import OAI.MathematicalPhysics.CriticalSK.SpinModel namespace OAI noncomputable section open scoped BigOperators Topology NNReal ENNReal open MeasureTheory ProbabilityTheory open scoped ENNReal NNReal open scoped BigOperators InnerProductSpace open Module open scoped BigOperators ENNReal NNReal Real Topology open MeasureTheory ProbabilityTheory Filter open scoped BigOperators NNReal open scoped BigOperators open Matrix Polynomial open scoped BigOperators Topology open Filter open scoped BigOperators NNReal ENNReal Topology Pointwise Matrix.Norms.Elementwise open Set Metric MeasureTheory MeasureTheory.Measure open scoped ENNReal NNReal Topology open MeasureTheory MeasureTheory.Measure Set Metric open scoped BigOperators ENNReal Topology open Set MeasureTheory open scoped BigOperators ENNReal open MeasureTheory namespace CriticalSK section variable {ι : Type*} [Fintype ι] [DecidableEq ι] lemma orthogonal_entry_bound (U : Matrix.orthogonalGroup ι ℝ) (i j : ι) : |(U : Matrix ι ι ℝ) i j| ≤ 1 := by have h : (∑ k : ι, ((U : Matrix ι ι ℝ) k j)^2) = 1 := by have he := congrArg (fun A : Matrix ι ι ℝ => A j j) (Unitary.star_mul_self_of_mem U.property) simpa only [Matrix.mul_apply, Matrix.star_apply, star_trivial, Matrix.one_apply_eq, ← sq] using he have hl : ((U : Matrix ι ι ℝ) i j)^2 ≤ 1 := by rw [← h] exact Finset.single_le_sum (f := fun k => ((U : Matrix ι ι ℝ) k j)^2) (fun k _ => sq_nonneg _) (Finset.mem_univ i) nlinarith [sq_abs ((U : Matrix ι ι ℝ) i j), abs_nonneg ((U : Matrix ι ι ℝ) i j)] instance orthogonalGroup_compactSpace : CompactSpace (Matrix.orthogonalGroup ι ℝ) := by apply isCompact_iff_compactSpace.mp apply Metric.isCompact_of_isClosed_isBounded (isClosed_unitary (R := Matrix ι ι ℝ)) apply Metric.isBounded_iff_subset_closedBall (0 : Matrix ι ι ℝ) |>.mpr refine ⟨1, ?_⟩ intro A hA rw [mem_closedBall, dist_zero_right] apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ) ≤ 1)).mpr intro i apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ) ≤ 1)).mpr intro j exact orthogonal_entry_bound ⟨A,hA⟩ i j instance orthogonalGroup_measurableSpace : MeasurableSpace (Matrix.orthogonalGroup ι ℝ) := borel _ instance orthogonalGroup_borelSpace : BorelSpace (Matrix.orthogonalGroup ι ℝ) := ⟨rfl⟩ def orthogonalHaar (ι : Type*) [Fintype ι] [DecidableEq ι] : Measure (Matrix.orthogonalGroup ι ℝ) := Measure.haarMeasure ⟨⟨univ, isCompact_univ⟩, by simpa only [interior_univ] using Set.univ_nonempty⟩ instance orthogonalHaar_probability : IsProbabilityMeasure (orthogonalHaar ι) := by constructor exact Measure.haarMeasure_self instance orthogonalHaar_leftInvariant : IsMulLeftInvariant (orthogonalHaar ι) := by unfold orthogonalHaar infer_instance instance orthogonalHaar_haar : IsHaarMeasure (orthogonalHaar ι) := by unfold orthogonalHaar infer_instance instance orthogonalHaar_rightInvariant : IsMulRightInvariant (orthogonalHaar ι) := by constructor intro g let ν := (orthogonalHaar ι).map (· * g) have hν : IsProbabilityMeasure ν := inferInstance have he := isMulInvariant_eq_smul_of_compactSpace ν (orthogonalHaar ι) have hf := congrArg (fun μ : Measure (Matrix.orthogonalGroup ι ℝ) => μ univ) he simp only [measure_univ, Measure.smul_apply, ENNReal.smul_def, smul_eq_mul, mul_one] at hf have hf' : haarScalarFactor ν (orthogonalHaar ι) = 1 := by exact_mod_cast hf.symm rw [hf', one_smul] at he exact he lemma euclideanMatrix_mul_apply (A B : Matrix ι ι ℝ) (x : EuclideanSpace ℝ ι) : (A*B).toEuclideanLin x = A.toEuclideanLin (B.toEuclideanLin x) := by exact congrArg (fun L : EuclideanSpace ℝ ι →ₗ[ℝ] EuclideanSpace ℝ ι => L x) (Matrix.toLpLin_mul 2 2 2 A B) lemma euclideanMatrix_one_apply (x : EuclideanSpace ℝ ι) : (1 : Matrix ι ι ℝ).toEuclideanLin x = x := by simp only [Matrix.toEuclideanLin, Matrix.toLpLin_one, LinearMap.id_apply] lemma orthogonal_inner_map (U : Matrix.orthogonalGroup ι ℝ) (x y : EuclideanSpace ℝ ι) : inner ℝ ((U : Matrix ι ι ℝ).toEuclideanLin x) ((U : Matrix ι ι ℝ).toEuclideanLin y) = inner ℝ x y := by rw [← LinearMap.adjoint_inner_right, ← Matrix.toEuclideanLin_conjTranspose_eq_adjoint, ← euclideanMatrix_mul_apply] rw [← Matrix.star_eq_conjTranspose, Unitary.star_mul_self_of_mem U.property, euclideanMatrix_one_apply] def orthogonalIsometry (U : Matrix.orthogonalGroup ι ℝ) : EuclideanSpace ℝ ι ≃ₗᵢ[ℝ] EuclideanSpace ℝ ι where toLinearMap := (U : Matrix ι ι ℝ).toEuclideanLin invFun := (star (U : Matrix ι ι ℝ)).toEuclideanLin left_inv x := by change (star (U : Matrix ι ι ℝ)).toEuclideanLin ((U : Matrix ι ι ℝ).toEuclideanLin x) = x rw [← euclideanMatrix_mul_apply, Unitary.star_mul_self_of_mem U.property, euclideanMatrix_one_apply] right_inv x := by change (U : Matrix ι ι ℝ).toEuclideanLin ((star (U : Matrix ι ι ℝ)).toEuclideanLin x) = x rw [← euclideanMatrix_mul_apply, Unitary.mul_star_self_of_mem U.property, euclideanMatrix_one_apply] norm_map' := (LinearMap.norm_map_iff_inner_map_map _).mpr (orthogonal_inner_map U) lemma orthogonalIsometry_apply (U : Matrix.orthogonalGroup ι ℝ) (x : EuclideanSpace ℝ ι) (i : ι) : orthogonalIsometry U x i = ∑ j, (U : Matrix ι ι ℝ) i j * x j := rfl lemma orthogonalIsometry_mul (U V : Matrix.orthogonalGroup ι ℝ) (x : EuclideanSpace ℝ ι) : orthogonalIsometry (U*V) x = orthogonalIsometry U (orthogonalIsometry V x) := euclideanMatrix_mul_apply _ _ _ lemma orthogonalIsometry_continuous : Continuous (fun p : Matrix.orthogonalGroup ι ℝ × EuclideanSpace ℝ ι => orthogonalIsometry p.1 p.2) := by change Continuous (fun p : Matrix.orthogonalGroup ι ℝ × EuclideanSpace ℝ ι => WithLp.toLp 2 (fun i => ∑ j, (p.1 : Matrix ι ι ℝ) i j * p.2 j)) apply (PiLp.continuous_toLp 2 (fun _ : ι => ℝ)).comp apply continuous_pi intro i apply continuous_finsetSum intro j _ have hU : Continuous (fun p : Matrix.orthogonalGroup ι ℝ × EuclideanSpace ℝ ι => (p.1 : Matrix ι ι ℝ)) := continuous_subtype_val.comp continuous_fst exact ((continuous_apply j).comp ((continuous_apply i).comp hU)).mul ((PiLp.continuous_apply 2 _ j).comp continuous_snd) lemma orthogonalIsometry_fixed_continuous (x : EuclideanSpace ℝ ι) : Continuous (fun U : Matrix.orthogonalGroup ι ℝ => orthogonalIsometry U x) := by change Continuous (fun U : Matrix.orthogonalGroup ι ℝ => WithLp.toLp 2 (fun i => ∑ j, (U : Matrix ι ι ℝ) i j * x j)) apply (PiLp.continuous_toLp 2 (fun _ : ι => ℝ)).comp apply continuous_pi intro i apply continuous_finsetSum intro j _ have hU : Continuous (fun U : Matrix.orthogonalGroup ι ℝ => (U : Matrix ι ι ℝ)) := continuous_subtype_val exact ((continuous_apply j).comp ((continuous_apply i).comp hU)).mul continuous_const def matrixOfIsometry (U : EuclideanSpace ℝ ι ≃ₗᵢ[ℝ] EuclideanSpace ℝ ι) : Matrix.orthogonalGroup ι ℝ := ⟨U.toMatrix (EuclideanSpace.basisFun ι ℝ).toBasis (EuclideanSpace.basisFun ι ℝ).toBasis, U.toMatrix_mem_unitaryGroup _ _⟩ lemma orthogonalIsometry_matrixOfIsometry (U : EuclideanSpace ℝ ι ≃ₗᵢ[ℝ] EuclideanSpace ℝ ι) : orthogonalIsometry (matrixOfIsometry U) = U := by ext x i change (Matrix.toEuclideanLin (LinearMap.toMatrix (EuclideanSpace.basisFun ι ℝ).toBasis (EuclideanSpace.basisFun ι ℝ).toBasis U.toLinearMap) x) i = U x i rw [Matrix.toEuclideanLin_eq_toLin_orthonormal, Matrix.toLin_toMatrix] rfl lemma orthogonal_transitive {x y : EuclideanSpace ℝ ι} (hxy : ‖x‖ = ‖y‖) : ∃ U : Matrix.orthogonalGroup ι ℝ, orthogonalIsometry U x = y := by refine ⟨matrixOfIsometry (Submodule.reflection (ℝ ∙ (x-y))ᗮ), ?_⟩ rw [orthogonalIsometry_matrixOfIsometry] exact Submodule.reflection_sub hxy end section variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] def sphereMap (U : E ≃ₗᵢ[ℝ] E) (u : sphere (0:E) 1) : sphere (0:E) 1 := ⟨U u.val, mem_sphere_zero_iff_norm.mpr (by rw [U.norm_map]; exact mem_sphere_zero_iff_norm.mp u.property)⟩ lemma sphereMap_continuous (U : E ≃ₗᵢ[ℝ] E) : Continuous (sphereMap U) := by exact (U.continuous.comp continuous_subtype_val).subtype_mk _ lemma sphereMap_symm (U : E ≃ₗᵢ[ℝ] E) (u : sphere (0:E) 1) : sphereMap U (sphereMap U.symm u) = u := by apply Subtype.ext exact U.apply_symm_apply u.val lemma sphereMap_cone_preimage (U : E ≃ₗᵢ[ℝ] E) (A : Set (sphere (0:E) 1)) : Ioo (0:ℝ) 1 • (Subtype.val '' (sphereMap U ⁻¹' A)) = U ⁻¹' (Ioo (0:ℝ) 1 • (Subtype.val '' A)) := by ext x constructor · rintro ⟨r, hr, y, ⟨u, hu, rfl⟩, rfl⟩ exact ⟨r, hr, U u.val, ⟨sphereMap U u, hu, rfl⟩, (U.map_smul r u.val).symm⟩ · rintro ⟨r, hr, y, ⟨u, hu, rfl⟩, hy⟩ refine ⟨r, hr, U.symm u.val, ⟨sphereMap U.symm u, ?