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import OAI.Probability.SKGap.Localization.ScalarEigenfunction
namespace OAI
section
open scoped BigOperators
namespace SKGapCutoff
noncomputable def centered {n : ℕ} (J : Interaction n) (f : Observables n) : Observables n :=
fun x => f x - gibbsExpectation J f
@[simp] lemma gibbsExpectation_centered {n : ℕ} (J : Interaction n) (f : Observables n) :
gibbsExpectation J (centered J f) = 0 := by
unfold centered
rw [gibbsExpectation_sub, gibbsExpectation_const, sub_self]
@[simp] lemma halfDiff_centered {n : ℕ} (J : Interaction n) (f : Observables n)
(i : Fin n) (x : Spin n) : halfDiff i (centered J f) x = halfDiff i f x := by
simp only [halfDiff, centered]
ring
lemma dirichlet_centered {n : ℕ} (J : Interaction n) (f g : Observables n) :
dirichlet J (centered J f) (centered J g) = dirichlet J f g := by
simp only [dirichlet, halfDiff_centered]
lemma centered_variance {n : ℕ} (J : Interaction n) (f : Observables n) :
stationaryInner J (centered J f) (centered J f) = gibbsVariance J f := by
simp only [stationaryInner, centered, gibbsVariance, pow_two]
lemma spectral_dirichlet {n : ℕ} (J : Interaction n)
(hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) (f : Observables n) :
dirichlet J f f = ∑ a, spinEigenvalues J hJ hdiag a *
((spinEigenbasis J hJ hdiag).repr (gibbsEuclideanEquiv J f) a) ^ 2 := by
rw [← gibbs_dirichlet J hJ hdiag]
change stationaryInner J f (-generator J f) = _
rw [← spectral_parseval J hJ hdiag]
simp_rw [spectral_generator_coordinates]
apply Finset.sum_congr rfl
intro a _
ring
lemma centered_spectral_coordinate_zero {n : ℕ} (J : Interaction n)
(hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) (f : Observables n)
(a : Fin (Fintype.card (Spin n))) (ha : spinEigenvalues J hJ hdiag a = 0) :
(spinEigenbasis J hJ hdiag).repr (gibbsEuclideanEquiv J (centered J f)) a = 0 := by
have hg : generator J (scalarEigenfunction J hJ hdiag a) = 0 := by
rw [generator_scalarEigenfunction, ha, neg_zero, zero_smul]
have hc (x : Spin n) : scalarEigenfunction J hJ hdiag a x =
scalarEigenfunction J hJ hdiag a (fun _ => false) :=
constant_of_generator_zero J hJ hdiag _ hg x _
rw [spectral_coordinates_inner]
simp only [stationaryInner, hc, gibbsExpectation_mul_const, gibbsExpectation_centered,
mul_zero]
lemma hasGap_of_positive_eigenvalues {n : ℕ} (J : Interaction n)
(hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) (γ : ℝ)
(hγ : ∀ a, 0 < spinEigenvalues J hJ hdiag a → γ ≤ spinEigenvalues J hJ hdiag a) :
HasGap J γ := by
intro f
rw [← centered_variance, ← dirichlet_centered J f f, spectral_dirichlet J hJ hdiag,
← spectral_parseval J hJ hdiag, Finset.mul_sum]
apply Finset.sum_le_sum
intro a _
by_cases ha : spinEigenvalues J hJ hdiag a = 0
· simp only [centered_spectral_coordinate_zero J hJ hdiag f a ha, mul_zero, zero_pow, ne_eq,
OfNat.ofNat_ne_zero, not_false_eq_true, le_refl]
· have hp := lt_of_le_of_ne (spinEigenvalues_nonneg J hJ hdiag a) (Ne.symm ha)
simpa only [pow_two, mul_assoc] using
mul_le_mul_of_nonneg_right (hγ a hp)
(sq_nonneg ((spinEigenbasis J hJ hdiag).repr (gibbsEuclideanEquiv J (centered J f)) a))
theorem finite_gap_pos {n : ℕ} (J : Interaction n)
(hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) :
∃ γ : ℝ, 0 < γ ∧ HasGap J γ := by
let values : Finset ℝ := insert 1 ((Finset.univ.image (spinEigenvalues J hJ hdiag)).filter (0 < ·))
have hne : values.Nonempty := ⟨1, Finset.mem_insert_self _ _⟩
