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arXiv · 1312.2892

Biorthogonal ensembles with two-particle interactions

Tom Claeys·Stefano Romano
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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 $θ\in\mathbb Q$, 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 $w(λ)=e^{-nV(λ)}$ in the one-cut case with and without hard edge.

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Tom Claeys, Stefano Romano. 2015-02-25. Biorthogonal ensembles with two-particle interactions. https://doi.org/10.1088/0951-7715%2F27%2F10%2F2419

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