The reciprocal Mahler ensembles of random polynomials

Abstract

We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree [Formula: see text] 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 [Formula: see text] 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 [Formula: see text]. Outside this region, the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from [Formula: see text]. These kernels as well as the scaling limits for the kernels in the bulk [Formula: see text] and at the endpoints [Formula: see text] 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.