Georgia Institute of Technology College of Sciences

The dimer model, i.e. random perfect matchings of a bipartite graph, is a classical object about which much is known. As soon as one biases the probability measure by edge weights which are themselves random, very little is known rigorously, though physicists have studied such models for several decades and made extensive predictions. I will discuss a new integrable model in this class (the Gamma-disordered Aztec diamond) which allows us to prove results on the free energy, and also exhibits surprising relations to integrable polymer models which lead to probabilistic results on tilings. Joint work with Maurice Duits (KTH), https://arxiv.org/abs/2512.03033.

We are going to discuss recent developments in the study of the local asymptotic behavior of the second correlation function of characteristic polynomials for several important examples of non-Hermitian random matrices including adjacency matrices of weighted Erdos-Renyi graphs and  band matrices, where non-zero entries are present only in the narrow strip around the main diagonal. The talk is based on the joint papers with Ievgenii Afanasiev and Maria Shcherbina.

A basic question in random matrix theory is how likely a matrix is to have a large rank. For many classical models, this probability decays extremely quickly, but the strongest results often rely on the entries being identically distributed. In this talk, I will discuss how to obtain the same exponential scale for matrices with fully independent, non-identically distributed entries under only a uniform anti-concentration assumption and explain the main ideas that make this possible.

Abstract: In the early 2000s, substantial work was devoted to understanding the generally distributed processor-sharing queue, which can be viewed as an idealization of round-robin or time-sharing protocols arising in computer-system applications. The fluid limit for a processor-sharing queue with reneging was established, but the diffusion limit for a processor-sharing queue with impatience was obtained only under a “soft deadlines” formulation, in which the patience times of jobs are tracked but jobs do not actually leave the queue when their patience times expire. This limitation was largely a consequence of the methodology underlying that research program: much of the analysis relied on heavy-traffic limits and the state-space-collapse framework introduced by Bramson and Williams, which is not naturally suited to systems with reneging. More recent work has developed methods for obtaining diffusion approximations of measure-valued queueing systems using classical central limit theorem analysis, martingale methods, and SPDE-valued limits (see work by Ramanan and by the author of this talk). Using these methods, the diffusion approximation for a processor-sharing queue with true reneging can be obtained.

Bio: Eva Loeser is a postdoctoral research associate in the Applied Probability Group in the Department of Statistics and Operations Research at the University of North Carolina at Chapel Hill. Her research interests include stochastic processes, particularly fluid and diffusion approximations of stochastic systems, high-dimensional stochastic processes, and universal limiting objects such as stochastic partial differential equations (SPDEs) and semimartingale reflecting Brownian motions (SRBMs). The models she studies arise in applications including systems biology, computer systems, queueing theory, operations research, and mathematical physics.