The reciprocal Mahler ensembles of random polynomials

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 functions (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. MSC2010: 15B52, 60B20, 60G55, 82B23, 15A15, 11C20 11R06