The reciprocal Mahler ensembles of random polynomials

Abstract

We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree $N$ whose Mahler measure is bounded by a constant. After a change of variables this reduces to a generalization of Ginibre's complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval $[-2,2]$ on the real axis in the complex plane. In the complex (real) case the random roots form a determinantal (Pfaffian) point process, and in both cases the empirical measure on roots converges weakly to the arcsine distribution supported on $[-2,2]$. Outside this region the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from $[-2,2]$. These kernels, as well as the scaling limits for the kernels in the bulk $(-2,2)$ and at the endpoints $\{-2,2\}$ are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non-reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels.