arXiv subjects

Janusz Wysoczański

Publications and source records attributed to Janusz Wysoczański.

6 recordsLinked to original sources

arXiv2025

General limit theorems for mixtures of free, monotone, and boolean independence

We study mixtures of free, monotone, and Boolean independence described by a directed graph $G = (V,E)$ in the context of $\mathcal{T}$-free convolutions of Jekel and Liu. We prove general limit theorems for the associated additive convolution operations $\boxplus_G$. For a sequence of digraphs $G_n = (V_n,E_n)$, we give sufficient conditions for the limit $\widehatμ = \lim_{n \to \infty} \boxplus_{G_n}(μ_n)$ to exist whenever the Boolean convolution powers $μ_n^{\uplus |V_n|}$ converge to some $μ$. This in particular includes central limit and Poisson limit theorems, as well as limit theorems for each classical domain of attraction. The hypothesis on the sequence of $G_n$ is that the normalized counts of digraph homomorphisms from rooted trees into $G_n$ converge as $n \to \infty$, and we verify this for several families of examples where the $G_n$'s converge in some sense to a continuum limit, or digraphon. In particular, we obtain a new limit theorem for multiregular digraphs, as well as recovering several limit theorems in prior work.

math.PR↗

arXiv2025

Finitely generated weakly monotone C*-algebra

We consider the $C^*$-algebra generated by finitely many annihilation operators acting on the weakly monotone Fock space, and we call it weakly monotone $C^*$-algebra. We give an abstract representation for this algebra, showing that it is isomorphic to a suitable quotient of a Cuntz-Krieger $C^*$-algebra $\mathcal{O}_A$ corresponding to a suitable matrix $A$. Furthermore, we show that the diagonal subalgebra of the weakly monotone $C^*$-algebra is a MASA and we give the detailed description of its Gelfand spectrum.

math.OA↗

arXiv2025

Weakly-monotone C*-algebras as Exel-Laca algebras

An abstract characterization of weakly monotone $C^*$-algebras, namely the concrete $C^*$-algebras generated by creators and annihilators acting on the so-called weakly monotone Fock spaces, is given in terms of (quotient of) suitable Exel-Laca algebras. The weakly monotone $C^*$-algebra indexed by $\mathbb{N}$ is shown to be a type-I $C^*$-algebra and its representation theory is entirely determined, whereas the weakly monotone $C^*$-algebra indexed by $\mathbb{Z}$ is shown not to be of type $I$.

math.OA↗

arXiv2023

Analogues of Poisson type limit theorems in discrete bm-Fock spaces

We present analogues of the Poisson limit distribution for the noncommutative bm-independence, which is associated with several positive symmetric cones. We construct related discrete Fock spaces with creation, annihilation and conservation operators, and prove Poisson type limit theorems for them. Properties of the positive cones, in particular the volume characteristic property they enjoy, and the combinatorics of labelled noncrossing partitions, play crucial role in these considerations.

math.CO↗

arXiv2017

Fock representations of $Q$-deformed commutation relations

We consider Fock representations of the $Q$-deformed commutation relations $$\partial_s\partial^†_t=Q(s,t)\partial_t^†\partial_s+δ(s,t), \quad s,t\in T.$$ Here $T:=\mathbb R^d$ (or more generally $T$ is a locally compact Polish space), the function $Q:T^2\to \mathbb C$ satisfies $|Q(s,t)|\le1$ and $Q(s,t)=\overline{Q(t,s)}$, and $$\int_{T^2}h(s)g(t)δ(s,t)\,σ(ds)σ(dt):=\int_T h(t)g(t)\,σ(dt),$$ $σ$ being a fixed reference measure on $T$. In the case where $|Q(s,t)|\equiv 1$, the $Q$-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev (1995). These generalized statistics contain anyon statistics as a special case (with $T=\mathbb R^2$ and a special choice of the function $Q$). The related $Q$-deformed Fock space $\mathcal F(\mathcal H)$ over $\mathcal H:=L^2(T\to\mathbb C,σ)$ is constructed. An explicit form of the orthogonal projection of $\mathcal H^{\otimes n}$ onto the $n$-particle space $\mathcal F_n(\mathcal H)$ is derived. A scalar product in $\mathcal F_n(\mathcal H)$ is given by an operator $\mathcal P_n\ge0$ in $\mathcal H^{\otimes n}$ which is strictly positive on $\mathcal F_n(\mathcal H)$. We realize the smeared operators $\partial_t^†$ and $\partial_t$ as creation and annihilation operators in $\mathcal F(\mathcal H)$, respectively. Additional $Q$-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form $\partial^†_s\partial^†_t=Q(t,s)\partial^†_t\partial^†_s$, $\partial_s\partial_t=Q(t,s)\partial_t\partial_s$, valid for those $s,t\in T$ for which $|Q(s,t)|=1$.

math-ph↗

arXiv2016

Rank two perturbations of matrices and operators and operator model for t-transformation of probability measures

Rank two parametric perturbations of operators and matrices are studied in various settings. In the finite dimensional case the formula for a characteristic polynomial is derived and the large parameter asymptotics of the spectrum is computed. The large parameter asymptotics of a rank one perturbation of singular values and condition number are discussed as well. In the operator case the formula for a rank two transformation of the spectral measure is derived and it appears to be the t-transformation of a probability measure, studied previously in the free probability context. New transformation of measures is studied and several examples are presented.

math.FA↗