y : Field n) : Matrix (Fin n) (Fin n) ℝ := |
| (-j/(n:ℝ)) • vecMulVec (fun _=>1) (spinVariance y)+ |
| (-2*j/(n:ℝ)) • vecMulVec (magnetization y) (fun i=>magnetization y i*spinVariance y i) |
|
|
| lemma plantedJacobianPerturbation_rank {n : ℕ} (j : ℝ) (y : Field n) : |
| (plantedJacobianPerturbation j y).rank ≤ 2 := by |
| have he : plantedJacobianPerturbation j y= |
| vecMulVec (fun _=>-j/(n:ℝ)) (spinVariance y)+ |
| vecMulVec (fun i=>(-2*j/(n:ℝ))*magnetization y i) (fun i=>magnetization y i*spinVariance y i) := by |
| ext i k |
| simp [plantedJacobianPerturbation,vecMulVec_apply,mul_assoc] |
| rw [he] |
| exact outer_add_rank_le_two _ _ _ _ |
|
|
| lemma plantedJacobianPerturbation_norm {n : ℕ} (hn : 0 < n) {j : ℝ} (hj : 0 ≤ j) (y : Field n) : |
| opNorm (plantedJacobianPerturbation j y) ≤ 3*j := by |
| have hn0 : (0:ℝ) < n := Nat.cast_pos.mpr hn |
| have h1 : opNorm (vecMulVec (fun _ : Fin n=>1) (spinVariance y)) ≤ (n:ℝ) := by |
| apply (operator_outer_vector_le _ _).trans |
| rw [vectorNorm_const_one] |
| exact (mul_le_mul_of_nonneg_left (spinVariance_norm_bound y) (sqrt_nonneg _)).trans_eq (mul_self_sqrt hn0.le) |
| have h2 : opNorm (vecMulVec (magnetization y) (fun i=>magnetization y i*spinVariance y i)) ≤ (n:ℝ) := by |
| apply (operator_outer_vector_le _ _).trans |
| exact (mul_le_mul (magnetization_norm_bound y) (weightedMagnetization_norm_bound y) |
| (vectorNorm_nonneg _) (sqrt_nonneg _)).trans_eq (mul_self_sqrt hn0.le) |
| have hh := real_opNorm_add ((-j/(n:ℝ)) • vecMulVec (fun _=>1) (spinVariance y)) |
| ((-2*j/(n:ℝ)) • vecMulVec (magnetization y) (fun i=>magnetization y i*spinVariance y i)) |
| change opNorm (plantedJacobianPerturbation j y) ≤ _ at hh |
| simp only [real_opNorm_smul,abs_div,abs_of_pos hn0,abs_neg,abs_of_nonneg hj] at hh |
| have hab : |-(2:ℝ)*j|=2*j := by rw [abs_mul,abs_neg,abs_of_nonneg (by norm_num : (0:ℝ) ≤ 2),abs_of_nonneg hj] |
| rw [hab] at hh |
| apply hh.trans |
| have ha := mul_le_mul_of_nonneg_left h1 (div_nonneg hj hn0.le) |
| have hb := mul_le_mul_of_nonneg_left h2 (div_nonneg (mul_nonneg (by norm_num : (0:ℝ) ≤ 2) hj) hn0.le) |
| have he1 : j/(n:ℝ)*(n:ℝ)=j := div_mul_cancel₀ j hn0.ne' |
| have he2 : 2*j/(n:ℝ)*(n:ℝ)=2*j := div_mul_cancel₀ (2*j) hn0.ne' |
| rw [he1] at ha |
| rw [he2] at hb |
| linarith |
|
|
| lemma fieldJacobian_planted_formula {n : ℕ} (hn : 0 < n) (j : ℝ) |
| (W : Matrix (Fin n) (Fin n) ℝ) (y : Field n) : |
| fieldJacobian j (plantedInteraction j W) y= |
| 1+((j*coordAverage (spinVariance y)) • 1-W)*diagonal (spinVariance y)+ |
| plantedJacobianPerturbation j y := by |
| have hb : coordAverage (spinVariance y)=1-overlap y := by |
| simpa only [coordAverage,Fintype.card_fin,varianceAverage] using (varianceAverage_eq_average hn y).symm |
| ext i k |
| simp only [Matrix.add_apply] |
| erw [Matrix.mul_diagonal] |
| simp only [fieldJacobian,hessianCore,plantedInteraction,plantedJacobianPerturbation, |
| Matrix.add_apply,Matrix.sub_apply,Matrix.smul_apply,smul_eq_mul,vecMulVec_apply,hb,onsager, |
| Matrix.one_apply] |
| by_cases he : i=k <;> simp only [he,↓reduceIte] <;> ring |
| end SKGap |
| end |
| end |
|
|
| end OAI |
|
|
