Free Bessel laws

Abstract

We introduce and study a remarkable family of real probability measures ${\pi }_{st}$, that we call free Bessel laws. These are related to the free Poisson law $\pi$ via the formulae ${\pi }_{s1}={\pi }^{⊠s}$ and ${\pi }_{1t}={\pi }^{⊞t}$. Our study includes: definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

40 pages

Cited by in corpus (28)

• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• Generating random density matrices
• Product of Ginibre matrices: Fuss-Catalan and Raney distributions
• Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits
• Singular value correlation functions for products of Wishart random matrices
• Eigenvalue statistics for product complex Wishart matrices
• The law of large numbers for the free multiplicative convolution
• Raney distributions and random matrix theory
• Local universality in biorthogonal Laguerre ensembles
• New spectral relations between products and powers of isotropic random matrices
• Random tensor networks with nontrivial links
• The difference between two random mixed quantum states: exact and asymptotic spectral analysis
• Spectral density of generalized Wishart matrices and free multiplicative convolution
• Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral
• Limit distributions of random matrices
• A note on the limiting mean distribution of singular values for products of two Wishart random matrices
• Summing free unitary random matrices
• Homogeneous quantum groups and their easiness level
• Schwinger-Dyson and loop equations for a product of square Ginibre random matrices
• On the joint distribution of the marginals of multipartite random quantum states
• On Powers of the Catalan Number Sequence
• Limit theorems for free Lévy processes
• Sparse Random Block Matrices
• Relations between generalised Wishart matrices, the Muttalib--Borodin model and matrix spherical functions
• Random matrix model for free Meixner laws
• Universality of the Wigner-Gurau limit for random tensors
• Quantum Bernstein's Theorem and the Hyperoctahedral Quantum Group
• $\lambda$-shaped random matrices, $\lambda$-plane trees, and $\lambda$-Dyck paths