The distribution of Mahler's measures of reciprocal polynomials

We study the distribution of Mahler's measures of reciprocal polynomials with complex coefficients and bounded even degree. We discover that the distribution function associated to Mahler's measure restricted to monic reciprocal polynomials is a reciprocal (or anti-reciprocal) Laurent polynomial on [1,\infty) and identically zero on [0,1). Moreover, the coefficients of this Laurent polynomial are rational numbers times a power of \pi. We are led to this discovery by the computation of the Mellin transform of the distribution function. This Mellin transform is an even (or odd) rational function with poles at small integers and residues that are rational numbers times a power of \pi. We also use this Mellin transform to show that the volume of the set of reciprocal polynomials with complex coefficients, bounded degree and Mahler's measure less than or equal to one is a rational number times a power of \pi.Comment: 13 pages. To be published in Int. J. Math. Math. Sc

The Ginibre ensemble of real random matrices and its scaling limits

We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a $2 \times 2$ matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large.Comment: 47 pages, 8 figure

Extremal laws for the real Ginibre ensemble

The real Ginibre ensemble refers to the family of $n\times n$ matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges in law to a Gumbel distribution as $n\rightarrow\infty$. This fact has been known to hold in the complex and quaternion analogues of the ensemble for some time, with simpler proofs. Along the way we establish a new form for the limit law of the largest real eigenvalue.Comment: Published in at http://dx.doi.org/10.1214/13-AAP958 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Equidistribution of Algebraic Numbers of Norm One in Quadratic Number Fields

Given a fixed quadratic extension K of Q, we consider the distribution of elements in K of norm 1 (denoted N). When K is an imaginary quadratic extension, N is naturally embedded in the unit circle in C and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in N can be written as \alpha/\bar{\alpha} for some \alpha \in O_K, which yields another ordering of \mathcal N given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that N is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that N is equidistributed with respect to norm, under the map \beta \mapsto \log| \beta | \bmod{\log | \epsilon^2 |} where \epsilon is a fundamental unit of O_K.Comment: 19 pages, 2 figures, comments welcome

Equidistribution of Elements of Norm 1 in Cyclic Extensions

Upon quotienting by units, the elements of norm 1 in a number field $K$ form a countable subset of a torus of dimension $r_1 + r_2 - 1$ where $r_1$ and $r_2$ are the numbers of real and pairs of complex embeddings. When $K$ is Galois with cyclic Galois group we demonstrate that this countable set is equidistributed in this torus with respect to a natural partial ordering.Comment: 7 page