_, rfl⟩, ?_⟩ · change sphereMap U (sphereMap U.symm u) ∈ A rwa [sphereMap_symm] · apply U.injective rw [U.map_smul, U.apply_symm_apply] exact hy variable [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] lemma sphereMap_preserving (U : E ≃ₗᵢ[ℝ] E) : MeasurePreserving (sphereMap U) (volume : Measure E).toSphere (volume : Measure E).toSphere := by constructor · exact (sphereMap_continuous U).measurable · ext A hA rw [Measure.map_apply (sphereMap_continuous U).measurable hA, Measure.toSphere_apply' _ ((sphereMap_continuous U).measurable hA), Measure.toSphere_apply' _ hA, sphereMap_cone_preimage, U.measurePreserving.measure_preimage_emb U.toHomeomorph.measurableEmbedding] end section variable {ι : Type*} [Fintype ι] [DecidableEq ι] abbrev unitSphere (ι : Type*) [Fintype ι] := Metric.sphere (0 : EuclideanSpace ℝ ι) 1 def orthogonalSphere (U : Matrix.orthogonalGroup ι ℝ) (u : unitSphere ι) : unitSphere ι := sphereMap (orthogonalIsometry U) u lemma orthogonalSphere_continuous : Continuous (fun p : Matrix.orthogonalGroup ι ℝ × unitSphere ι => orthogonalSphere p.1 p.2) := by exact (orthogonalIsometry_continuous.comp (continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd))).subtype_mk _ lemma orthogonalSphere_mul (U V : Matrix.orthogonalGroup ι ℝ) (u : unitSphere ι) : orthogonalSphere (U*V) u = orthogonalSphere U (orthogonalSphere V u) := by apply Subtype.ext exact orthogonalIsometry_mul U V u.val lemma orthogonalSphere_transitive (u w : unitSphere ι) : ∃ U : Matrix.orthogonalGroup ι ℝ, orthogonalSphere U u = w := by obtain ⟨U,hU⟩ := orthogonal_transitive ((mem_sphere_zero_iff_norm.mp u.property).trans (mem_sphere_zero_iff_norm.mp w.property).symm) exact ⟨U, Subtype.ext hU⟩ lemma orthogonalSphere_integral (U : Matrix.orthogonalGroup ι ℝ) (f : unitSphere ι → ℝ) : (∫ u, f (orthogonalSphere U u) ∂(volume : Measure (EuclideanSpace ℝ ι)).toSphere) = ∫ u, f u ∂(volume : Measure (EuclideanSpace ℝ ι)).toSphere := by apply (sphereMap_preserving (orthogonalIsometry U)).integral_comp apply (sphereMap_continuous (orthogonalIsometry U)).measurableEmbedding intro u w h apply Subtype.ext exact (orthogonalIsometry U).injective (congrArg Subtype.val h) lemma orthogonal_orbit_integral_eq (f : unitSphere ι → ℝ) (u w : unitSphere ι) : (∫ U, f (orthogonalSphere U u) ∂orthogonalHaar ι) = ∫ U, f (orthogonalSphere U w) ∂orthogonalHaar ι := by obtain ⟨V,hV⟩ := orthogonalSphere_transitive u w rw [← hV] simp_rw [← orthogonalSphere_mul] exact (integral_mul_right_eq_self (fun U => f (orthogonalSphere U u)) V).symm variable [Nonempty ι] def sphereArea : ℝ := (volume : Measure (EuclideanSpace ℝ ι)).toSphere.real Set.univ def sphereAverage (f : unitSphere ι → ℝ) : ℝ := (∫ u, f u ∂(volume : Measure (EuclideanSpace ℝ ι)).toSphere) / sphereArea (ι := ι) omit [DecidableEq ι] in lemma sphereArea_pos : 0 < sphereArea (ι := ι) := by exact ENNReal.toReal_pos ((measure_univ_pos.mpr (Measure.toSphere_ne_zero _)).ne') (measure_ne_top _ _) lemma orthogonal_first_moment (f : unitSphere ι → ℝ) (hf : Continuous f) (u : unitSphere ι) : (∫ U, f (orthogonalSphere U u) ∂orthogonalHaar ι) = sphereAverage f := by let σ := (volume : Measure (EuclideanSpace ℝ ι)).toSphere let J := fun w : unitSphere ι => ∫ U, f (orthogonalSphere U w) ∂orthogonalHaar ι have hc : Continuous (fun p : Matrix.orthogonalGroup ι ℝ × unitSphere ι => f (orthogonalSphere p.1 p.2)) := hf.comp orthogonalSphere_continuous have hi : Integrable (fun p : Matrix.orthogonalGroup ι ℝ × unitSphere ι => f (orthogonalSphere p.1 p.2)) ((orthogonalHaar ι).prod σ) := by simpa only [integrableOn_univ] using hc.continuousOn.integrableOn_compact isCompact_univ have he : sphereArea (ι := ι) * J u = ∫ w, f w ∂σ := by calc _ = ∫ _w : unitSphere ι, J u ∂σ := by rw [integral_const, smul_eq_mul] rfl _ = ∫ w : unitSphere ι, J w ∂σ := integral_congr_ae (Filter.Eventually.of_forall (fun w => orthogonal_orbit_integral_eq f u w)) _ = ∫ U, ∫ w : unitSphere ι, f (orthogonalSphere U w) ∂σ ∂orthogonalHaar ι := (integral_integral_swap hi).symm _ = ∫ _U : Matrix.orthogonalGroup ι ℝ, ∫ w : unitSphere ι, f w ∂σ ∂orthogonalHaar ι := by apply integral_congr_ae exact Filter.Eventually.of_forall (fun U => orthogonalSphere_integral U f) _ = _ := by simp exact (eq_div_iff (ne_of_gt (sphereArea_pos (ι := ι)))).mpr (by simpa only [mul_comm] using he) end section variable {ι : Type*} [Fintype ι] [DecidableEq ι] [Nonempty ι] def sphereUniform : Measure (unitSphere ι) := ((volume : Measure (EuclideanSpace ℝ ι)).toSphere Set.univ)⁻¹ • (volume : Measure (EuclideanSpace ℝ ι)).toSphere instance sphereUniform_probability : IsProbabilityMeasure (sphereUniform (ι := ι)) := by constructor rw [sphereUniform, Measure.smul_apply, smul_eq_mul] exact ENNReal.inv_mul_cancel (measure_univ_pos.mpr (Measure.toSphere_ne_zero _)).ne' (measure_ne_top _ _) omit [DecidableEq ι] [Nonempty ι] in lemma sphereUniform_integral (f : unitSphere ι → ℝ) : (∫ u, f u ∂sphereUniform) = sphereAverage f := by rw [sphereUniform, integral_smul_measure, ENNReal.toReal_inv, smul_eq_mul] exact inv_mul_eq_div _ _ lemma orthogonal_sphere_preserving (u : unitSphere ι) : MeasurePreserving (fun U => orthogonalSphere U u) (orthogonalHaar ι) sphereUniform := by have hc : Continuous (fun U : Matrix.orthogonalGroup ι ℝ => orthogonalSphere U u) := by apply Continuous.subtype_mk exact orthogonalIsometry_fixed_continuous u.val refine ⟨hc.measurable, ?_⟩ apply ext_of_forall_integral_eq_of_IsFiniteMeasure intro f rw [integral_map hc.measurable.aemeasurable f.continuous.measurable.aestronglyMeasurable, sphereUniform_integral] exact orthogonal_first_moment f f.continuous u end section variable {ι : Type*} [Fintype ι] [DecidableEq ι] lemma orthogonal_pair_transitive {x y x' y' : EuclideanSpace ℝ ι} (hx : ‖x‖ = ‖x'‖) (hy : ‖y‖ = ‖y'‖) (hxy : inner ℝ x y = inner ℝ x' y') : ∃ U : Matrix.orthogonalGroup ι ℝ, orthogonalIsometry U x = x' ∧ orthogonalIsometry U y = y' := by let A := Submodule.reflection (ℝ ∙ (x-x'))ᗮ have hA : A x = x' := Submodule.reflection_sub hx let B := Submodule.reflection (ℝ ∙ (A y-y'))ᗮ have hBx : B x' = x' := by apply Submodule.reflection_mem_subspace_eq_self rw [Submodule.mem_orthogonal_singleton_iff_inner_left, inner_sub_right] have he : inner ℝ x' (A y) = inner ℝ x y := by rw [← hA, A.inner_map_map] rw [he, hxy, sub_self] have hBy : B (A y) = y' := Submodule.reflection_sub ((A.norm_map y).trans hy) refine ⟨matrixOfIsometry (A.trans B), ?_, ?