import Mathlib import OAI.Probability.SKGap.Localization.BilinearCoefficient import OAI.Probability.SKGap.Gaussian.SquaredGaussianFourth namespace OAI section open scoped BigOperators open scoped BigOperators open scoped BigOperators open scoped BigOperators open scoped BigOperators open scoped BigOperators NNReal open MeasureTheory ProbabilityTheory open MeasureTheory ProbabilityTheory Filter open scoped BigOperators NNReal open MeasureTheory ProbabilityTheory open scoped BigOperators NNReal ENNReal open MeasureTheory ProbabilityTheory Filter open scoped BigOperators NNReal ENNReal open MeasureTheory ProbabilityTheory open scoped BigOperators Matrix Matrix.Norms.Elementwise open scoped BigOperators open MeasureTheory ProbabilityTheory open scoped BigOperators Matrix Matrix.Norms.Elementwise open scoped BigOperators open scoped BigOperators NNReal ENNReal open MeasureTheory Metric Set open scoped BigOperators NNReal ENNReal open MeasureTheory ProbabilityTheory Filter Set open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator open MeasureTheory ProbabilityTheory Filter Set open scoped BigOperators Matrix.Norms.L2Operator open MeasureTheory ProbabilityTheory Filter Set namespace SKGapCutoff lemma tanh_cubic_remainder (d : ℝ) : |Real.tanh d - d| ≤ |d|^3 := by let f (z : ℝ) := Real.tanh z - z have hd (z : ℝ) : HasDerivAt f (-(Real.tanh z)^2) z := by convert! (tanh_hasDerivAt z).sub (hasDerivAt_id z) using 1 unfold scalarVariance ring have hb (z : ℝ) (hz : z ∈ segment ℝ 0 d) : ‖-(Real.tanh z)^2‖ ≤ |d|^2 := by have h1 : |Real.tanh z| ≤ |z| := by simpa using tanh_lipschitz 0 z have h2 : |z| ≤ |d| := by simpa only [sub_zero, Real.norm_eq_abs] using norm_sub_le_of_mem_segment hz rw [norm_neg, norm_pow, Real.norm_eq_abs] exact pow_le_pow_left₀ (abs_nonneg _) (h1.trans h2) 2 have h := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le (fun z (_ : z ∈ segment ℝ 0 d) => (hd z).hasDerivWithinAt) hb (convex_segment 0 d) (left_mem_segment ℝ 0 d) (right_mem_segment ℝ 0 d) simpa only [f, Real.tanh_zero, sub_zero, Real.norm_eq_abs, pow_succ] using h lemma normalized_tanh_difference (h d : ℝ) : (Real.tanh h - Real.tanh (h-d)) / scalarVariance h = Real.tanh d / (1 - Real.tanh h * Real.tanh d) := by have hc : Real.cosh (h-d) ≠ 0 := (Real.cosh_pos _).ne' have hh : Real.cosh h ≠ 0 := (Real.cosh_pos _).ne' have hd : Real.cosh d ≠ 0 := (Real.cosh_pos _).ne' have hm : Real.cosh h * Real.cosh d - Real.sinh h * Real.sinh d ≠ 0 := by simpa only [Real.cosh_sub] using hc rw [tanh_difference_eq, show h-(h-d)=d by ring, scalarVariance_eq] simp only [Real.tanh_eq_sinh_div_cosh] rw [Real.cosh_sub] field_simp lemma rational_cubic_remainder (m u d : ℝ) (hm : |m| ≤ 1) (hd : |d| ≤ 1/2) (hu : |u| ≤ |d|) (hud : |u-d| ≤ |d|^3) : |u/(1-m*u) - d - m*d^2| ≤ 4*|d|^3 := by have hmud : |m*u| ≤ |d| := by rw [abs_mul] exact (mul_le_mul hm hu (abs_nonneg _) (by norm_num)).trans_eq (one_mul _) have hden : 1/2 ≤ 1-m*u := by linarith [le_abs_self (m*u)] have hdenpos : 0 < 1-m*u := by linarith have hd0 := abs_nonneg d have he : u/(1-m*u) - d - m*d^2 = (u-d) + m*(u^2-d^2) + m^2*u^3/(1-m*u) := by apply mul_right_cancel₀ hdenpos.ne' simp only [sub_mul, add_mul, div_mul_cancel₀ _ hdenpos.ne'] ring