Probability
Organizers: Amir Dembo
Upcoming Events
We investigate fluctuation phenomena for the graph distance and the number of cut points associated with random media arising from the range of a random walk. Our results demonstrate a sequence of dimension-dependent phase transitions in the scaling behavior of these fluctuations, leading to…
The dimer model refers to the study of random dimer covers (or perfect matchings) of a bipartite graph. A remarkable feature of these models is the emergence of limit shapes: in large periodic graphs, a random matching concentrates around a deterministic shape. Although general dimer models…
Past Events
Polymers in disordered media are examples of random Gibbs measures on directed paths moving through a random environment. Such disordered systems often exhibit complex landscape behavior rich with multiple valleys which act as metastable states. This generic property manifests in multiple forms…
In this talk, I will discuss the construction of the Yang-Mills-Higgs measure on the two-dimensional torus. In the 1980s, Parisi and Wu proposed a dynamical approach to constructing such measure via the corresponding Langevin dynamics, aiming to sidestep the difficulty of fixing a global gauge.…
In this talk we discuss some results describing the fluctuation scaling limits for some interacting particle systems.
Dimer models (random lozenge or domino tilings) on large planar domains exhibit universality behavior: local convergence to translation-invariant Gibbs measures, global fluctuations described by the Gaussian Free Field (GFF), and Airy line ensemble at the edges. In this talk, I discuss two…
In this talk, we consider the asymmetric simple exclusion process with open boundaries (open ASEP). We give an overview on recent results on mixing times for the open ASEP. In particular, we discuss mixing times for the open ASEP at the triple point.
This talk is based on joint work with…
We study uniformly random lozenge tilings of polygonal domains as a model of random stepped surfaces, encoded by their height functions. Our analysis relies on an exact correspondence between vertical sections of the tiling and discrete beta-ensembles with repulsive interactions, providing a…
Sparse random graphs are widely viewed as discrete models of chaotic physical systems. Heuristically, this suggests that eigenvectors of the adjacency operator should exhibit Gaussian statistics. We prove that a broad class of random graphs, including both random regular graphs and irregular…
We consider the problem of efficiently optimizing random (spherical or Ising) perceptron models with general bounded Lipschitz activation. We focus on a class of algorithms with Lipschitz dependence on the disorder: this includes gradient descent, Langevin dynamics, approximate message passing,…
We study a natural one-parameter family of random bipolar-oriented planar maps which lies in the $\gamma$-Liouville quantum gravity (LQG) universality class for $\gamma \in (0, \sqrt{4/3}]$. For these maps, we identify exact scaling exponents for directed graph distances. Writing $n$ for the…
The Toda lattice is a particle system of classical mechanics, discovered by Toda in 1967. Due to its integrability, the Toda lattice with N particles possesses N independent conserved quantities. In this work, we show that under a certain class of random initial data with a constant particle…