arXiv subjects

Marek Bożejko

Publications and source records attributed to Marek Bożejko.

9 recordsLinked to original sources

arXiv2025

A Poisson Type Operator Deformed by Generalized Fibonacci Numbers and Its Combinatorial Moment Formula

We introduce a two-parameter deformation of the classical Poisson distribution from the viewpoint of noncommutative probability theory, by defining a $(q,t)$-Poisson type operator (random variable) on the $(q,t)$-Fock space \cite{Bl12} (See also \cite{BY06, AY20}). From the analogous viewpoint of the classical Poisson limit theorem in probability theory, we are naturally led to a family of orthogonal polynomials, which we call the $(q,t)$-Charlier polynomials. These generalize the $q$-Charlier polynomials of Saitoh-Yoshida \cite{SY00a, SY00b} and reflect deeper combinatorial symmetries through the additional deformation parameter $t$. A central feature of this paper is the derivation of a combinatorial moment formula of the $(q,t)$-Poisson type operator and the $(q,t)$-Poisson distribution. This is accomplished by means of a card arrangement technique, which encodes set partitions together with crossing and nesting statistics. The resulting expression naturally exhibits a duality between these statistics, arising from a structure rooted in generalized Fibonacci numbers. Our approach provides a concrete framework where methods in combinatorics and theory of orthogonal polynomials are used to investigate the probabilistic properties arising from the $(q,t)$-deformation.

math.CO↗

arXiv2023

The Double Fock Space of Type B

In this article, we introduce the notion of a double Fock space of type B. We will show that this new construction is compatible with combinatorics of counting positive and negative inversions on a hyperoctahedral group.

math.FA↗

arXiv2021

Reflection length with two parameters in the asymptotic representation theory of type B/C and applications

We introduce a two-parameter function $ϕ_{q_+,q_-}$ on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length keeping track of the long and the short reflections separately. We show that this signed reflection function $ϕ_{q_+,q_-}$ is positive definite if and only if it is an extreme character of the infinite hyperoctahedral group and we classify the corresponding set of parameters $q_+,q_-$. We construct the corresponding representations through a natural action of the hyperoctahedral group $B(n)$ on the tensor product of $n$ copies of a vector space, which gives a two-parameter analog of the classical construction of Schur--Weyl. We apply our classification to construct a cyclic Fock space of type B generalizing the one-parameter construction in type A found previously by Bożejko and Guta. We also construct a new Gaussian operator acting on the cyclic Fock space of type B and we relate its moments with the Askey--Wimp--Kerov distribution by using the notion of cycles on pair-partitions, which we introduce here. Finally, we explain how to solve the analogous problem for the Coxeter groups of type D by using our main result.

math.RT↗

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↗

arXiv2017

Positive definite functions on Coxeter groups with applications to operator spaces and noncommutative probability

A new class of positive definite functions related to colour-length function on arbitrary Coxeter group is introduced. Extensions of positive definite functions, called the Riesz-Coxeter product, from the Riesz product on the Rademacher (Abelian Coxeter) group to arbitrary Coxeter group is obtained. Applications to harmonic analysis, operator spaces and noncommutative probability is presented. Characterization of radial and colour-radial functions on dihedral groups and infinite permutation group are shown.

math.OA↗

arXiv2017

Noncommutative probability of type D

We construct a deformed Fock space and a Brownian motion coming from Coxeter groups of type D. The construction is analogous to that of the $q$-Fock space (of type A) and the $(α,q)$-Fock space (of type B).

math.FA↗

arXiv2016

Approximation of a free Poisson process by systems of freely independent particles

Let $σ$ be a non-atomic, infinite Radon measure on $\mathbb R^d$, for example, $dσ(x)=z\,dx$ where $z>0$. We consider a system of freely independent particles $x_1,\dots,x_N$ in a bounded set $Λ\subset\mathbb R^d$, where each particle $x_i$ has distribution $\frac1{σ(Λ)}\,σ$ on $Λ$ and the number of particles, $N$, is random and has Poisson distribution with parameter $σ(Λ)$. If the particles were classically independent rather than freely independent, this particle system would be the restriction to $Λ$ of the Poisson point process on $\mathbb R^d$ with intensity measure $σ$. In the case of free independence, this particle system is not the restriction of the free Poisson process on $\mathbb R^d$ with intensity measure $σ$. Nevertheless, we prove that this is true in an approximative sense: if bounded sets $Λ^{(n)}$ ($n\in\mathbb N$) are such that $Λ^{(1)}\subsetΛ^{(2)}\subsetΛ^{(3)}\subset\dotsm$ and $\bigcup_{n=1}^\infty Λ^{(n)}=\mathbb R^d$, then the corresponding particle system in $Λ^{(n)}$ converges (as $n\to\infty$) to the free Poisson process on $\mathbb R^d$ with intensity measure $σ$. We also prove the following $N/V$-limit: Let $N^{(n)}$ be a determinstic sequence of natural numbers such that $\lim_{n\to\infty}N^{(n)}/σ(Λ^{(n)})=1$. Then the system of $N^{(n)}$ freely independent particles in $Λ^{(n)}$ converges (as $n\to\infty$) to the free Poisson process. We finally extend these results to the case of a free Lévy white noise (in particular, a free Lévy process) without free Gaussian part.

math.PR↗

arXiv2016

Radial Bargmann representation for the Fock space of type B

Let $ν_{α,q}$ be the probability and orthogonality measure for the $q$-Meixner-Pollaczek orthogonal polynomials, which has appeared in \cite{BEH15} as the distribution of the $(α,q)$-Gaussian process (the Gaussian process of type B) over the $(α,q)$-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of $ν_{α,q}$. Our main results cover not only the representation of $q$-Gaussian distribution by \cite{LM95}, but also of $q^2$-Gaussian and symmetric free Meixner distributions on $\mathbb R$. In addition, non-trivial commutation relations satisfied by $(α,q)$-operators are presented.

math.FA↗

arXiv2014

Fock space associated to Coxeter group of type B

In this article we construct a generalized Gaussian process coming from Coxeter groups of type B. It is given by creation and annihilation operators on an $(α,q)$-Fock space, which satisfy the commutation relation $$ b_{α,q}(x)b_{α,q}^\ast(y)-qb_{α,q}^\ast(y)b_{α,q}(x)=\langle x, y\rangle I+α\langle \overline{x}, y \rangle q^{2N}, $$ where $x,y$ are elements of a complex Hilbert space with a self-adjoint involution $x\mapsto\bar{x}$ and $N$ is the number operator with respect to the grading on the $(α,q)$-Fock space. We give an estimate of the norms of creation operators. We show that the distribution of the operators $b_{α,q}(x)+b_{α,q}^\ast(x)$ with respect to the vacuum expectation becomes a generalized Gaussian distribution, in the sense that all mixed moments can be calculated from the second moments with the help of a combinatorial formula related with set partitions. Our generalized Gaussian distribution associates the orthogonal polynomials called the $q$-Meixner-Pollaczek polynomials, yielding the $q$-Hermite polynomials when $α=0$ and free Meixner polynomials when $q=0$.

math.FA↗