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TauCeti.Probability.FreeProbability.Cumulant.Free

Freeness is the vanishing of mixed free cumulants #

Let φ : A →ₗ[R] R be a linear functional on an algebra A over a commutative ring R, and let B i be unital subalgebras of A. A free cumulant κ_n(a₁, …, aₙ) is mixed for the family (B i) when aⱼ ∈ B iⱼ for labels iⱼ that are not all equal. For a normalised φ, the family is freely independent exactly when every mixed free cumulant vanishes. This turns freeness, a condition on centred alternating words, into the vanishing of the off-diagonal cumulants, which is what makes cumulants additive on free families and underlies the free convolution.

The generated-algebra test refines this: freeness of the unital subalgebras generated by sets G i only needs the vanishing of the mixed free cumulants whose letters are generators, and in particular elements are freely independent exactly when their own mixed free cumulants vanish.

The words are indexed by a finite set s of positions in a linear order, so that the results apply to every subword of a word; the cumulants κ_n are the case s = Finset.univ : Finset (Fin n).

Main results #

References #

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theorem TauCeti.IsFreelyIndependent.freeCumulant_eq_zero {R : Type u_1} {A : Type u_2} {ι : Type u_3} {α : Type u_4} [CommRing R] [Semiring A] [Algebra R A] [LinearOrder α] {φ : A →ₗ[R] R} {B : ι → Subalgebra R A} {s : Finset α} {ℓ : α → ι} (h : IsFreelyIndependent φ B) (hφ : φ 1 = 1) {a : α → A} (ha : ∀ x ∈ s, a x ∈ B (ℓ x)) {x y : α} (hx : x ∈ s) (hy : y ∈ s) (hxy : ℓ x ≠ ℓ y) :
freeCumulant (⇑φ) a s = 0

Free cumulants detect freeness, the vanishing direction: if the subalgebras B i are freely independent for a normalised φ, then the free cumulant κ_s(a) of a word whose letters satisfy a x ∈ B (ℓ x) vanishes as soon as two of its labels differ.

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theorem TauCeti.isFreelyIndependent_of_freeCumulant_eq_zero {R : Type u_1} {A : Type u_2} {ι : Type u_3} [CommRing R] [Semiring A] [Algebra R A] {φ : A →ₗ[R] R} {B : ι → Subalgebra R A} (h : ∀ (n : ℕ) (a : Fin n → A) (ℓ : Fin n → ι), (∀ (k : Fin n), a k ∈ B (ℓ k)) → ∀ (k k' : Fin n), ℓ k ≠ ℓ k' → freeCumulant (⇑φ) a Finset.univ = 0) :
IsFreelyIndependent φ B

Free cumulants detect freeness, the converse direction: if every mixed free cumulant κ_n(a₁, …, aₙ), with letters aₖ ∈ B (ℓ k) whose labels are not all equal, vanishes, then the subalgebras B i are freely independent. No normalisation of φ is needed.

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theorem TauCeti.isFreelyIndependent_iff_freeCumulant_eq_zero {R : Type u_1} {A : Type u_2} {ι : Type u_3} [CommRing R] [Semiring A] [Algebra R A] {φ : A →ₗ[R] R} {B : ι → Subalgebra R A} (hφ : φ 1 = 1) :
IsFreelyIndependent φ B ↔ ∀ (n : ℕ) (a : Fin n → A) (ℓ : Fin n → ι), (∀ (k : Fin n), a k ∈ B (ℓ k)) → ∀ (k k' : Fin n), ℓ k ≠ ℓ k' → freeCumulant (⇑φ) a Finset.univ = 0

Free cumulants detect freeness: for a normalised φ, the subalgebras B i are freely independent exactly when every mixed free cumulant κ_n(a₁, …, aₙ), with letters aₖ ∈ B (ℓ k) whose labels are not all equal, vanishes.

The generated-algebra test #

For a normalised φ, freeness of the unital subalgebras generated by sets G i can be tested on the generators alone: it suffices that the mixed free cumulants with letters in the sets G i vanish. By multilinearity it is enough to consider letters that are products of generators, and such a product letter is split into two adjacent letters, one generator at a time; by the formula for cumulants with products as arguments, splitting does not change a mixed cumulant once the mixed cumulants of the shorter words vanish.

