Convergence of the largest singular value of a polynomial in independent Wigner matrices

For polynomials in independent Wigner matrices, we prove convergence of the largest singular value to the operator norm of the corresponding polynomial in free semicircular variables, under fourth moment hypotheses. We actually prove a more general result of the form "no eigenvalues outside the support of the limiting eigenvalue distribution." We build on ideas of Haagerup-Schultz-Thorbj{\o}rnsen on the one hand and Bai-Silverstein on the other. We refine the linearization trick so as to preserve self-adjointness and we develop a secondary trick bearing on the calculation of correction terms. Instead of Poincar\'{e}-type inequalities, we use a variety of matrix identities and $L^p$ estimates. The Schwinger-Dyson equation controls much of the analysis.Comment: Published in at http://dx.doi.org/10.1214/11-AOP739 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Abeliants and their application to an elementary construction of Jacobians

The {\em abeliant} is a polynomial rule for producing an $n$ by $n$ matrix with entries in a given ring from an $n$ by $n$ by $n+2$ array of elements of that ring. The theory of abeliants, first introduced in an earlier paper of the author, is redeveloped here in a simpler way. Then this theory is exploited to give an explicit elementary construction of the Jacobian of a nonsingular projective algebraic curve defined over an algebraically closed field. The standard of usefulness and aptness we strive toward is that set by Mumford's elementary construction of the Jacobian of a hyperelliptic curve. This paper has appeared as Advances in Math 172 (2002) 169-205

Interpolation of hypergeometric ratios in a global field of positive characteristic

In connection with each global field of positive characteristic we exhibit many examples of two-variable algebraic functions possessing properties consistent with a conjectural refinement of the Stark conjecture in the function field case recently proposed by the author (math.NT/0407535). Most notably, all examples are Coleman units. We obtain our results by studying rank one shtukas in which both zero and pole are generic, i.~e., shtukas not associated to any Drinfeld module.Comment: 25 pages, LaTe

A local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices

Our main result is a local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices, modeled on the local semicircle law. Our approach is to adapt some techniques from one of the recent papers of Erd\"os-Yau-Yin. We also use an algebraic description of the law of the anticommutator of free semicircular variables due to Nica-Speicher, a self-adjointness-preserving variant of the linearization trick due to Haagerup-Schultz-Thorbj\o rnsen, and the Schwinger-Dyson equation. A byproduct of our work is a relatively simple deterministic version of the local semicircle law.Comment: 33 pages, LaTeX, 2 figures. In v2 (this version) we make minor revisions, add references and correct typo

Preservation of algebraicity in free probability

We show that any matrix-polynomial combination of free noncommutative random variables each having an algebraic law has again an algebraic law. Our result answers a question raised by a recent paper of Shlyakhtenko and Skoufranis. The result belongs to a family of results with origins outside free probability theory, including a result of Aomoto asserting algebraicity of the Green function of random walk of quite general type on a free group.Comment: 41 pages, LaTeX, no figures. In v2, we added references, corrected typos, reorganized some material, and added explanations. Main results remain the same. In this version, v3, we added references and explanation, and simplified the second half of the proof of the main resul

Collapsing Sub-Critical Bubbles

In the standard scenario, the electroweak phase transition is a first order phase transition which completes by the nucleation of critical bubbles. Recently, there has been speculation that the standard picture of the electroweak phase transition is incorrect. Instead, it has been proposed that throughout the phase transition appreciable amounts of both broken and unbroken phases of $SU(2)$ coexist in equilibrium. I argue that this can not be the case. General principles insure that the universe will remain in a homogenous state of unbroken $SU(2)$ until the onset of critical bubble production.Comment: 7 pages plus three figures. OHSTPY-HEP-T-92-016 A topdrawer file of the figures is appended to the en