have hs : |u^2-d^2| ≤ |d|^3 := by have hh : |u^2-d^2| = |u-d| * |u+d| := by rw [← abs_mul]; congr 1; ring rw [hh] have ha : |u+d| ≤ 1 := (abs_add_le _ _).trans (by linarith) nlinarith [mul_le_mul hud ha (abs_nonneg _) (pow_nonneg hd0 3)] have ht : |m*(u^2-d^2)| ≤ |d|^3 := by rw [abs_mul] simpa only [one_mul] using mul_le_mul hm hs (abs_nonneg _) (by norm_num : (0:ℝ) ≤ 1) have hq : |m^2*u^3/(1-m*u)| ≤ 2*|d|^3 := by rw [abs_div, abs_mul, abs_pow, abs_pow, abs_of_pos hdenpos, div_le_iff₀ hdenpos] have hm2 : |m|^2 ≤ 1 := by nlinarith [abs_nonneg m] have hu3 : |u|^3 ≤ |d|^3 := pow_le_pow_left₀ (abs_nonneg _) hu 3 have hp := mul_le_mul hm2 hu3 (pow_nonneg (abs_nonneg _) _) (by norm_num : (0:ℝ) ≤ 1) nlinarith [mul_le_mul_of_nonneg_left hden (pow_nonneg hd0 3)] rw [he] exact (abs_add_le _ _).trans ((add_le_add ((abs_add_le _ _).trans (add_le_add hud ht)) hq).trans (by ring_nf; rfl)) lemma normalized_tanh_cubic (h d : ℝ) (hd : |d| ≤ 1/2) : |(Real.tanh h - Real.tanh (h-d))/scalarVariance h - d - Real.tanh h*d^2| ≤ 4*|d|^3 := by rw [normalized_tanh_difference] exact rational_cubic_remainder _ _ _ (Real.abs_tanh_lt_one h).le hd (by simpa using tanh_lipschitz 0 d) (tanh_cubic_remainder d) lemma normalized_influence_cubic {n : ℕ} (J : Interaction n) (x : Spin n) (i j : Fin n) (hJ : |J j i| ≤ 1/4) : |influence J x i j / siteVariance J x j - J j i - 2*spin x i*mean J x j*(J j i)^2| ≤ 16*|J j i|^3 := by have ha : |2*J j i*spin x i| = 2*|J j i| := by rw [abs_mul, abs_mul, abs_spin_eq_one]; norm_num have hd : |2*J j i*spin x i| ≤ 1/2 := by rw [ha]; linarith have hh := normalized_tanh_cubic (field J x j) (2*J j i*spin x i) hd have hv : siteVariance J x j ≠ 0 := (siteVariance_pos J x j).ne' have he : influence J x i j / siteVariance J x j - J j i - 2*spin x i*mean J x j*(J j i)^2 = (spin x i / 2) * ((Real.tanh (field J x j) - Real.tanh (field J x j - 2*J j i*spin x i))/scalarVariance (field J x j) - 2*J j i*spin x i - Real.tanh (field J x j)*(2*J j i*spin x i)^2) := by simp only [influence, halfDiff_as_flip, mean, field_flip_general, scalarVariance, siteVariance] have hs := spin_sq x i have ht : 1 - Real.tanh (field J x j)^2 ≠ 0 := (scalarVariance_pos _).ne' field_simp [ht] linear_combination (2 * J j i * (1 - Real.tanh (field J x j)^2) + 4 * spin x i * Real.tanh (field J x j) * (J j i)^2 * (1 - Real.tanh (field J x j)^2)) * hs rw [he, abs_mul, abs_div, abs_spin_eq_one] rw [ha] at hh norm_num only [abs_of_pos (show (0:ℝ) < 2 by norm_num)] nlinarith lemma matrix_opNorm_le_schur {n : ℕ} (A : Interaction n) (R : ℝ) (hR : 0 ≤ R) (hr : ∀ i, ∑ j, |A i j| ≤ R) (hc : ∀ j, ∑ i, |A i j| ≤ R) : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) A‖ ≤ R := by apply ContinuousLinearMap.opNorm_le_of_re_inner_le hR intro x y hx hy change inner (𝕜 := ℝ) (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) A x) y ≤ R rw [real_inner_comm, Matrix.inner_toEuclideanCLM] change (∑ i, y i * ∑ j, A i j*x j) ≤ R simp only [Finset.mul_sum] calc _ ≤ ∑ i, ∑ j, |A i j| * ((y i)^2 + (x j)^2)/2 := by apply Finset.sum_le_sum intro i _ apply Finset.sum_le_sum intro j _ have hab : 2*|y i| *|x j| ≤ (y i)^2+(x j)^2 := by nlinarith [sq_nonneg (|y i|-|x j|), sq_abs (y i), sq_abs (x j)] have