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theorem TauCeti.isFreelyIndependent_adjoin_of_freeCumulant_eq_zero {R : Type u_1} {A : Type u_2} {ι : Type u_3} [CommRing R] [Semiring A] [Algebra R A] {φ : A →ₗ[R] R} {G : ι → Set A} (hφ : φ 1 = 1) (h : ∀ (n : ℕ) (a : Fin n → A) (ℓ : Fin n → ι), (∀ (k : Fin n), a k ∈ G (ℓ k)) → ∀ (k k' : Fin n), ℓ k ≠ ℓ k' → freeCumulant (⇑φ) a Finset.univ = 0) :
IsFreelyIndependent φ fun (i : ι) => Algebra.adjoin R (G i)

The generated-algebra test for freeness: for a normalised φ, if every mixed free cumulant κ_n(a₁, …, aₙ) with letters aₖ ∈ G (ℓ k) in the generating sets, whose labels are not all equal, vanishes, then the unital subalgebras generated by the sets G i are freely independent.

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theorem TauCeti.isFreelyIndependent_adjoin_iff_freeCumulant_eq_zero {R : Type u_1} {A : Type u_2} {ι : Type u_3} [CommRing R] [Semiring A] [Algebra R A] {φ : A →ₗ[R] R} {G : ι → Set A} (hφ : φ 1 = 1) :
(IsFreelyIndependent φ fun (i : ι) => Algebra.adjoin R (G i)) ↔ ∀ (n : ℕ) (a : Fin n → A) (ℓ : Fin n → ι), (∀ (k : Fin n), a k ∈ G (ℓ k)) → ∀ (k k' : Fin n), ℓ k ≠ ℓ k' → freeCumulant (⇑φ) a Finset.univ = 0

The generated-algebra test for freeness: for a normalised φ, the unital subalgebras generated by the sets G i are freely independent exactly when every mixed free cumulant κ_n(a₁, …, aₙ) with letters aₖ ∈ G (ℓ k) in the generating sets, whose labels are not all equal, vanishes.

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theorem TauCeti.isFreelyIndependentElements_iff_freeCumulant_eq_zero {R : Type u_1} {A : Type u_2} {ι : Type u_3} [CommRing R] [Semiring A] [Algebra R A] {φ : A →ₗ[R] R} (hφ : φ 1 = 1) {c : ι → A} :
IsFreelyIndependentElements φ c ↔ ∀ (n : ℕ) (w : Fin n → ι) (k k' : Fin n), w k ≠ w k' → freeCumulant (⇑φ) (c ∘ w) Finset.univ = 0

Free cumulants detect free elements: for a normalised φ, the elements c i are freely independent exactly when every mixed free cumulant κ_n(c_{w 1}, …, c_{w n}) of the elements, for a word w whose labels are not all equal, vanishes.

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theorem TauCeti.IsFreelyIndependent.freeCumulant_sum {R : Type u_1} {A : Type u_2} {ι : Type u_3} {α : Type u_4} [CommRing R] [Semiring A] [Algebra R A] [LinearOrder α] {φ : A →ₗ[R] R} {B : ι → Subalgebra R A} {s : Finset α} (h : IsFreelyIndependent φ B) (hφ : φ 1 = 1) {T : Finset ι} {a : ι → α → A} (ha : ∀ k ∈ T, ∀ x ∈ s, a k x ∈ B k) (hs : s.Nonempty) :
freeCumulant (⇑φ) (fun (x : α) => ∑ k ∈ T, a k x) s = ∑ k ∈ T, freeCumulant (⇑φ) (a k) s

Free cumulants are additive on free families: if the letters of the words a k lie in the freely independent subalgebras B k, for a normalised φ, then the free cumulant of the sum of these words is the sum of their free cumulants, κ_s(∑ k ∈ T, a k) = ∑ k ∈ T, κ_s(a k) for nonempty s.

