arXiv subjects
Publications and source records attributed to Piotr Sniady.
We study asymptotics of the Itzykson-Zuber integrals in the scaling when one of the matrices has a small rank compared to the full rank. We show that the result is basically the same as in the case when one of the matrices has a fixed rank. In this way we extend the recent results of Guionnet and Maida who showed that for a latter scaling the Itzykson-Zuber integral is given in terms of the Voiculescu's R-transform of the full rank matrix.
We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haar-distributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson-Zuber type.
We study the von Neumann algebra generated by q--deformed Gaussian elements l_i+l_i^* where operators l_i fulfill the q--deformed canonical commutation relations l_i l_j^*-q l_j^* l_i=delta_{ij} for -1<q<1. We show that if the number of generators is finite, greater than some constant depending on q, it is a II_1 factor which does not have the property Gamma. Our technique can be used for proving factoriality of many examples of von Neumann algebras arising from some generalized Brownian motions, both for type II_1 and type III case.
We discuss the meaning of renormalization used for deriving quadratic bosonic commutation relations introduced by Accardi and find a representation of these relations on an interacting Fock space. Also, we investigate classical stochastic processes which can be constructed from noncommutative quadratic white noise. We postulate quadratic free white noise commutation relations and find their representation on an interacting Fock space.
We prove a family of new identities fulfilled by multinomial coefficients, which were conjectured by Dykema and Haagerup. Our method bases on a study of the, so-called, triangular operator T by the means of the free probability theory.
We study Voiculescu's microstate free entropy for a single non-selfadjoint random variable. The main result is that certain additional constraints on eigenvalues of microstates do not change the free entropy. Our tool is the method of random regularization of Brown measure which was studied recently by Haagerup and the author. As a simple application we present an upper bound for the free entropy of a single non--selfadjoint operator in terms of its Brown measure and the second moment. We furthermore show that this inequality becomes an equality for a class of DT-operators which was introduced recently by Dykema and Haagerup.
We generalize a recent result of Haagerup; namely we show that a convolution with a standard Gaussian random matrix regularizes behaviour of Kadison--Fuglede determinant and Brown spectral distribution measure. In this way it is possible to establish a connection between limit eigenvalues distributions of a wide class of random matrices and the Brown measure of the corresponding limits.
This is an introductory text to differential geometry (written in Polish) aimed for high-school students.
We show that every R-diagonal operator x has a continuous family of invariant subspaces relative to the von Neumann algebra generated by x. This allows us to find the Brown measure of x and to find a new conceptual proof that Voiculescu's S-transform is multiplicative. Our considerations base on a new concept of R-diagonality with amalgamation, for which we give several equivalent characterizations.
In this paper we consider a very general U(1)-invariant field theory such that a field operator commutes with its adjoint, what corresponds to a theory of a charged bosonic particle. We show that from such an invariance follows the existence of particles and antiparticles associated to the same physical state. The field operator always turns out to be a sum of a particle creation and an antiparticle annihilation operators. We study in detail the case when creation and annihilation operators factorize and show that such operators are closely related to q-deformed commutation relations.
In this paper we investigate a quantum stochastic calculus build of creation, annihilation and number of particles operators which fulfill some deformed commutation relations. Namely, we introduce a deformation of a number of particles operator which has simple commutation relations with well known q-deformed creation and annihilation operators. Since all operators considered in this theory are bounded we do not deal with some difficulties of a non deformed theory of Hudson and Parthasarathy. We define stochastic integrals and estimate them in the operator norm. We prove Ito's formula as well.
We construct a family of random matrix models for the q-deformed Gaussian random variables G_μ=a_μ+a^\star_μwhere the annihilation operators a_μand creation operators a^\star_νfulfil the q-deformed commutation relation a_μa^\star_ν-q a^\star_νa_μ=Γ_{μν}, Γ_{μν} is the covariance and 0<q<1 is a given number. Important feature of considered random matrices is that the joint distribution of their entries is Gaussian.