Sinclair, Christopher D.Yattselev, Maxim L.2019-08-232019-08-232018Sinclair, C. D., & Yattselev, M. L. (2018). The reciprocal Mahler ensembles of random polynomials. Random Matrices: Theory and Applications, 1950012. https://doi.org/10.1142/S2010326319500126https://hdl.handle.net/1805/20546We 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.enPublisher PolicyMahler measurerandom polynomialsasymmetric random matrixArticle