Abstract:  This talk explores how models of random motion in a disordered medium give rise to natural notions of boundaries at infinity. Recent results on existence and uniqueness are presented for a class of models that generalizes first- and last-passage percolation, random walks in random environments, and directed polymers. The resulting boundary structures include jointly stationary distributions, geodesic rays, Busemann functions, harmonic functions and the associated Martin boundary, and extremal Gibbs-DLR measures. 

Based on joint works with Sean Groathouse, Sergazy Nurbavliyev, Firas Rassoul-Agha, and Timo Seppäläinen

Abstract: In this talk, a novel framework for deriving diffusion limit equations of queueing models with high-dimensional state descriptors will be discussed. The method hinges on a martingale decomposition of dynamics driven by time-changed renewal processes, which are a common feature of many queueing models. A central limit theorem for models decomposed in this way, which gives the form of stochastic differential equations (SDEs) that will be satisfied in the diffusion limit of such a system, will be presented. Unlike existing approaches, the framework does not require state space collapse or tractable workload representations, enabling its application to systems with features such as reneging. The approach will be demonstrated on a multiclass, multi-server random order of service queue with reneging and generally distributed interarrival, service, and patience times. The results offer a broadly applicable method for diffusion approximations in complex queueing systems without relying on dimension reductions or model-specific structure.

Abstract: The KPZ fixed point is a (1+1)-dimensional space-time random field conjectured to be the universal limit for models within the Kardar-Parisi-Zhang (KPZ) universality class. We consider the KPZ fixed point with the narrow-wedge initial condition, conditioning on a large value at a specific point. By zooming in the neighborhood of this high point appropriately, we obtain a limiting random field which we call an upper tail field of the KPZ fixed point. Different from the KPZ fixed point, where the time parameter has to be nonnegative, the upper tail field is defined in the full 2-dimensional space. Particularly, if we zoom out the upper tail field appropriately, it behaves like a Brownian-type field in the negative time regime and the KPZ fixed point in the positive time regime. One main ingredient of the proof is an upper tail estimate of the joint tail probability functions of the KPZ fixed point near the given point, which generalizes the well-known one-point upper tail estimate of the GUE Tracy-Widom distribution. This is a joint work with Zhipeng Liu.

Abstract: In this talk, we will focus on the distributions of superlevel sets of the exponential sum \sum_{n=1}^N a_n(nx+n^2 t). First, we look at the case when all coefficients a_n=1. We have the optimal pointwise estimate at each point on the torus. Everything is well understood. One special feature is that those points (x,t) that take the absolute value N^{5/6} do not distribute randomly in the torus, they cluster in the vertical direction and in the horizontal direction. This is not predicted by decoupling theory, but it is expected to be true for general sequence a_n. I will formulate conjectures, explain why current methods do not work, and mention some partial results. 

Abstract:  I will describe a newly introduced toolbox that connects two areas of Probability Theory: Schramm-Loewner Evolutions (SLE) and Random Matrix Theory. This machinery opens new avenues of research that allow the use of techniques from one field to another. One aspect of this research direction is centered in an interacting particle systems model, namely the Dyson Brownian motion. In the first part of the talk, I will introduce basic ideas of SLE theory, then I will describe the connection with Random Matrix Theory via a first application of our method. I will finish the talk with some open problems that emerge using this newly introduced toolbox. This is a joint work with A. Campbell and K. Luh.

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