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theorem TauCeti.IsFreelyIndependent.freeCumulant_add {R : Type u_1} {A : Type u_2} {ι : Type u_3} {α : Type u_4} [CommRing R] [Semiring A] [Algebra R A] [LinearOrder α] {φ : A →ₗ[R] R} {B : ι → Subalgebra R A} {s : Finset α} (h : IsFreelyIndependent φ B) (hφ : φ 1 = 1) {i j : ι} (hij : i ≠ j) {a b : α → A} (ha : ∀ x ∈ s, a x ∈ B i) (hb : ∀ x ∈ s, b x ∈ B j) (hs : s.Nonempty) :
freeCumulant (⇑φ) (fun (x : α) => a x + b x) s = freeCumulant (⇑φ) a s + freeCumulant (⇑φ) b s

Free cumulants are additive on two free families: if the letters of a lie in B i and those of b in B j for i ≠ j, with B freely independent for a normalised φ, then κ_s(a + b) = κ_s(a) + κ_s(b) for nonempty s.

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theorem TauCeti.IsFreelyIndependentElements.freeCumulant_sum {R : Type u_1} {A : Type u_2} {ι : Type u_3} {α : Type u_4} [CommRing R] [Semiring A] [Algebra R A] [LinearOrder α] {φ : A →ₗ[R] R} {s : Finset α} {c : ι → A} (h : IsFreelyIndependentElements φ c) (hφ : φ 1 = 1) (T : Finset ι) (hs : s.Nonempty) :
freeCumulant (⇑φ) (fun (x : α) => ∑ k ∈ T, c k) s = ∑ k ∈ T, freeCumulant (⇑φ) (fun (x : α) => c k) s

Free cumulants are additive on free elements: if the elements c k are freely independent for a normalised φ, then κ_n(∑ k ∈ T, c k) = ∑ k ∈ T, κ_n(c k) for n ≥ 1, where κ_n(c) is the free cumulant of the constant word (c, …, c) along a nonempty set of n positions.

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theorem TauCeti.IsFreelyIndependentElements.freeCumulant_add {R : Type u_1} {A : Type u_2} {ι : Type u_3} {α : Type u_4} [CommRing R] [Semiring A] [Algebra R A] [LinearOrder α] {φ : A →ₗ[R] R} {s : Finset α} {c : ι → A} (h : IsFreelyIndependentElements φ c) (hφ : φ 1 = 1) {i j : ι} (hij : i ≠ j) (hs : s.Nonempty) :
freeCumulant (⇑φ) (fun (x : α) => c i + c j) s = freeCumulant (⇑φ) (fun (x : α) => c i) s + freeCumulant (⇑φ) (fun (x : α) => c j) s

Free cumulants are additive on two free elements: if c i and c j, for i ≠ j, belong to a freely independent family for a normalised φ, then κ_n(c i + c j) = κ_n(c i) + κ_n(c j) for n ≥ 1, where κ_n(c) is the free cumulant of the constant word (c, …, c) along a nonempty set of n positions.

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theorem TauCeti.IsFreelyIndependentElements.apply_sum_pow_eq_sum_prod_sum_freeCumulant {R : Type u_1} {A : Type u_2} {ι : Type u_3} [CommRing R] [Semiring A] [Algebra R A] {φ : A →ₗ[R] R} {c : ι → A} (h : IsFreelyIndependentElements φ c) (hφ : φ 1 = 1) (T : Finset ι) (n : ℕ) :
φ ((∑ k ∈ T, c k) ^ n) = ∑ π : NoncrossingPartition Finset.univ, ∏ B ∈ (↑π).parts, ∑ k ∈ T, freeCumulant (⇑φ) (fun (x : Fin n) => c k) B

Moments of a sum of free elements: if the elements c k are freely independent for a normalised φ, then the moments of their sum over T are given by the moment–cumulant formula with the summed cumulants of the individual elements, φ ((∑ k ∈ T, c k) ^ n) = ∑_{π ∈ NC(n)} ∏_{B ∈ π} ∑ k ∈ T, κ_B(c k), where κ_B(c k) is the free cumulant of the constant word (c k, …, c k) along the part B.