refine ⟨values.min' hne, ?_, hasGap_of_positive_eigenvalues J hJ hdiag _ ?_⟩
· have hm := Finset.min'_mem values hne
rcases Finset.mem_insert.mp hm with h | h
· rw [h]; norm_num
· exact (Finset.mem_filter.mp h).2
· intro a ha
apply Finset.min'_le
exact Finset.mem_insert_of_mem (Finset.mem_filter.mpr
⟨Finset.mem_image.mpr ⟨a, Finset.mem_univ _, rfl⟩, ha⟩)
end SKGapCutoff
namespace SKGapCutoff
lemma scalarEigenfunction_centered {n : ℕ} (J : Interaction n)
(hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0)
(a : Fin (Fintype.card (Spin n))) (ha : spinEigenvalues J hJ hdiag a ≠ 0) :
gibbsExpectation J (scalarEigenfunction J hJ hdiag a) = 0 := by
have hz := gibbs_generator_zero J hJ hdiag (scalarEigenfunction J hJ hdiag a)
rw [generator_scalarEigenfunction] at hz
change gibbsExpectation J (fun x => -spinEigenvalues J hJ hdiag a *
scalarEigenfunction J hJ hdiag a x) = 0 at hz
rw [gibbsExpectation_mul_const] at hz
exact (mul_eq_zero.mp hz).resolve_left (neg_ne_zero.mpr ha)
lemma scalarEigenfunction_inner_self {n : ℕ} (J : Interaction n)
(hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0)
(a : Fin (Fintype.card (Spin n))) :
stationaryInner J (scalarEigenfunction J hJ hdiag a)
(scalarEigenfunction J hJ hdiag a) = 1 := by
rw [← gibbsEuclideanEquiv_inner, lift_scalarEigenfunction,
real_inner_self_eq_norm_sq, (spinEigenbasis J hJ hdiag).orthonormal.norm_eq_one, one_pow]
lemma gap_le_positive_eigenvalue {n : ℕ} (J : Interaction n)
(hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0)
{γ : ℝ} (hγ : HasGap J γ) (a : Fin (Fintype.card (Spin n)))
(ha : 0 < spinEigenvalues J hJ hdiag a) : γ ≤ spinEigenvalues J hJ hdiag a := by
have hv : gibbsVariance J (scalarEigenfunction J hJ hdiag a) = 1 := by
rw [gibbsVariance, scalarEigenfunction_centered J hJ hdiag a (ne_of_gt ha)]
simp only [sub_zero, pow_two]
exact scalarEigenfunction_inner_self J hJ hdiag a
have hd : dirichlet J (scalarEigenfunction J hJ hdiag a)
(scalarEigenfunction J hJ hdiag a) = spinEigenvalues J hJ hdiag a := by
rw [← gibbs_dirichlet J hJ hdiag, generator_scalarEigenfunction]
simp only [Pi.smul_apply, smul_eq_mul, neg_mul, neg_neg]
simp_rw [mul_left_comm (scalarEigenfunction J hJ hdiag a _) (spinEigenvalues J hJ hdiag a)]
rw [gibbsExpectation_mul_const]
change spinEigenvalues J hJ hdiag a * stationaryInner J
(scalarEigenfunction J hJ hdiag a) (scalarEigenfunction J hJ hdiag a) = _
rw [scalarEigenfunction_inner_self, mul_one]
simpa only [hv, hd, mul_one] using hγ (scalarEigenfunction J hJ hdiag a)
theorem hasGap_iff_positive_eigenvalues {n : ℕ} (J : Interaction n)
(hJ : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) (γ : ℝ) :
HasGap J γ ↔ ∀ a, 0 < spinEigenvalues J hJ hdiag a →
γ ≤ spinEigenvalues J hJ hdiag a :=
⟨fun h a ha => gap_le_positive_eigenvalue J hJ hdiag h a ha,
hasGap_of_positive_eigenvalues J hJ hdiag γ⟩
theorem quenchedGap_iff_spectral (β : ℝ) : QuenchedGap β ↔
∃ γ : ℝ, 0 < γ ∧ Filter.Tendsto (fun n => (disorderLaw β n)
{g | ∀ a, 0 < spinEigenvalues (sampledInteraction g)
(sampledInteraction_symm g) (sampledInteraction_diag g) a →
γ ≤ spinEigenvalues (sampledInteraction g)
(sampledInteraction_symm g) (sampledInteraction_diag g) a})
Filter.atTop (nhds 1) := by
simp only [QuenchedGap, hasGap_iff_positive_eigenvalues (sampledInteraction _)
(sampledInteraction_symm _) (sampledInteraction_diag _)]
end SKGapCutoff
end
end OAI
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