_⟩ · rw [orthogonalIsometry_matrixOfIsometry] change B (A x) = x' rw [hA, hBx] · rw [orthogonalIsometry_matrixOfIsometry] exact hBy lemma orthogonal_pair_integral_eq (f : EuclideanSpace ℝ ι → EuclideanSpace ℝ ι → ℝ) {x y x' y' : EuclideanSpace ℝ ι} (hx : ‖x‖ = ‖x'‖) (hy : ‖y‖ = ‖y'‖) (hxy : inner ℝ x y = inner ℝ x' y') : (∫ U, f (orthogonalIsometry U x) (orthogonalIsometry U y) ∂orthogonalHaar ι) = ∫ U, f (orthogonalIsometry U x') (orthogonalIsometry U y') ∂orthogonalHaar ι := by obtain ⟨V,hVx,hVy⟩ := orthogonal_pair_transitive hx hy hxy rw [← hVx, ← hVy] simp_rw [← orthogonalIsometry_mul] exact (integral_mul_right_eq_self (fun U => f (orthogonalIsometry U x) (orthogonalIsometry U y)) V).symm lemma orthogonal_pair_swap (f : EuclideanSpace ℝ ι → EuclideanSpace ℝ ι → ℝ) {x y : EuclideanSpace ℝ ι} (hxy : ‖x‖ = ‖y‖) : (∫ U, f (orthogonalIsometry U x) (orthogonalIsometry U y) ∂orthogonalHaar ι) = ∫ U, f (orthogonalIsometry U y) (orthogonalIsometry U x) ∂orthogonalHaar ι := by exact orthogonal_pair_integral_eq f hxy hxy.symm (real_inner_comm _ _) def diagonalEnergy (lam : ι → ℝ) (x : EuclideanSpace ℝ ι) : ℝ := ∑ i, lam i * x i ^ 2 omit [DecidableEq ι] in lemma diagonalEnergy_continuous (lam : ι → ℝ) : Continuous (diagonalEnergy lam) := by apply continuous_finsetSum intro i _ exact continuous_const.mul ((PiLp.continuous_apply 2 _ i).pow 2) lemma orthogonalHaar_integrable {f : Matrix.orthogonalGroup ι ℝ → ℝ} (hf : Continuous f) : Integrable f (orthogonalHaar ι) := by simpa only [integrableOn_univ] using hf.continuousOn.integrableOn_compact isCompact_univ def pairIntegrand (lam : ι → ℝ) (x y : EuclideanSpace ℝ ι) (q : ℝ) (U : Matrix.orthogonalGroup ι ℝ) : ℝ := Real.exp (((1+q)*diagonalEnergy lam (orthogonalIsometry U x) + (1-q)*diagonalEnergy lam (orthogonalIsometry U y))/2) def pairMoment (lam : ι → ℝ) (x y : EuclideanSpace ℝ ι) (q : ℝ) : ℝ := ∫ U, pairIntegrand lam x y q U ∂orthogonalHaar ι lemma pairIntegrand_continuous (lam : ι → ℝ) (x y : EuclideanSpace ℝ ι) (q : ℝ) : Continuous (pairIntegrand lam x y q) := by have hx := (diagonalEnergy_continuous lam).comp (orthogonalIsometry_fixed_continuous x) have hy := (diagonalEnergy_continuous lam).comp (orthogonalIsometry_fixed_continuous y) exact ((continuous_const.mul hx).add (continuous_const.mul hy)).div_const 2 |>.rexp lemma pairMoment_nonneg (lam : ι → ℝ) (x y : EuclideanSpace ℝ ι) (q : ℝ) : 0 ≤ pairMoment lam x y q := integral_nonneg (fun _ => (Real.exp_pos _).le) lemma pairMoment_even (lam : ι → ℝ) {x y : EuclideanSpace ℝ ι} (hxy : ‖x‖ = ‖y‖) (q : ℝ) : pairMoment lam x y (-q) = pairMoment lam x y q := by unfold pairMoment pairIntegrand rw [orthogonal_pair_swap (fun a b => Real.exp (((1+-q)*diagonalEnergy lam a + (1- -q)*diagonalEnergy lam b)/2)) hxy] apply integral_congr_ae filter_upwards [] with U congr 1 ring lemma exp_pair_cosh (A D q : ℝ) : Real.exp ((A+D)/2) * Real.cosh (q*(A-D)/2) = (Real.exp (((1+q)*A+(1-q)*D)/2) + Real.exp (((1-q)*A+(1+q)*D)/2))/2 := by rw [Real.cosh_eq, ← mul_div_assoc, mul_add, ← Real.exp_add, ← Real.exp_add] congr 2 <;> ring_nf lemma pairMoment_cosh (lam : ι → ℝ) {x y : EuclideanSpace ℝ ι} (hxy : ‖x‖ = ‖y‖) (q : ℝ) : pairMoment lam x y q = ∫ U, Real.exp ((diagonalEnergy lam (orthogonalIsometry U x) + diagonalEnergy lam (orthogonalIsometry U y))/2) * Real.cosh (q*(diagonalEnergy lam (orthogonalIsometry U x) - diagonalEnergy lam (orthogonalIsometry U y))/2) ∂orthogonalHaar ι := by simp_rw [exp_pair_cosh] have hf (U : Matrix.orthogonalGroup ι ℝ) : Real.exp (((1-q)*diagonalEnergy lam (orthogonalIsometry U x)+ (1+q)*diagonalEnergy lam (orthogonalIsometry U y))/2) = pairIntegrand lam x y (-q) U := by unfold pairIntegrand congr 1 ring simp_rw [hf] rw [integral_div] change pairMoment lam x y q = (∫ U, pairIntegrand lam x y q U + pairIntegrand lam x y (-q) U ∂orthogonalHaar ι)/2 rw [integral_add (orthogonalHaar_integrable (pairIntegrand_continuous lam x y q)) (orthogonalHaar_integrable (pairIntegrand_continuous lam x y (-q)))] change pairMoment lam x y q = (pairMoment lam x y q + pairMoment lam x y (-q))/2 rw [pairMoment_even lam hxy] ring lemma pairMoment_mono_abs (lam : ι → ℝ) {x y : EuclideanSpace ℝ ι} (hxy : ‖x‖ = ‖y‖) {q r : ℝ} (hqr : |q| ≤ |r|) : pairMoment lam x y q ≤ pairMoment lam x y r := by rw [pairMoment_cosh lam hxy q, pairMoment_cosh lam hxy r] have hx := (diagonalEnergy_continuous lam).comp (orthogonalIsometry_fixed_continuous x) have hy := (diagonalEnergy_continuous lam).comp (orthogonalIsometry_fixed_continuous y) have hc (t : ℝ) : Continuous (fun U : Matrix.orthogonalGroup ι ℝ => Real.exp ((diagonalEnergy lam (orthogonalIsometry U x) + diagonalEnergy lam (orthogonalIsometry U y))/2) * Real.cosh (t*(diagonalEnergy lam (orthogonalIsometry U x) - diagonalEnergy lam (orthogonalIsometry U y))/2)) := by exact ((hx.add hy).div_const 2).rexp.mul (Real.continuous_cosh.comp ((continuous_const.mul (hx.sub hy)).div_const 2)) apply integral_mono (orthogonalHaar_integrable (hc q)) (orthogonalHaar_integrable (hc r)) intro U apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le apply Real.cosh_le_cosh.mpr simp only [abs_div, abs_mul] exact div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_right hqr (abs_nonneg _)) (abs_nonneg _) end section variable {ι : Type*} [Fintype ι] [DecidableEq ι] [Nonempty ι] def spherePartition (lam : ι → ℝ) (a r : ℝ) : ℝ := sphereAverage (fun u => Real.exp (a / 2 * diagonalEnergy lam (r • u.val))) omit [DecidableEq ι] [Nonempty ι] in lemma sphere_energy_continuous (lam : ι → ℝ) (a r : ℝ) : Continuous (fun u : unitSphere ι => Real.exp (a / 2 * diagonalEnergy lam (r • u.val))) := by have hscale : Continuous (fun u : unitSphere ι => r • (u.val : EuclideanSpace ℝ ι)) := (continuous_const_smul r).comp continuous_subtype_val have henergy := (diagonalEnergy_continuous lam).comp hscale exact (henergy.const_mul (a / 2)).rexp lemma orthogonal_scaled_partition (lam : ι → ℝ) (a r : ℝ) (u : unitSphere ι) : (∫ U, Real.exp (a / 2 * diagonalEnergy lam (orthogonalIsometry U (r • u.val))) ∂orthogonalHaar ι) = spherePartition lam a r := by calc _ = ∫ U, Real.exp (a / 2 * diagonalEnergy lam (r • (orthogonalSphere U u).val)) ∂orthogonalHaar ι := by apply integral_congr_ae filter_upwards [] with U rw [(orthogonalIsometry U).map_smul] rfl _ = _ := orthogonal_first_moment _ (sphere_energy_continuous lam a r) u lemma orthogonal_radial_partition (lam : ι → ℝ) (a : ℝ) {r : ℝ} (hr : 0 < r) (x : EuclideanSpace ℝ ι) (hx : ‖x‖ = r) : (∫ U, Real.exp (a/2 * diagonalEnergy lam (orthogonalIsometry U x)) ∂orthogonalHaar ι) = spherePartition lam a r := by have hu : ‖r⁻¹ • x‖ = 1 := by rw [norm_smul, Real.norm_of_nonneg (inv_nonneg.mpr hr.le), hx, inv_mul_cancel₀ hr.ne'] let u : unitSphere ι := ⟨r⁻¹ • x, mem_sphere_zero_iff_norm.mpr hu⟩ have he : r • u.val = x := by change r • (r⁻¹ • x) = x rw [smul_smul, mul_inv_cancel₀ hr.ne', one_smul] calc _ = ∫ U, Real.exp (a/2 * diagonalEnergy lam (orthogonalIsometry U (r • u.val))) ∂orthogonalHaar ι := by simp only [he] _ = _ := orthogonal_scaled_partition lam a r u end section lemma cubeVector_norm {n : ℕ} (x : Spin n) : ‖cubeVector x‖ = Real.sqrt n := by apply (sq_eq_sq₀ (norm_nonneg _) (Real.sqrt_nonneg _)).mp rw [Real.sq_sqrt (Nat.cast_nonneg _), EuclideanSpace.real_norm_sq_eq] change ∑ i : Fin n, spinValue (x i)^2 = (n : ℝ) simp only [spinValue_sq, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, mul_one] def rotatedCubePartition {n : ℕ} (lam : Fin n → ℝ) (a : ℝ) (U : Matrix.orthogonalGroup (Fin n) ℝ) : ℝ := (Fintype.card (Spin n) : ℝ)⁻¹ * ∑ x : Spin n, Real.exp (a/2 * diagonalEnergy lam (orthogonalIsometry U (cubeVector x))) lemma rotatedCubePartition_first_moment {n : ℕ} (hn : 0 < n) (lam : Fin n → ℝ) (a : ℝ) : (∫ U, rotatedCubePartition lam a U ∂orthogonalHaar (Fin n)) = spherePartition lam a (Real.sqrt n) := by let : Nonempty (Fin n) := ⟨⟨0, hn⟩⟩ unfold rotatedCubePartition rw [integral_const_mul, integral_finsetSum] · simp_rw [orthogonal_radial_partition lam a (Real.sqrt_pos.mpr (Nat.cast_pos.mpr hn)) _ (cubeVector_norm _)] rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, ← mul_assoc, inv_mul_cancel₀ (by exact_mod_cast Fintype.card_ne_zero (α := Spin n)), one_mul] · intro x _ apply orthogonalHaar_integrable exact (continuous_const.mul ((diagonalEnergy_continuous lam).comp (orthogonalIsometry_fixed_continuous (cubeVector x)))).rexp variable {ι : Type*} [Fintype ι] [DecidableEq ι] def overlapPlus (u v : EuclideanSpace ℝ ι) (q : ℝ) : EuclideanSpace ℝ ι := Real.sqrt ((1+q)/2) • u + Real.sqrt ((1-q)/2) • v def overlapMinus (u v : EuclideanSpace ℝ ι) (q : ℝ) : EuclideanSpace ℝ ι := Real.sqrt ((1+q)/2) • u - Real.sqrt ((1-q)/2) • v omit [DecidableEq ι] in lemma overlapPair_gram {u v : EuclideanSpace ℝ ι} {r q : ℝ} (hu : ‖u‖ = r) (hv : ‖v‖ = r) (horth : inner ℝ u v = 0) (hq : |q| ≤ 1) : ‖overlapPlus u v q‖ = r ∧ ‖overlapMinus u v q‖ = r ∧ inner ℝ (overlapPlus u v q) (overlapMinus u v q) = q * r^2 := by have hp : 0 ≤ (1+q)/2 := by linarith [(abs_le.mp hq).1] have hm : 0 ≤ (1-q)/2 := by linarith [(abs_le.mp hq).2] have hpu := Real.sq_sqrt hp have hmu := Real.sq_sqrt hm have horth' : inner ℝ v u = 0 := by rw [real_inner_comm, horth] have huu : inner ℝ u u = r^2 := by rw [real_inner_self_eq_norm_sq, hu] have hvv : inner ℝ v v = r^2 := by rw [real_inner_self_eq_norm_sq, hv] have hr : 0 ≤ r := hu ▸ norm_nonneg u have hplus : ‖overlapPlus u v q‖^2 = r^2 := by rw [← real_inner_self_eq_norm_sq] simp only [overlapPlus, inner_add_left, inner_add_right, real_inner_smul_left, real_inner_smul_right, horth, horth', huu, hvv] nlinarith have hminus : ‖overlapMinus u v q‖^2 = r^2 := by rw [← real_inner_self_eq_norm_sq] simp only [overlapMinus, inner_sub_left, inner_sub_right, real_inner_smul_left, real_inner_smul_right, horth, horth', huu, hvv] nlinarith refine ⟨(sq_eq_sq₀ (norm_nonneg _) hr).mp hplus, (sq_eq_sq₀ (norm_nonneg _) hr).mp hminus, ?