Biorthogonal ensembles with two-particle interactions

Abstract

We investigate determinantal point processes on $[0,+\infty )$ of the form \begin{equation*}\label{probability distribution} \frac{1}{Z_n}\prod_{1\leq i<j\leq n}(λ_j-λ_i)\prod_{1\leq i<j\leq n}(λ_j^θ-λ_i^θ) \prod_{j=1}^n w(λ_j)dλ_j,\qquad θ\geq 1. \end{equation*} We prove that the biorthogonal polynomials associated to such models satisfy a recurrence relation and a Christoffel-Darboux formula if $\theta \in \mathbb{Q}$, and that they can be characterized in terms of $1\times 2$ vector-valued Riemann-Hilbert problems which exhibit some non-standard properties. In addition, we obtain expressions for the equilibrium measure associated to our model if $w(\lambda )=e^{-nV(\lambda )}$ in the one-cut case with and without hard edge.

28 pages, 6 figures

References in corpus (2)

• Energy correlations for a random matrix model of disordered bosons
• Moment determinants as isomonodromic tau functions

Cited by in corpus (14)

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• Local universality in biorthogonal Laguerre ensembles
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• Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary
• The local universality of Muttalib-Borodin biorthogonal ensembles with parameter $\theta =\frac{1}{2}$
• On Wright's generalized Bessel kernel
• Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral
• Matrix-valued orthogonal polynomials related to hexagon tilings
• A Vector Equilibrium Problem for Muttalib-Borodin Biorthogonal Ensembles
• Generalized random matrix model with additional interactions
• The local universality of Muttalib-Borodin ensembles when the parameter $\theta$ is the reciprocal of an integer
• Nonmonotonic confining potential and eigenvalue density transition for generalized random matrix model
• Relations between generalised Wishart matrices, the Muttalib--Borodin model and matrix spherical functions
• Global rigidity and exponential moments for soft and hard edge point processes