The reciprocal Mahler ensembles of random polynomials

摘要

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.