_⟩ simp only [overlapPlus, overlapMinus, inner_add_left, inner_sub_right, real_inner_smul_left, real_inner_smul_right, horth, horth', huu, hvv] nlinarith omit [DecidableEq ι] in lemma diagonalEnergy_overlap (lam : ι → ℝ) (u v : EuclideanSpace ℝ ι) {q : ℝ} (hq : |q| ≤ 1) : diagonalEnergy lam (overlapPlus u v q) + diagonalEnergy lam (overlapMinus u v q) = (1+q)*diagonalEnergy lam u + (1-q)*diagonalEnergy lam v := by have hp : 0 ≤ (1+q)/2 := by linarith [(abs_le.mp hq).1] have hm : 0 ≤ (1-q)/2 := by linarith [(abs_le.mp hq).2] have hpu := Real.sq_sqrt hp have hmu := Real.sq_sqrt hm unfold diagonalEnergy rw [← Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum, ← Finset.sum_add_distrib] apply Finset.sum_congr rfl intro i _ simp only [overlapPlus, overlapMinus, PiLp.add_apply, PiLp.sub_apply, PiLp.smul_apply, smul_eq_mul] calc _ = 2 * lam i * (Real.sqrt ((1+q)/2))^2 * (u i)^2 + 2 * lam i * (Real.sqrt ((1-q)/2))^2 * (v i)^2 := by ring _ = _ := by rw [hpu, hmu]; ring lemma pair_energy_orbit (lam : ι → ℝ) {x y u v : EuclideanSpace ℝ ι} {r q : ℝ} (hx : ‖x‖ = r) (hy : ‖y‖ = r) (hu : ‖u‖ = r) (hv : ‖v‖ = r) (horth : inner ℝ u v = 0) (hq : |q| ≤ 1) (hxy : inner ℝ x y = q*r^2) : (∫ U, Real.exp ((diagonalEnergy lam (orthogonalIsometry U x) + diagonalEnergy lam (orthogonalIsometry U y))/2) ∂orthogonalHaar ι) = pairMoment lam u v q := by obtain ⟨hplus,hminus,hgram⟩ := overlapPair_gram hu hv horth hq rw [orthogonal_pair_integral_eq (fun a b => Real.exp ((diagonalEnergy lam a + diagonalEnergy lam b)/2)) (hx.trans hplus.symm) (hy.trans hminus.symm) (hxy.trans hgram.symm)] unfold pairMoment pairIntegrand apply integral_congr_ae filter_upwards [] with U have hplus' : orthogonalIsometry U (overlapPlus u v q) = overlapPlus (orthogonalIsometry U u) (orthogonalIsometry U v) q := by simp only [overlapPlus, map_add, map_smul] have hminus' : orthogonalIsometry U (overlapMinus u v q) = overlapMinus (orthogonalIsometry U u) (orthogonalIsometry U v) q := by simp only [overlapMinus, map_sub, map_smul] rw [hplus', hminus', diagonalEnergy_overlap lam _ _ hq] end section variable {ι : Type*} [Fintype ι] [DecidableEq ι] lemma pairMoment_continuous (lam : ι → ℝ) (u v : EuclideanSpace ℝ ι) : Continuous (pairMoment lam u v) := by have hu : Continuous (fun p : ℝ × Matrix.orthogonalGroup ι ℝ => diagonalEnergy lam (orthogonalIsometry p.2 u)) := ((diagonalEnergy_continuous lam).comp (orthogonalIsometry_fixed_continuous u)).comp continuous_snd have hv : Continuous (fun p : ℝ × Matrix.orthogonalGroup ι ℝ => diagonalEnergy lam (orthogonalIsometry p.2 v)) := ((diagonalEnergy_continuous lam).comp (orthogonalIsometry_fixed_continuous v)).comp continuous_snd have hc : Continuous (fun p : ℝ × Matrix.orthogonalGroup ι ℝ => pairIntegrand lam u v p.1 p.2) := ((((continuous_const.add continuous_fst).mul hu).add ((continuous_const.sub continuous_fst).mul hv)).div_const 2).rexp have hi := continuous_parametric_integral_of_continuous (μ := orthogonalHaar ι) (f := fun q U => pairIntegrand lam u v q U) hc (s := Set.univ) isCompact_univ change Continuous (fun q : ℝ => ∫ U, pairIntegrand lam u v q U ∂orthogonalHaar ι) simpa only [setIntegral_univ] using hi variable [Nonempty ι] omit [Nonempty ι] in lemma sphereUniform_rotation_preserving (U : Matrix.orthogonalGroup ι ℝ) : MeasurePreserving (orthogonalSphere U) (sphereUniform (ι := ι)) sphereUniform := (sphereMap_preserving (orthogonalIsometry U)).smul_measure _ omit [Nonempty ι] in lemma orthogonalSphere_injective (U : Matrix.orthogonalGroup ι ℝ) : Function.Injective (orthogonalSphere U) := by intro x y h apply Subtype.ext exact (orthogonalIsometry U).injective (congrArg Subtype.val h) omit [Nonempty ι] in lemma orthogonalSphere_fixed_continuous (U : Matrix.orthogonalGroup ι ℝ) : Continuous (orthogonalSphere U) := sphereMap_continuous (orthogonalIsometry U) lemma spherePair_rotation_integral (U : Matrix.orthogonalGroup ι ℝ) (f : unitSphere ι × unitSphere ι → ℝ) : (∫ p, f (orthogonalSphere U p.1, orthogonalSphere U p.2) ∂(sphereUniform.prod sphereUniform)) = ∫ p, f p ∂(sphereUniform.prod sphereUniform) := by have hp := (sphereUniform_rotation_preserving U).prod (sphereUniform_rotation_preserving U) have hc : Continuous (Prod.map (orthogonalSphere U) (orthogonalSphere U)) := (orthogonalSphere_fixed_continuous U).prodMap (orthogonalSphere_fixed_continuous U) have hinj : Function.Injective (Prod.map (orthogonalSphere U) (orthogonalSphere U)) := by intro a b h exact Prod.ext (orthogonalSphere_injective U (congrArg Prod.fst h)) (orthogonalSphere_injective U (congrArg Prod.snd h)) exact hp.integral_comp (hc.measurableEmbedding hinj) f omit [Fintype ι] [DecidableEq ι] [Nonempty ι] in lemma continuous_pair_first {A B C D : Type*} [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] {f : A × B → D} (hf : Continuous f) : Continuous (fun z : A × (B × C) => f (z.1, z.2.1)) := hf.comp (continuous_fst.prodMk (continuous_fst.comp continuous_snd)) omit [Fintype ι] [DecidableEq ι] [Nonempty ι] in lemma continuous_pair_second {A B C D : Type*} [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] {f : A × C → D} (hf : Continuous f) : Continuous (fun z : A × (B × C) => f (z.1, z.2.2)) := hf.comp (continuous_fst.prodMk (continuous_snd.comp continuous_snd)) omit [Nonempty ι] in lemma spherePair_first_continuous : Continuous (fun z : Matrix.orthogonalGroup ι ℝ × (unitSphere ι × unitSphere ι) => orthogonalSphere z.1 z.2.1) := continuous_pair_first (orthogonalSphere_continuous (ι := ι)) omit [Nonempty ι] in lemma spherePair_second_continuous : Continuous (fun z : Matrix.orthogonalGroup ι ℝ × (unitSphere ι × unitSphere ι) => orthogonalSphere z.1 z.2.2) := continuous_pair_second (orthogonalSphere_continuous (ι := ι)) omit [Nonempty ι] in lemma spherePair_action_continuous : Continuous (fun z : Matrix.orthogonalGroup ι ℝ × (unitSphere ι × unitSphere ι) => (orthogonalSphere z.1 z.2.1, orthogonalSphere z.1 z.2.2)) := (spherePair_first_continuous (ι := ι)).prodMk (spherePair_second_continuous (ι := ι)) lemma spherePair_haar_average (f : unitSphere ι × unitSphere ι → ℝ) (hf : Continuous f) : (∫ p, ∫ U, f (orthogonalSphere U p.1, orthogonalSphere U p.2) ∂orthogonalHaar ι ∂(sphereUniform.prod sphereUniform)) = ∫ p, f p ∂(sphereUniform.prod sphereUniform) := by have hc : Continuous (fun z : Matrix.orthogonalGroup ι ℝ × (unitSphere ι × unitSphere ι) => f (orthogonalSphere z.1 z.2.1, orthogonalSphere z.1 z.2.2)) := hf.comp spherePair_action_continuous have hi : Integrable (fun z : Matrix.orthogonalGroup ι ℝ × (unitSphere ι × unitSphere ι) => f (orthogonalSphere z.1 z.2.1, orthogonalSphere z.1 z.2.2)) ((orthogonalHaar ι).prod (sphereUniform.prod sphereUniform)) := by simpa only [integrableOn_univ] using hc.continuousOn.integrableOn_compact isCompact_univ rw [← integral_integral_swap hi] simp only [spherePair_rotation_integral, integral_const, probReal_univ, one_smul] omit [DecidableEq ι] [Nonempty ι] in lemma sphere_inner_abs_le_one (x y : unitSphere ι) : |inner ℝ x.val y.val| ≤ 1 := by have h := abs_real_inner_le_norm x.val y.val simpa only [mem_sphere_zero_iff_norm.mp x.property, mem_sphere_zero_iff_norm.mp y.property, one_mul] using h def spherePairEnergy (lam : ι → ℝ) (r : ℝ) (p : unitSphere ι × unitSphere ι) : ℝ := Real.exp ((diagonalEnergy lam (r • p.1.val) + diagonalEnergy lam (r • p.2.val))/2) omit [DecidableEq ι] [Nonempty ι] in lemma spherePairEnergy_factor (lam : ι → ℝ) (r : ℝ) (p : unitSphere ι × unitSphere ι) : spherePairEnergy lam r p = Real.exp (1 / 2 * diagonalEnergy lam (r • p.1.val)) * Real.exp (1 / 2 * diagonalEnergy lam (r • p.2.val)) := by rw [← Real.exp_add] unfold spherePairEnergy congr 1 ring omit [DecidableEq ι] [Nonempty ι] in lemma spherePairEnergy_continuous (lam : ι → ℝ) (r : ℝ) : Continuous (spherePairEnergy lam r) := by have hf := sphere_energy_continuous lam 1 r have he : spherePairEnergy lam r = fun p : unitSphere ι × unitSphere ι => Real.exp (1 / 2 * diagonalEnergy lam (r • p.1.val)) * Real.exp (1 / 2 * diagonalEnergy lam (r • p.2.val)) := funext (spherePairEnergy_factor lam r) rw [he] exact (hf.comp continuous_fst).mul (hf.comp continuous_snd) omit [DecidableEq ι] in lemma spherePairEnergy_integral (lam : ι → ℝ) (r : ℝ) : (∫ p, spherePairEnergy lam r p ∂(sphereUniform.prod sphereUniform)) = (spherePartition lam 1 r)^2 := by simp_rw [spherePairEnergy_factor] let f : unitSphere ι → ℝ := fun u => Real.exp (1 / 2 * diagonalEnergy lam (r • u.val)) change (∫ p, f p.1 * f p.2 ∂((sphereUniform (ι := ι)).prod sphereUniform)) = _ rw [integral_prod_mul f f, sphereUniform_integral] exact (pow_two _).symm lemma spectrum_pair_second_moment (lam : ι → ℝ) {u v : EuclideanSpace ℝ ι} {r : ℝ} (hu : ‖u‖ = r) (hv : ‖v‖ = r) (horth : inner ℝ u v = 0) : (∫ p : unitSphere ι × unitSphere ι, pairMoment lam u v (inner ℝ p.1.val p.2.val) ∂(sphereUniform.prod sphereUniform)) = (spherePartition lam 1 r)^2 := by have hr : 0 ≤ r := hu ▸ norm_nonneg u have hnorm (x : unitSphere ι) : ‖r • x.val‖ = r := by rw [norm_smul, Real.norm_of_nonneg hr, mem_sphere_zero_iff_norm.mp x.property, mul_one] have he (p : unitSphere ι × unitSphere ι) : pairMoment lam u v (inner ℝ p.1.val p.2.val) = ∫ U, spherePairEnergy lam r (orthogonalSphere U p.1, orthogonalSphere U p.2) ∂orthogonalHaar ι := by have hxy : inner ℝ (r • p.1.val) (r • p.2.val) = inner ℝ p.1.val p.2.val * r^2 := by rw [real_inner_smul_left, real_inner_smul_right] ring rw [← pair_energy_orbit lam (hnorm p.1) (hnorm p.2) hu hv horth (sphere_inner_abs_le_one p.1 p.2) hxy] apply integral_congr_ae filter_upwards [] with U simp only [spherePairEnergy, orthogonalSphere, sphereMap, map_smul] simp_rw [he] rw [spherePair_haar_average _ (spherePairEnergy_continuous lam r), spherePairEnergy_integral] def sphereOverlapLaw (ι : Type*) [Fintype ι] : Measure ℝ := ((sphereUniform (ι := ι)).prod sphereUniform).map (fun p : unitSphere ι × unitSphere ι => inner ℝ p.1.val p.2.val) lemma sphereOverlap_second_moment (lam : ι → ℝ) {u v : EuclideanSpace ℝ ι} {r : ℝ} (hu : ‖u‖ = r) (hv : ‖v‖ = r) (horth : inner ℝ u v = 0) : (∫ q, pairMoment lam u v q ∂sphereOverlapLaw ι) = (spherePartition lam 1 r)^2 := by have hc : Continuous (fun p : unitSphere ι × unitSphere ι => inner ℝ p.1.val p.2.val) := (continuous_subtype_val.comp continuous_fst).inner (continuous_subtype_val.comp continuous_snd) rw [sphereOverlapLaw, integral_map hc.measurable.aemeasurable (pairMoment_continuous lam u v).aestronglyMeasurable] exact spectrum_pair_second_moment lam hu hv horth end section def spinProduct {n : ℕ} (x y : Spin n) : Spin n := fun i => x i == y i lemma spinValue_beq (a b : Bool) : spinValue (a == b) = spinValue a * spinValue b := by cases a <;> cases b <;> norm_num [spinValue] lemma spinProduct_involution {n : ℕ} (x : Spin n) : Function.Involutive (spinProduct x) := by intro y funext i change (x i == (x i == y i)) = y i cases x i <;> cases y i <;> rfl def spinProductEquiv {n : ℕ} (x : Spin n) : Spin n ≃ Spin n := (spinProduct_involution x).toPerm def signMean {n : ℕ} (x : Spin n) : ℝ := (∑ i, spinValue (x i)) / n lemma cubeOverlap_signMean {n : ℕ} (x y : Spin n) : cubeOverlap x y = signMean (spinProduct x y) := by unfold cubeOverlap signMean cubeVector rw [EuclideanSpace.inner_toLp_toLp] simp only [star_trivial, dotProduct, spinProduct, spinValue_beq, mul_comm] lemma cubeOverlap_abs_le_one {n : ℕ} (x y : Spin n) : |cubeOverlap x y| ≤ 1 := by rcases Nat.eq_zero_or_pos n with rfl | hn · simp [cubeOverlap] have h := abs_real_inner_le_norm (cubeVector x) (cubeVector y) rw [cubeVector_norm, cubeVector_norm, ← sq, Real.sq_sqrt (Nat.cast_nonneg _)] at h unfold cubeOverlap rw [abs_div, show |(n : ℝ)| = n from abs_of_nonneg (Nat.cast_nonneg n)] exact (div_le_one (Nat.cast_pos.mpr hn)).mpr h lemma cubeOverlap_gram {n : ℕ} (hn : 0 < n) (x y : Spin n) : inner ℝ (cubeVector x) (cubeVector y) = cubeOverlap x y * (Real.sqrt n)^2 := by rw [Real.sq_sqrt (Nat.cast_nonneg _)] exact (div_mul_cancel₀ _ (Nat.cast_ne_zero.mpr hn.ne')).symm lemma cubeOverlap_sum {n : ℕ} (x : Spin n) (f : ℝ → ℝ) : (∑ y : Spin n, f (cubeOverlap x y)) = ∑ y : Spin n, f (signMean y) := by simp_rw [cubeOverlap_signMean] exact (spinProductEquiv x).sum_comp (fun y => f (signMean y)) lemma cubeOverlap_average {n : ℕ} (f : ℝ → ℝ) : (Fintype.card (Spin n) : ℝ)⁻¹ ^ 2 * ∑ x : Spin n, ∑ y : Spin n, f (cubeOverlap x y) = (Fintype.card (Spin n) : ℝ)⁻¹ * ∑ y : Spin n, f (signMean y) := by simp_rw [cubeOverlap_sum] rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul] have hn : (Fintype.card (Spin n) : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero field_simp def spinFinsetEquiv (n : ℕ) : Spin n ≃ Finset (Fin n) where toFun x := Finset.univ.filter fun i => x i = true invFun s := fun i => decide (i ∈ s) left_inv x := by funext i simp right_inv s := by ext i simp lemma signMean_finset {n : ℕ} (hn : 0 < n) (x : Spin n) : signMean x = -1 + 2 * ((spinFinsetEquiv n x).card : ℝ) / n := by have he : (∑ i : Fin n, spinValue (x i)) = 2 * ((spinFinsetEquiv n x).card : ℝ) - n := by have hv (i : Fin n) : spinValue (x i) = (if x i = true then 2 else 0) - 1 := by cases x i <;> norm_num [spinValue] simp_rw [hv, Finset.sum_sub_distrib] simp [spinFinsetEquiv, Finset.sum_ite, mul_comm] rw [signMean, he] field_simp [Nat.cast_ne_zero.mpr hn.ne'] ring lemma finset_card_sum (n : ℕ) (f : ℕ → ℝ) : (∑ s : Finset (Fin n), f s.card) = ∑ k ∈ Finset.range (n+1), (Nat.choose n k : ℝ) * f k := by have hm (s : Finset (Fin n)) (_ : s ∈ Finset.univ) : s.card ∈ Finset.range (n+1) := by simp only [Finset.mem_range] exact Nat.lt_succ_of_le (by simpa using s.card_le_univ) rw [← Finset.sum_fiberwise_of_maps_to' hm f] simp only [Finset.sum_const, Finset.univ_filter_card_eq, Finset.card_powersetCard, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] lemma cubeOverlap_binomial_sum {n : ℕ} (hn : 0 < n) (f : ℝ → ℝ) : (Fintype.card (Spin n) : ℝ)⁻¹ ^ 2 * ∑ x : Spin n, ∑ y : Spin n, f (cubeOverlap x y) = (2 : ℝ)⁻¹ ^ n * ∑ k ∈ Finset.range (n+1), (Nat.choose n k : ℝ) * f (-1 + 2 * (k : ℝ) / n) := by rw [cubeOverlap_average] simp_rw [signMean_finset hn] rw [(spinFinsetEquiv n).sum_comp (fun s => f (-1 + 2 * (s.card : ℝ) / n)), finset_card_sum n (fun k => f (-1 + 2 * (k : ℝ) / n))] congr 1 simp [Spin, inv_pow] lemma rotatedCubePartition_continuous {n : ℕ} (lam : Fin n → ℝ) (a : ℝ) : Continuous (rotatedCubePartition lam a) := by apply Continuous.const_mul apply continuous_finsetSum intro x _ exact (continuous_const.mul ((diagonalEnergy_continuous lam).comp (orthogonalIsometry_fixed_continuous (cubeVector x)))).rexp lemma rotatedCubePartition_square {n : ℕ} (lam : Fin n → ℝ) (U : Matrix.orthogonalGroup (Fin n) ℝ) : rotatedCubePartition lam 1 U ^ 2 = (Fintype.card (Spin n) : ℝ)⁻¹ ^ 2 * ∑ x : Spin n, ∑ y : Spin n, Real.exp ((diagonalEnergy lam (orthogonalIsometry U (cubeVector x)) + diagonalEnergy lam (orthogonalIsometry U (cubeVector y)))/2) := by unfold rotatedCubePartition rw [mul_pow, pow_two (∑ _, _), Finset.sum_mul_sum] simp_rw [← Real.exp_add] congr 1 apply Finset.sum_congr rfl intro x _ apply Finset.sum_congr rfl intro y _ congr 1 ring lemma cube_pair_energy_continuous {n : ℕ} (lam : Fin n → ℝ) (x y : Spin n) : Continuous (fun U : Matrix.orthogonalGroup (Fin n) ℝ => Real.exp ((diagonalEnergy lam (orthogonalIsometry U (cubeVector x)) + diagonalEnergy lam (orthogonalIsometry U (cubeVector y)))/2)) := by have hx := (diagonalEnergy_continuous lam).comp (orthogonalIsometry_fixed_continuous (cubeVector x)) have hy := (diagonalEnergy_continuous lam).comp (orthogonalIsometry_fixed_continuous (cubeVector y)) exact ((hx.add hy).div_const 2).rexp lemma rotatedCubePartition_second_moment {n : ℕ} (hn : 0 < n) (lam : Fin n → ℝ) {u v : EuclideanSpace ℝ (Fin n)} (hu : ‖u‖ = Real.sqrt n) (hv : ‖v‖ = Real.sqrt n) (horth : inner ℝ u v = 0) : (∫ U, rotatedCubePartition lam 1 U ^ 2 ∂orthogonalHaar (Fin n)) = (2 : ℝ)⁻¹ ^ n * ∑ k ∈ Finset.range (n+1), (Nat.choose n k : ℝ) * pairMoment lam u v (-1 + 2 * (k : ℝ) / n) := by have hi (x y : Spin n) := orthogonalHaar_integrable (cube_pair_energy_continuous lam x y) simp_rw [rotatedCubePartition_square] rw [integral_const_mul] rw [integral_finsetSum _ (fun x _ => integrable_finsetSum _ (fun y _ => hi x y))] have hinner (x : Spin n) : (∫ U, ∑ y : Spin n, Real.exp ((diagonalEnergy lam (orthogonalIsometry U (cubeVector x)) + diagonalEnergy lam (orthogonalIsometry U (cubeVector y)))/2) ∂orthogonalHaar (Fin n)) = ∑ y : Spin n, pairMoment lam u v (cubeOverlap x y) := by rw [integral_finsetSum _ (fun y _ => hi x y)] apply Finset.sum_congr rfl intro y _ exact pair_energy_orbit lam (cubeVector_norm x) (cubeVector_norm y) hu hv horth (cubeOverlap_abs_le_one x y) (cubeOverlap_gram hn x y) simp_rw [hinner] exact cubeOverlap_binomial_sum hn _ section variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E] lemma integral_polar (μ : Measure E) [μ.IsAddHaarMeasure] (f : E → ℝ) : (∫ x, f x ∂μ) = ∫ p : sphere (0 : E) 1 × Ioi (0 : ℝ), f (p.2.val • p.1.val) ∂μ.toSphere.prod (volumeIoiPow (Module.finrank ℝ E - 1)) := by calc _ = ∫ x : ({(0)}ᶜ : Set E), f x.val ∂(μ.comap (↑)) := by rw [integral_subtype_comap (measurableSet_singleton _).compl f, restrict_compl_singleton] _ = _ := by simpa only [Function.comp_def, Homeomorph.symm_apply_apply, homeomorphUnitSphereProd_symm_apply_coe] using μ.measurePreserving_homeomorphUnitSphereProd.integral_comp (Homeomorph.measurableEmbedding _) (fun p : sphere (0 : E) 1 × Ioi (0 : ℝ) => f ((homeomorphUnitSphereProd E).symm p).val) lemma radial_map_preserving (μ : Measure E) [μ.IsAddHaarMeasure] : MeasurePreserving (fun p : sphere (0 : E) 1 × Ioi (0 : ℝ) => p.2.val • p.1.val) (μ.toSphere.prod (volumeIoiPow (Module.finrank ℝ E - 1))) μ := by have hs : MeasurePreserving (Subtype.val : ({(0)}ᶜ : Set E) → E) (μ.comap Subtype.val) μ := by simpa only [restrict_compl_singleton] using (measurePreserving_subtype_coe (μa := μ) (measurableSet_singleton (0 : E)).compl) exact hs.comp (MeasurePreserving.symm (homeomorphUnitSphereProd E).toMeasurableEquiv μ.measurePreserving_homeomorphUnitSphereProd) lemma lintegral_polar (μ : Measure E) [μ.IsAddHaarMeasure] {f : E → ℝ≥0∞} (hf : Measurable f) : (∫⁻ x, f x ∂μ) = ∫⁻ p : sphere (0 : E) 1 × Ioi (0 : ℝ), f (p.2.val • p.1.val) ∂μ.toSphere.prod (volumeIoiPow (Module.finrank ℝ E - 1)) := ((radial_map_preserving μ).lintegral_comp hf).symm end section def radialGaussian (n : ℕ) : ℝ≥0∞ := ∫⁻ r : ℝ in Ioi 0, ENNReal.ofReal (r^(n-1) * Real.exp (-r^2)) lemma radialGaussian_finite (n : ℕ) : radialGaussian n < ⊤ := by apply Integrable.lintegral_lt_top have h := integrableOn_rpow_mul_exp_neg_mul_sq (by norm_num : (0 : ℝ) < 1) (lt_of_lt_of_le (by norm_num : (-1 : ℝ) < 0) (Nat.cast_nonneg (n-1))) simp only [Real.rpow_natCast, neg_one_mul] at h change Integrable _ (volume.restrict (Ioi 0)) at h exact h lemma radialGaussian_pos (n : ℕ) : 0 < radialGaussian n := by rw [radialGaussian, setLIntegral_pos_iff (by fun_prop)] have he : Function.support (fun r : ℝ => ENNReal.ofReal (r^(n-1) * Real.exp (-r^2))) ∩ Ioi 0 = Ioi 0 := by apply inter_eq_right.mpr intro r hr exact (ENNReal.ofReal_pos.mpr (mul_pos (pow_pos hr _) (Real.exp_pos _))).ne' rw [he, Real.volume_Ioi] exact ENNReal.zero_lt_top lemma lintegral_scale_pos {a : ℝ} (ha : 0 < a) (f : ℝ → ℝ≥0∞) : (∫⁻ x, f x) = ∫⁻ t : ℝ, ENNReal.ofReal a * f (a*t) := by have hi : (fun t : ℝ => a*t) '' Set.univ = Set.univ := Set.image_univ_of_surjective (fun x => ⟨x/a, mul_div_cancel₀ x ha.ne'⟩) have h := lintegral_image_eq_lintegral_abs_deriv_mul (s := Set.univ) (f' := fun _ => a) MeasurableSet.univ (fun t _ => by convert! ((hasDerivAt_id t).const_mul a).hasDerivWithinAt (s := Set.univ) using 1; simp only [mul_one]) (mul_right_injective₀ ha.ne').injOn f simpa only [hi, setLIntegral_univ, abs_of_pos ha] using h lemma lintegral_scale_pos_Ioi {a : ℝ} (ha : 0 < a) (f : ℝ → ℝ≥0∞) : (∫⁻ x in Ioi 0, f x) = ∫⁻ t in Ioi 0, ENNReal.ofReal a * f (a*t) := by have hi : (fun t : ℝ => a*t) '' Ioi 0 = Ioi 0 := by ext x constructor · rintro ⟨t, ht, rfl⟩ exact mul_pos ha ht · intro hx exact ⟨x/a, div_pos hx ha, mul_div_cancel₀ x ha.ne'⟩ have h := lintegral_image_eq_lintegral_abs_deriv_mul (s := Ioi (0 : ℝ)) (f' := fun _ => a) measurableSet_Ioi (fun t _ => by convert! ((hasDerivAt_id t).const_mul a).hasDerivWithinAt (s := Ioi (0 : ℝ)) using 1; simp only [mul_one]) (mul_right_injective₀ ha.ne').injOn f simpa only [hi, abs_of_pos ha] using h lemma radialGaussian_scaled (k : ℕ) {c : ℝ} (hc : 0 < c) : (∫⁻ r : ℝ in Ioi 0, ENNReal.ofReal (r^k * Real.exp (-(c*r)^2))) = ENNReal.ofReal (c⁻¹^(k+1)) * radialGaussian (k+1) := by rw [lintegral_scale_pos_Ioi (inv_pos.mpr hc)] have he (r : ℝ) : ENNReal.ofReal c⁻¹ * ENNReal.ofReal ((c⁻¹*r)^k * Real.exp (-(c*(c⁻¹*r))^2)) = ENNReal.ofReal (c⁻¹^(k+1)) * ENNReal.ofReal (r^k * Real.exp (-r^2)) := by rw [mul_inv_cancel_left₀ hc.ne', mul_pow, ← ENNReal.ofReal_mul (inv_nonneg.mpr hc.le), ← ENNReal.ofReal_mul (pow_nonneg (inv_nonneg.mpr hc.le) _)] congr 1 rw [pow_succ] ring simp_rw [he] rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top] rfl variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E] lemma radialGaussian_polar (μ : Measure E) [μ.IsAddHaarMeasure] {g : E → ℝ≥0∞} (hg : Measurable g) (hhom : ∀ (r : ℝ), 0 < r → ∀ x : E, g (r • x) = g x) : (∫⁻ x, ENNReal.ofReal (Real.exp (-‖x‖^2)) * g x ∂μ) = (∫⁻ u : sphere (0 : E) 1, g u.val ∂μ.toSphere) * radialGaussian (Module.finrank ℝ E) := by rw [lintegral_polar μ (by fun_prop)] have he (p : sphere (0 : E) 1 × Ioi (0 : ℝ)) : ENNReal.ofReal (Real.exp (-‖p.2.val • p.1.val‖^2)) * g (p.2.val • p.1.val) = g p.1.val * ENNReal.ofReal (Real.exp (-p.2.val^2)) := by rw [hhom _ p.2.property, norm_smul, Real.norm_of_nonneg p.2.property.le, mem_sphere_zero_iff_norm.mp p.1.property, mul_one, mul_comm] simp_rw [he] rw [lintegral_prod_mul (f := fun u : sphere (0 : E) 1 => g u.val) (g := fun r : Ioi (0 : ℝ) => ENNReal.ofReal (Real.exp (-r.val^2))) (by fun_prop) (by fun_prop)] congr 1 rw [volumeIoiPow, lintegral_withDensity_eq_lintegral_mul (f := fun r : Ioi (0 : ℝ) => ENNReal.ofReal (r.val^(Module.finrank ℝ E-1))) (g := fun r : Ioi (0 : ℝ) => ENNReal.ofReal (Real.exp (-r.val^2))) _ (by fun_prop) (by fun_prop)] calc _ = ∫⁻ r : Ioi (0 : ℝ), ENNReal.ofReal (r.val^(Module.finrank ℝ E-1) * Real.exp (-r.val^2)) ∂volume.comap Subtype.val := by apply lintegral_congr intro r exact (ENNReal.ofReal_mul (pow_nonneg r.property.le _)).symm _ = _ := lintegral_subtype_comap (μ := volume) (s := Ioi (0 : ℝ)) measurableSet_Ioi (fun r : ℝ => ENNReal.ofReal (r^(Module.finrank ℝ E-1) * Real.exp (-r^2))) end lemma volumeIoiPow_lintegral (k : ℕ) {f : ℝ → ℝ≥0∞} (hf : Measurable f) : (∫⁻ r : Ioi (0 : ℝ), f r.val ∂volumeIoiPow k) = ∫⁻ r : ℝ in Ioi 0, ENNReal.ofReal (r^k) * f r := by rw [volumeIoiPow, lintegral_withDensity_eq_lintegral_mul _ (f := fun r : Ioi (0 : ℝ) => ENNReal.ofReal (r.val^k)) (g := fun r : Ioi (0 : ℝ) => f r.val) (by fun_prop) (by fun_prop)] exact lintegral_subtype_comap (μ := volume) (s := Ioi (0 : ℝ)) measurableSet_Ioi (fun r : ℝ => ENNReal.ofReal (r^k) * f r) variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E] lemma lintegral_radial_norm (μ : Measure E) [μ.IsAddHaarMeasure] {f : ℝ → ℝ≥0∞} (hf : Measurable f) : (∫⁻ x, f ‖x‖ ∂μ) = μ.toSphere Set.univ * ∫⁻ r : ℝ in Ioi 0, ENNReal.ofReal (r^(Module.finrank ℝ E-1)) * f r := by rw [lintegral_polar μ (f := fun x => f ‖x‖) (by fun_prop)] have he (p : sphere (0 : E) 1 × Ioi (0 : ℝ)) : f ‖p.2.val • p.1.val‖ = (1 : ℝ≥0∞) * f p.2.val := by rw [norm_smul, Real.norm_of_nonneg p.2.property.le, mem_sphere_zero_iff_norm.mp p.1.property, mul_one, one_mul] simp_rw [he] rw [lintegral_prod_mul (f := fun _ : sphere (0 : E) 1 => (1 : ℝ≥0∞)) (g := fun r : Ioi (0 : ℝ) => f r.val) (by fun_prop) (by fun_prop), lintegral_const, one_mul, volumeIoiPow_lintegral _ hf] end def sphereRatio (t : ℝ) : ℝ := t / Real.sqrt (1+t^2) def sphereRatioInv (q : ℝ) : ℝ := q / Real.sqrt (1-q^2) lemma sphereRatio_den_pos (t : ℝ) : 0 < 1+t^2 := by positivity lemma sphereRatio_sq (t : ℝ) : sphereRatio t ^ 2 = t^2/(1+t^2) := by rw [sphereRatio, div_pow, Real.sq_sqrt (sphereRatio_den_pos t).le] lemma one_sub_sphereRatio_sq (t : ℝ) : 1-sphereRatio t ^ 2 = 1/(1+t^2) := by rw [sphereRatio_sq] field_simp ring lemma sphereRatio_mem (t : ℝ) : sphereRatio t ∈ Ioo (-1 : ℝ) 1 := by have hp : 0 < 1-sphereRatio t^2 := by rw [one_sub_sphereRatio_sq]; positivity constructor <;> nlinarith [sq_nonneg (sphereRatio t + 1), sq_nonneg (sphereRatio t - 1)] lemma sphereRatioInv_den_pos {q : ℝ} (hq : q ∈ Ioo (-1 : ℝ) 1) : 0 < 1-q^2 := by nlinarith [hq.1, hq.2] lemma one_add_sphereRatioInv_sq {q : ℝ} (hq : q ∈ Ioo (-1 : ℝ) 1) : 1+sphereRatioInv q ^ 2 = 1/(1-q^2) := by rw [sphereRatioInv, div_pow, Real.sq_sqrt (sphereRatioInv_den_pos hq).le] field_simp [(sphereRatioInv_den_pos hq).ne'] ring lemma sphereRatio_left_inverse (t : ℝ) : sphereRatioInv (sphereRatio t) = t := by unfold sphereRatioInv rw [one_sub_sphereRatio_sq, Real.sqrt_div (by norm_num), Real.sqrt_one] unfold sphereRatio simpa only [div_one] using div_div_div_cancel_right₀ (Real.sqrt_ne_zero'.mpr (sphereRatio_den_pos t)) t 1 lemma sphereRatio_right_inverse {q : ℝ} (hq : q ∈ Ioo (-1 : ℝ) 1) : sphereRatio (sphereRatioInv q) = q := by unfold sphereRatio rw [one_add_sphereRatioInv_sq hq, Real.sqrt_div (by norm_num), Real.sqrt_one] unfold sphereRatioInv simpa only [div_one] using div_div_div_cancel_right₀ (Real.sqrt_ne_zero'.mpr (sphereRatioInv_den_pos hq)) q 1 lemma sphereRatio_injective : Function.Injective sphereRatio := Function.LeftInverse.injective sphereRatio_left_inverse lemma sphereRatioInv_injOn : Set.InjOn sphereRatioInv (Ioo (-1 : ℝ) 1) := by intro q hq r hr he have := congrArg sphereRatio he simpa only [sphereRatio_right_inverse hq, sphereRatio_right_inverse hr] using this lemma sphereRatioInv_image : sphereRatioInv '' Ioo (-1 : ℝ) 1 = Set.univ := by apply Set.eq_univ_of_forall intro t exact ⟨sphereRatio t, sphereRatio_mem t, sphereRatio_left_inverse t⟩ lemma sphereRatioInv_hasDerivAt {q : ℝ} (hq : q ∈ Ioo (-1 : ℝ) 1) : HasDerivAt sphereRatioInv (1/(Real.sqrt (1-q^2))^3) q := by have hder := (hasDerivAt_id q).div (((hasDerivAt_const q 1).sub ((hasDerivAt_id q).pow 2)).sqrt (sphereRatioInv_den_pos hq).ne') (Real.sqrt_ne_zero'.mpr (sphereRatioInv_den_pos hq)) convert! hder using 1 have hs := Real.sq_sqrt (sphereRatioInv_den_pos hq).le have hn := (Real.sqrt_ne_zero'.mpr (sphereRatioInv_den_pos hq)) dsimp only [Pi.sub_apply, Pi.pow_apply, id_eq] norm_num only [Nat.cast_ofNat, Nat.reduceSub, pow_one, one_mul, mul_one, zero_sub] field_simp [hn] nlinarith lemma sphereRatioInv_continuousOn : ContinuousOn sphereRatioInv (Ioo (-1 : ℝ) 1) := fun _ hq => (sphereRatioInv_hasDerivAt hq).continuousAt.continuousWithinAt def sphereOverlapShape (n : ℕ) (q : ℝ) : ℝ := (Real.sqrt (1-q^2))^(n-3) lemma sphereRatioInv_jacobian_shape {n : ℕ} (hn : 3 ≤ n) {q : ℝ} (hq : q ∈ Ioo (-1 : ℝ) 1) : |1/(Real.sqrt (1-q^2))^3| * (Real.sqrt (1+sphereRatioInv q^2))⁻¹ ^ n = sphereOverlapShape n q := by have hs : 0 < Real.sqrt (1-q^2) := Real.sqrt_pos.mpr (sphereRatioInv_den_pos hq) rw [abs_of_pos (by positivity), one_add_sphereRatioInv_sq hq, Real.sqrt_div (by norm_num), Real.sqrt_one] simp only [one_div, inv_inv] rw [mul_comm, ← div_eq_mul_inv] exact (pow_sub₀ _ hs.ne' hn).symm lemma sphereRatio_integral {n : ℕ} (hn : 3 ≤ n) (f : ℝ → ℝ≥0∞) : (∫⁻ t : ℝ, ENNReal.ofReal ((Real.sqrt (1+t^2))⁻¹ ^ n) * f (sphereRatio t)) = ∫⁻ q in Ioo (-1 : ℝ) 1, ENNReal.ofReal (sphereOverlapShape n q) * f q := by have h := lintegral_image_eq_lintegral_abs_deriv_mul measurableSet_Ioo (fun q hq => (sphereRatioInv_hasDerivAt hq).hasDerivWithinAt) sphereRatioInv_injOn (fun t => ENNReal.ofReal ((Real.sqrt (1+t^2))⁻¹ ^ n) * f (sphereRatio t)) rw [sphereRatioInv_image, setLIntegral_univ] at h rw [h] apply setLIntegral_congr_fun measurableSet_Ioo intro q hq dsimp only rw [sphereRatio_right_inverse hq, ← mul_assoc, ← ENNReal.ofReal_mul (abs_nonneg _), sphereRatioInv_jacobian_shape hn hq] def cylinderKernel (f : ℝ → ℝ≥0∞) (s r : ℝ) : ℝ≥0∞ := ENNReal.ofReal (Real.exp (-(s^2+r^2))) * f (s / Real.sqrt (s^2+r^2)) lemma cylinderKernel_measurable {f : ℝ → ℝ≥0∞} (hf : Measurable f) : Measurable (Function.uncurry (cylinderKernel f)) := by unfold cylinderKernel Function.uncurry fun_prop lemma scaled_ratio {r : ℝ} (hr : 0 < r) (t : ℝ) : r*t / Real.sqrt ((r*t)^2+r^2) = sphereRatio t := by rw [show (r*t)^2+r^2 = r^2*(1+t^2) by ring, Real.sqrt_mul (sq_nonneg r), Real.sqrt_sq hr.le] exact mul_div_mul_left t _ hr.ne' lemma cylinderKernel_scaled {r : ℝ} (hr : 0 < r) (t : ℝ) (f : ℝ → ℝ≥0∞) : cylinderKernel f (r*t) r = ENNReal.ofReal (Real.exp (-(Real.sqrt (1+t^2)*r)^2)) * f (sphereRatio t) := by unfold cylinderKernel rw [scaled_ratio hr] congr 3 simp only [mul_pow, Real.sq_sqrt (sphereRatio_den_pos t).le] ring lemma cylinder_scalar_scale {f : ℝ → ℝ≥0∞} {r : ℝ} (hr : 0 < r) : (∫⁻ s : ℝ, cylinderKernel f s r) = ENNReal.ofReal r * ∫⁻ t : ℝ, ENNReal.ofReal (Real.exp (-(Real.sqrt (1+t^2)*r)^2)) * f (sphereRatio t) := by rw [lintegral_scale_pos hr] simp_rw [cylinderKernel_scaled hr] exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top lemma cylinder_radius_integral (k : ℕ) {f : ℝ → ℝ≥0∞} (hf : Measurable f) : (∫⁻ r : ℝ in Ioi 0, ENNReal.ofReal (r^k) * ∫⁻ s : ℝ, cylinderKernel f s r) = radialGaussian (k+2) * ∫⁻ t : ℝ, ENNReal.ofReal ((Real.sqrt (1+t^2))⁻¹^(k+2)) * f (sphereRatio t) := by calc _ = ∫⁻ r : ℝ in Ioi 0, ∫⁻ t : ℝ, ENNReal.ofReal (r^(k+1)) * (ENNReal.ofReal (Real.exp (-(Real.sqrt (1+t^2)*r)^2)) * f (sphereRatio t)) := by apply setLIntegral_congr_fun measurableSet_Ioi intro r hr dsimp only rw [cylinder_scalar_scale hr, ← mul_assoc, ← ENNReal.ofReal_mul (pow_nonneg hr.le _), ← pow_succ, ← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top] _ = ∫⁻ t : ℝ, ∫⁻ r : ℝ in Ioi 0, ENNReal.ofReal (r^(k+1)) * (ENNReal.ofReal (Real.exp (-(Real.sqrt (1+t^2)*r)^2)) * f (sphereRatio t)) := by apply lintegral_lintegral_swap unfold sphereRatio fun_prop _ = ∫⁻ t : ℝ, ENNReal.ofReal ((Real.sqrt (1+t^2))⁻¹^(k+2)) * radialGaussian (k+2) * f (sphereRatio t) := by apply lintegral_congr intro t calc _ = (∫⁻ r : ℝ in Ioi 0, ENNReal.ofReal (r^(k+1) * Real.exp (-(Real.sqrt (1+t^2)*r)^2))) * f (sphereRatio t) := by rw [← lintegral_mul_const _ (by fun_prop)] apply setLIntegral_congr_fun measurableSet_Ioi intro r hr dsimp only rw [ENNReal.ofReal_mul (pow_nonneg hr.le _), mul_assoc] _ = _ := by rw [radialGaussian_scaled _ (Real.sqrt_pos.mpr (sphereRatio_den_pos t))] _ = _ := by