hp := mul_le_mul_of_nonneg_left hab (abs_nonneg (A i j)) have hh : y i*(A i j*x j) ≤ |y i| *|A i j| *|x j| := by simpa only [abs_mul, mul_assoc] using le_abs_self (y i*(A i j*x j)) nlinarith _ = ((∑ i, (y i)^2 * ∑ j, |A i j|) + ∑ j, (x j)^2 * ∑ i, |A i j|)/2 := by simp only [mul_add, add_div, Finset.sum_add_distrib, ← Finset.sum_div] rw [Finset.sum_comm (f := fun i j => |A i j| * (x j)^2)] simp only [← Finset.sum_mul] congr 2 <;> apply Finset.sum_congr rfl <;> intro i _ <;> ring _ ≤ ((∑ i, (y i)^2 * R) + ∑ j, (x j)^2*R)/2 := by apply div_le_div_of_nonneg_right _ (by norm_num) exact add_le_add (Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hr i) (sq_nonneg _))) (Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hc i) (sq_nonneg _))) _ = R := by rw [← Finset.sum_mul, ← Finset.sum_mul, ← EuclideanSpace.real_norm_sq_eq, ← EuclideanSpace.real_norm_sq_eq, hx, hy] ring lemma matrix_diagonal_opNorm_le {n : ℕ} (d : Fin n → ℝ) (R : ℝ) (hR : 0 ≤ R) (hd : ∀ i, |d i| ≤ R) : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.diagonal d)‖ ≤ R := by apply matrix_opNorm_le_schur _ R hR · intro i simpa [Matrix.diagonal, apply_ite, abs_zero] using hd i · intro i simpa [Matrix.diagonal, apply_ite, abs_zero] using hd i noncomputable def normalizedInfluence {n : ℕ} (J : Interaction n) (x : Spin n) : Interaction n := fun i j => influence J x i j / siteVariance J x j noncomputable def influenceCubicError {n : ℕ} (J : Interaction n) (x : Spin n) : Interaction n := fun i j => normalizedInfluence J x i j - J j i - 2*spin x i*mean J x j*(J j i)^2 lemma influenceCubicError_norm_le {n : ℕ} (J : Interaction n) (hsymm : ∀ i j, J i j=J j i) (x : Spin n) (c B : ℝ) (hc : 0 ≤ c) (hc1 : c ≤ 1/4) (hB : 0 ≤ B) (hmax : ∀ i j, |J i j| ≤ c) (hcol : ∀ i, ∑ j, J j i^2 ≤ B) : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (influenceCubicError J x)‖ ≤ 16*c*B := by have hp (i j : Fin n) : |influenceCubicError J x i j| ≤ 16*c*(J j i)^2 := by have hh := normalized_influence_cubic J x i j ((hmax j i).trans hc1) change |influenceCubicError J x i j| ≤ _ at hh apply hh.trans have hmul := mul_le_mul_of_nonneg_right (hmax j i) (sq_nonneg |J j i|) nlinarith [sq_abs (J j i)] apply matrix_opNorm_le_schur _ (16*c*B) (by positivity) · intro i calc _ ≤ ∑ j, 16*c*(J j i)^2 := Finset.sum_le_sum (fun j _ => hp i j) _ = 16*c*∑ j, J j i^2 := by rw [Finset.mul_sum] _ ≤ _ := mul_le_mul_of_nonneg_left (hcol i) (by positivity) · intro j calc _ ≤ ∑ i, 16*c*(J j i)^2 := Finset.sum_le_sum (fun i _ => hp i j) _ = 16*c*∑ i, J i j^2 := by simp only [hsymm j, Finset.mul_sum] _ ≤ _ := mul_le_mul_of_nonneg_left (hcol j) (by positivity) noncomputable def normalizedInfluenceApprox {n : ℕ} (β : ℝ) (J : Interaction n) (x : Spin n) : Interaction n := fun i j => J i j + 2*β^2/(n:ℝ)*spin x i*mean J x j lemma normalizedInfluence_sub_approx {n : ℕ} (β : ℝ) (g : GaussianCoordinates n) (x : Spin n) : normalizedInfluence (sampledInteraction g) x - normalizedInfluenceApprox β (sampledInteraction g) x = influenceCubicError (sampledInteraction g) x + (2:ℝ) • (Matrix.diagonal (spin x) * RandomMatrix.sampledSquaredDeviation β g * Matrix.diagonal (mean (sampledInteraction g) x)) := by ext i j simp only [Matrix.sub_apply, Matrix.add_apply, Matrix.smul_apply, smul_eq_mul, Matrix.mul_diagonal, Matrix.diagonal_mul, normalizedInfluenceApprox, influenceCubicError, RandomMatrix.sampledSquaredDeviation, sampledInteraction_symm g j i] ring lemma normalizedInfluence_approx_norm_le {n : ℕ} (β : ℝ) (g : GaussianCoordinates n) (x : Spin n) (c B : ℝ) (hc : 0 ≤ c) (hc1 : c ≤ 1/4) (hB : 0 ≤ B) (hmax : ∀ i j, |sampledInteraction g i j| ≤ c) (hcol : ∀ i, ∑ j, sampledInteraction g j i^2 ≤ B) : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluence (sampledInteraction g) x - normalizedInfluenceApprox β (sampledInteraction g) x)‖ ≤ 16*c*B+2*‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (RandomMatrix.sampledSquaredDeviation β g)‖ := by have hs : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.diagonal (spin x))‖ ≤ 1 := matrix_diagonal_opNorm_le _ 1 (by norm_num) (fun i => (abs_spin_eq_one x i).le) have hm : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.diagonal (mean (sampledInteraction g) x))‖ ≤ 1 := matrix_diagonal_opNorm_le _ 1 (by norm_num) (fun i => (Real.abs_tanh_lt_one _).le) have hp : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.diagonal (spin x) * RandomMatrix.sampledSquaredDeviation β g * Matrix.diagonal (mean (sampledInteraction g) x))‖ ≤ ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (RandomMatrix.sampledSquaredDeviation β g)‖ := by rw [map_mul, map_mul] calc _ ≤ _ * _ := norm_mul_le _ _ _ ≤ (‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.diagonal (spin x))‖ * ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (RandomMatrix.sampledSquaredDeviation β g)‖) * ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.diagonal (mean (sampledInteraction g) x))‖ := by exact mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _) _ ≤ (1 * ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (RandomMatrix.sampledSquaredDeviation β g)‖) * 1 := by gcongr _ = _ := by ring rw [normalizedInfluence_sub_approx, map_add] calc _ ≤ _ + _ := norm_add_le _ _ _ ≤ _ := add_le_add (influenceCubicError_norm_le _ (sampledInteraction_symm g) x c B hc hc1 hB hmax hcol) (by rw [map_smul, norm_smul]; norm_num only [Real.norm_eq_abs, abs_of_pos (show (0:ℝ)<2 by norm_num)]; exact mul_le_mul_of_nonneg_left hp (by norm_num)) lemma normalizedInfluence_approx_tendsto (β ε : ℝ) (hε : 0 < ε) : Tendsto (fun n => disorderLaw β n {g | ∃ x : Spin n, ε < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluence (sampledInteraction g) x - normalizedInfluenceApprox β (sampledInteraction g) x)‖}) atTop (nhds 0) := by let B : ℝ := (2*(β^2+10))^2 let c : ℝ := min (1/4) (ε/(64*(B+1))) have hB : 0 ≤ B := sq_nonneg _ have hc : 0 < c := lt_min (by norm_num) (by positivity) have hc1 : c ≤ 1/4 := min_le_left _ _ have hcd : 16*c*B ≤ ε/4 := by calc _ ≤ 16*(ε/(64*(B+1)))*(B+1) := mul_le_mul (mul_le_mul_of_nonneg_left (min_le_right _ _) (by norm_num)) (by linarith) hB (by positivity) _ = ε/4 := by field_simp; ring have hl := (RandomMatrix.sampled_opNorm_tendsto β).add ((RandomMatrix.sampled_maxEntry_tendsto β c hc).add (RandomMatrix.sampledSquaredDeviation_tendsto β (ε/8) (by positivity))) simp only [add_zero] at hl apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hl (Eventually.of_forall (fun _ => bot_le)) apply Eventually.of_forall intro n let N : Set (GaussianCoordinates n) := {g | 2*(β^2+10) < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (sampledInteraction g)‖} let E : Set (GaussianCoordinates n) := {g | ∃ i j, c ≤ |sampledInteraction g i j|} let V : Set (GaussianCoordinates n) := {g | ε/8 < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (RandomMatrix.sampledSquaredDeviation β g)‖} have hs : {g | ∃ x : Spin n, ε < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluence (sampledInteraction g) x - normalizedInfluenceApprox β (sampledInteraction g) x)‖} ⊆ N ∪ (E ∪ V) := by intro g hg by_cases hN : g ∈ N · exact Or.inl hN by_cases hE : g ∈ E · exact Or.inr (Or.inl hE) by_cases hV : g ∈ V · exact Or.inr (Or.inr hV) have hn : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (sampledInteraction g)‖ ≤ 2*(β^2+10) := le_of_not_gt hN have he (i j : Fin n) : |sampledInteraction g i j| ≤ c := (not_le.mp (fun h => hE ⟨i,j,h⟩)).le have hv : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (RandomMatrix.sampledSquaredDeviation β g)‖ ≤ ε/8 := le_of_not_gt hV obtain ⟨x,hx⟩ := hg have hp := normalizedInfluence_approx_norm_le β g x c B hc.le hc1 hB he (fun i => (RandomMatrix.matrix_column_sq_le _ i).trans (pow_le_pow_left₀ (norm_nonneg _) hn 2)) exfalso linarith exact (measure_mono hs).trans ((measure_union_le N (E ∪ V)).trans (add_le_add le_rfl (measure_union_le E V))) noncomputable def influenceApprox {n : ℕ} (β : ℝ) (J : Interaction n) (x : Spin n) : Interaction n := normalizedInfluenceApprox β J x * Matrix.diagonal (siteVariance J x) lemma influence_eq_normalized_mul {n : ℕ} (J : Interaction n) (x : Spin n) : influence J x = normalizedInfluence J x * Matrix.diagonal (siteVariance J x) := by ext i j simp only [Matrix.mul_diagonal, normalizedInfluence] rw [div_mul_cancel₀ _ (siteVariance_pos J x j).ne'] lemma influenceApprox_apply {n : ℕ} (β : ℝ) (J : Interaction n) (x : Spin n) (i j : Fin n) : influenceApprox β J x i j = J i j*siteVariance J x j + 2*β^2/(n:ℝ)*spin x i*(mean J x j*siteVariance J x j) := by simp only [influenceApprox, Matrix.mul_diagonal, normalizedInfluenceApprox] ring lemma influence_approx_error_le {n : ℕ} (β : ℝ) (J : Interaction n) (x : Spin n) : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (influence J x - influenceApprox β J x)‖ ≤ ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluence J x - normalizedInfluenceApprox β J x)‖ := by have hv : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.diagonal (siteVariance J x))‖ ≤ 1 := by apply matrix_diagonal_opNorm_le _ 1 (by norm_num) intro i rw [abs_of_pos (siteVariance_pos J x i)] exact siteVariance_le_one J x i rw [influence_eq_normalized_mul, influenceApprox, ← Matrix.sub_mul, map_mul] exact (norm_mul_le _ _).trans (by simpa only [mul_one] using mul_le_mul_of_nonneg_left hv (norm_nonneg _)) lemma influence_approx_tendsto (β ε : ℝ) (hε : 0 < ε) : Tendsto (fun n => disorderLaw β n {g | ∃ x : Spin n, ε < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (influence (sampledInteraction g) x - influenceApprox β (sampledInteraction g) x)‖}) atTop (nhds 0) := by apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds (normalizedInfluence_approx_tendsto β ε hε) (Eventually.of_forall (fun _ => bot_le)) apply Eventually.of_forall intro n apply measure_mono rintro g ⟨x,hx⟩ exact ⟨x,hx.trans_le (influence_approx_error_le β (sampledInteraction g) x)⟩ lemma normalizedInfluenceApprox_norm_le {n : ℕ} (β : ℝ) (J : Interaction n) (x : Spin n) : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluenceApprox β J x)‖ ≤ ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) J‖ + 2*β^2 := by let R : Interaction n := fun i j => 2*β^2/(n:ℝ)*spin x i*mean J x j have he : normalizedInfluenceApprox β J x = J+R := rfl have hr : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) R‖ ≤ 2*β^2 := by by_cases hn : n=0 · subst n have hz : R=0 := Subsingleton.elim _ _ rw [hz, map_zero, norm_zero]; positivity have hn0 : 0 < (n:ℝ) := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn) have hp (i j : Fin n) : |R i j| ≤ 2*β^2/(n:ℝ) := by change |2*β^2/(n:ℝ)*spin x i*mean J x j| ≤ _ rw [abs_mul, abs_mul, abs_spin_eq_one, mul_one, abs_of_nonneg (by positivity : 0 ≤ 2*β^2/(n:ℝ))] exact (mul_le_mul_of_nonneg_left (Real.abs_tanh_lt_one _).le (by positivity)).trans_eq (mul_one _) apply matrix_opNorm_le_schur _ (2*β^2) (by positivity) · intro i calc _ ≤ ∑ j : Fin n, 2*β^2/(n:ℝ) := Finset.sum_le_sum (fun j _ => hp i j) _ = _ := by simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]; field_simp · intro j calc _ ≤ ∑ i : Fin n, 2*β^2/(n:ℝ) := Finset.sum_le_sum (fun i _ => hp i j) _ = _ := by simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]; field_simp rw [he, map_add] exact (norm_add_le _ _).trans (add_le_add le_rfl hr) lemma normalizedInfluence_opNorm_tight (β : ℝ) : Tendsto (fun n => disorderLaw β n {g | ∃ x : Spin n, 2*(β^2+10)+2*β^2+1 < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluence (sampledInteraction g) x)‖}) atTop (nhds 0) := by have hl := (RandomMatrix.sampled_opNorm_tendsto β).add (normalizedInfluence_approx_tendsto β 1 (by norm_num)) simp only [add_zero] at hl apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hl (Eventually.of_forall (fun _ => bot_le)) apply Eventually.of_forall intro n refine (measure_mono (show {g | ∃ x : Spin n, 2*(β^2+10)+2*β^2+1 < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluence (sampledInteraction g) x)‖} ⊆ {g | 2*(β^2+10) < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (sampledInteraction g)‖} ∪ {g | ∃ x : Spin n, 1 < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluence (sampledInteraction g) x - normalizedInfluenceApprox β (sampledInteraction g) x)‖} from ?_)).trans (measure_union_le _ _) rintro g ⟨x,hx⟩ by_cases hJ : 2*(β^2+10) < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (sampledInteraction g)‖ · exact Or.inl hJ apply Or.inr refine ⟨x,?_⟩ have hh := normalizedInfluenceApprox_norm_le β (sampledInteraction g) x have ht := norm_le_norm_sub_add (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluence (sampledInteraction g) x)) (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluenceApprox β (sampledInteraction g) x)) rw [← map_sub] at ht linarith [le_of_not_gt hJ] end SKGapCutoff open scoped BigOperators Matrix Matrix.Norms.Elementwise open MeasureTheory ProbabilityTheory Filter Set end end OAI