Probability

Permutons, which are probability measures on [0, 1]^2 with uniform marginals, are the natural scaling limits for sequences of (random) permutations. In particular, various classes of permutations arising from combinatorics and pattern avoidance have been shown to converge to universal permuton…

A mathematically precise definition of quantum field theories is a major and mostly unsolved goal of mathematical physics. One possible route to reaching this goal is through the program of constructive field theory, which is probabilistic at its core. This is one reason why there has been a…

A k-index model is a classical statistical model describing the dependency of a response variable y onto an input vector of covariates x. It posits that y depends on x only via its projection onto a k-dimensional subspace. Learning in this model boils down to estimating this subspace from data,…

In 2017, Miller computed the character tables of S_n for all n up to 38 and looked at various statistical properties of the entries. Characters of symmetric groups take only integer values, and based on his computations, Miller conjectured that almost all entries of the character table of S_n…

There is a close connection between certain models of random integer partitions and random growth models in the KPZ universality class. I will give an introduction to these connections before discussing some new work establishing non-trivial symmetries of two particular models of random…

Exponential random graph models (ERGMs) are exponential tilts of Erdos–Renyi models where higher-order interactions promote the presence of small subgraphs like triangles. These models exhibit metastable behavior at low temperatures (strong interaction) and decompose as mixtures of phase…

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…

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…

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.…