simp_rw [mul_assoc, mul_left_comm _ (radialGaussian (k+2))] exact lintegral_const_mul' _ _ (radialGaussian_finite _).ne def euclidSplit (n : ℕ) (x : EuclideanSpace ℝ (Fin (n+1))) : ℝ × EuclideanSpace ℝ (Fin n) := (x 0, WithLp.toLp 2 (fun i => x i.succ)) lemma euclidSplit_preserving (n : ℕ) : MeasurePreserving (euclidSplit n) volume (volume.prod volume) := by have hs := (measurePreserving_piFinSuccAbove (fun _ : Fin (n+1) => (volume : Measure ℝ)) 0).comp (PiLp.volume_preserving_ofLp (Fin (n+1))) have ht := (MeasurePreserving.id (volume : Measure ℝ)).prod (PiLp.volume_preserving_toLp (Fin n)) exact ht.comp hs lemma euclidSplit_norm_sq (n : ℕ) (x : EuclideanSpace ℝ (Fin (n+1))) : ‖x‖^2 = (euclidSplit n x).1^2 + ‖(euclidSplit n x).2‖^2 := by rw [EuclideanSpace.norm_sq_eq, Fin.sum_univ_succ, EuclideanSpace.norm_sq_eq] simp only [euclidSplit, WithLp.ofLp_toLp, Real.norm_eq_abs, sq_abs] lemma euclidSplit_norm (n : ℕ) (x : EuclideanSpace ℝ (Fin (n+1))) : ‖x‖ = Real.sqrt ((euclidSplit n x).1^2 + ‖(euclidSplit n x).2‖^2) := by rw [← euclidSplit_norm_sq, Real.sqrt_sq (norm_nonneg x)] lemma euclidSplit_integral (n : ℕ) {f : ℝ → ℝ≥0∞} (hf : Measurable f) : (∫⁻ x : EuclideanSpace ℝ (Fin (n+1)), ENNReal.ofReal (Real.exp (-‖x‖^2)) * f (x 0 / ‖x‖)) = ∫⁻ p : ℝ × EuclideanSpace ℝ (Fin n), ENNReal.ofReal (Real.exp (-(p.1^2 + ‖p.2‖^2))) * f (p.1 / Real.sqrt (p.1^2 + ‖p.2‖^2)) ∂volume.prod volume := by have hm : Measurable (fun p : ℝ × EuclideanSpace ℝ (Fin n) => ENNReal.ofReal (Real.exp (-(p.1^2 + ‖p.2‖^2))) * f (p.1 / Real.sqrt (p.1^2 + ‖p.2‖^2))) := by fun_prop rw [← (euclidSplit_preserving n).lintegral_comp hm] apply lintegral_congr intro x rw [← euclidSplit_norm_sq, Real.sqrt_sq (norm_nonneg x)] rfl lemma gaussian_cylinder_volume (n : ℕ) {f : ℝ → ℝ≥0∞} (hf : Measurable f) : (∫⁻ x : EuclideanSpace ℝ (Fin (n+3)), ENNReal.ofReal (Real.exp (-‖x‖^2)) * f (x 0 / ‖x‖)) = (volume : Measure (EuclideanSpace ℝ (Fin (n+2)))).toSphere Set.univ * radialGaussian (n+3) * ∫⁻ t : ℝ, ENNReal.ofReal ((Real.sqrt (1+t^2))⁻¹^(n+3)) * f (sphereRatio t) := by rw [euclidSplit_integral (n+2) hf] change (∫⁻ p : ℝ × EuclideanSpace ℝ (Fin (n+2)), cylinderKernel f p.1 ‖p.2‖ ∂volume.prod volume) = _ rw [lintegral_prod _ (by unfold cylinderKernel; fun_prop)] have ht (s : ℝ) : (∫⁻ y : EuclideanSpace ℝ (Fin (n+2)), cylinderKernel f s ‖y‖) = (volume : Measure (EuclideanSpace ℝ (Fin (n+2)))).toSphere Set.univ * ∫⁻ r : ℝ in Ioi 0, ENNReal.ofReal (r^(n+1)) * cylinderKernel f s r := by simpa only [finrank_euclideanSpace, Fintype.card_fin, show n+2-1=n+1 by omega] using (lintegral_radial_norm (volume : Measure (EuclideanSpace ℝ (Fin (n+2)))) (f := fun r => cylinderKernel f s r) (by unfold cylinderKernel; fun_prop)) simp_rw [ht] rw [lintegral_const_mul' _ _ (measure_ne_top _ _)] calc _ = (volume : Measure (EuclideanSpace ℝ (Fin (n+2)))).toSphere Set.univ * ∫⁻ r : ℝ in Ioi 0, ENNReal.ofReal (r^(n+1)) * ∫⁻ s : ℝ, cylinderKernel f s r := by congr 1 rw [lintegral_lintegral_swap (by unfold cylinderKernel Function.uncurry; fun_prop)] apply lintegral_congr intro r exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top _ = _ := by rw [cylinder_radius_integral _ hf, ← mul_assoc] lemma gaussian_coordinate_polar (n : ℕ) {f : ℝ → ℝ≥0∞} (hf : Measurable f) : (∫⁻ x : EuclideanSpace ℝ (Fin (n+3)), ENNReal.ofReal (Real.exp (-‖x‖^2)) * f (x 0 / ‖x‖)) = (∫⁻ u : sphere (0 : EuclideanSpace ℝ (Fin (n+3))) 1, f (u.val 0) ∂(volume : Measure (EuclideanSpace ℝ (Fin (n+3)))).toSphere) * radialGaussian (n+3) := by have hm : Measurable (fun x : EuclideanSpace ℝ (Fin (n+3)) => f (x 0 / ‖x‖)) := by fun_prop have hhom (r : ℝ) (hr : 0 < r) (x : EuclideanSpace ℝ (Fin (n+3))) : f ((r • x) 0 / ‖r • x‖) = f (x 0 / ‖x‖) := by rw [norm_smul, Real.norm_of_nonneg hr.le] change f (r * x 0 / (r * ‖x‖)) = _ rw [mul_div_mul_left _ _ hr.ne'] have h := radialGaussian_polar volume hm hhom simp only [finrank_euclideanSpace, Fintype.card_fin, mem_sphere_zero_iff_norm.mp (Subtype.property _), div_one] at h exact h lemma sphere_coordinate_lintegral (n : ℕ) {f : ℝ → ℝ≥0∞} (hf : Measurable f) : (∫⁻ u : sphere (0 : EuclideanSpace ℝ (Fin (n+3))) 1, f (u.val 0) ∂(volume : Measure (EuclideanSpace ℝ (Fin (n+3)))).toSphere) = (volume : Measure (EuclideanSpace ℝ (Fin (n+2)))).toSphere Set.univ * ∫⁻ q in Ioo (-1 : ℝ) 1, ENNReal.ofReal (sphereOverlapShape (n+3) q) * f q := by apply (ENNReal.mul_left_inj (radialGaussian_pos (n+3)).ne' (radialGaussian_finite (n+3)).ne).mp rw [← gaussian_coordinate_polar n hf, gaussian_cylinder_volume n hf, sphereRatio_integral (by omega : 3 ≤ n+3)] ac_rfl def overlapShapeMeasure (n : ℕ) : Measure ℝ := (volume.restrict (Ioo (-1 : ℝ) 1)).withDensity (fun q => ENNReal.ofReal (sphereOverlapShape n q)) def normalizedOverlapShape (n : ℕ) : Measure ℝ := ((overlapShapeMeasure n) Set.univ)⁻¹ • overlapShapeMeasure n lemma overlapShapeMeasure_lintegral (n : ℕ) {f : ℝ → ℝ≥0∞} (hf : Measurable f) : (∫⁻ q, f q ∂overlapShapeMeasure n) = ∫⁻ q in Ioo (-1 : ℝ) 1, ENNReal.ofReal (sphereOverlapShape n q) * f q := by rw [overlapShapeMeasure, lintegral_withDensity_eq_lintegral_mul _ (by unfold sphereOverlapShape; fun_prop) hf] rfl lemma overlapShapeMeasure_univ (n : ℕ) : overlapShapeMeasure n Set.univ = ∫⁻ q in Ioo (-1 : ℝ) 1, ENNReal.ofReal (sphereOverlapShape n q) := by rw [← lintegral_one, overlapShapeMeasure_lintegral n measurable_const] simp only [mul_one] lemma sphereUniform_coordinate_lintegral (n : ℕ) {f : ℝ → ℝ≥0∞} (hf : Measurable f) : (∫⁻ u : unitSphere (Fin (n+3)), f (u.val 0) ∂sphereUniform) = ∫⁻ q, f q ∂normalizedOverlapShape (n+3) := by let A := (volume : Measure (EuclideanSpace ℝ (Fin (n+2)))).toSphere Set.univ have hA0 : A ≠ 0 := (measure_univ_pos.mpr (Measure.toSphere_ne_zero _)).ne' have hAt : A ≠ ⊤ := measure_ne_top _ _ have hfull : (volume : Measure (EuclideanSpace ℝ (Fin (n+3)))).toSphere Set.univ = A * overlapShapeMeasure (n+3) Set.univ := by have h := sphere_coordinate_lintegral n (f := fun _ => 1) measurable_const simpa only [lintegral_one, mul_one, overlapShapeMeasure_univ, A] using h rw [sphereUniform, lintegral_smul_measure, smul_eq_mul, sphere_coordinate_lintegral n hf, hfull, ENNReal.mul_inv (Or.inl hA0) (Or.inl hAt), normalizedOverlapShape, lintegral_smul_measure, smul_eq_mul, overlapShapeMeasure_lintegral _ hf] change (A⁻¹ * (overlapShapeMeasure (n+3) Set.univ)⁻¹) * (A * _) = _ rw [mul_assoc, mul_left_comm _ A, ENNReal.inv_mul_cancel_left hA0 hAt] def firstUnit (n : ℕ) : unitSphere (Fin (n+1)) := ⟨EuclideanSpace.single 0 1, mem_sphere_zero_iff_norm.mpr (by simp)⟩ lemma sphere_inner_coordinate_lintegral (n : ℕ) {f : ℝ → ℝ≥0∞} (hf : Measurable f) (u : unitSphere (Fin (n+1))) : (∫⁻ v : unitSphere (Fin (n+1)), f (inner ℝ u.val v.val) ∂sphereUniform) = ∫⁻ v : unitSphere (Fin (n+1)), f (v.val 0) ∂sphereUniform := by obtain ⟨U, hU⟩ := orthogonalSphere_transitive (firstUnit n) u have hm : Measurable (fun v : unitSphere (Fin (n+1)) => f (inner ℝ u.val v.val)) := by fun_prop rw [← (sphereUniform_rotation_preserving U).lintegral_comp hm] apply lintegral_congr intro v congr 1 rw [← hU] change inner ℝ (orthogonalIsometry U (firstUnit n).val) (orthogonalIsometry U v.val) = _ rw [(orthogonalIsometry U).inner_map_map] simp only [firstUnit, EuclideanSpace.inner_single_left, map_one, one_mul] lemma sphereOverlapLaw_lintegral (n : ℕ) {f : ℝ → ℝ≥0∞} (hf : Measurable f) : (∫⁻ q, f q ∂sphereOverlapLaw (Fin (n+3))) = ∫⁻ q, f q ∂normalizedOverlapShape (n+3) := by rw [sphereOverlapLaw, lintegral_map hf (by fun_prop), lintegral_prod _ (by fun_prop)] simp_rw [sphere_inner_coordinate_lintegral (n+2) hf] rw [lintegral_const, measure_univ, mul_one, sphereUniform_coordinate_lintegral n hf] lemma sphereOverlapLaw_density (n : ℕ) : sphereOverlapLaw (Fin (n+3)) = normalizedOverlapShape (n+3) := by apply Measure.ext_of_lintegral exact fun _ hf => sphereOverlapLaw_lintegral n hf end CriticalSK end end OAI