A polymer measure rewards or punishes paths based on the sum of random weights along their trajectories. Their large-scale behavior balances two competing effects -- energy and entropy. In this talk we will consider what happens when the starting and ending points of such a model are free. Focusing on a special two-dimensional directed model -- the log-gamma polymer -- we demonstrate a phase transition, controlled by the mean of the weights, between long polymers with highly localized starting and ending points, and short polymers with highly delocalized starting and ending points. The advantage of working with this model is that its free energy can be studied using certain tools from integrable probability such as moderate deviation bounds and Gibbsian line ensemble. We highlight the roles of these tools and how they combine to yield our phase diagram. This talk is based on joint works with Guillaume Barraquand and Evgeni Dimitrov.

The Schramm-Loewner evolution (SLE) is a one-parameter family of random fractal curves which describe the scaling limit of statistical physics models. We derive an explicit formula for the moments of the derivative of a particular uniformizing conformal map associated with an SLE. Our proof is based on conformal welding of Liouville quantum gravity surfaces along with integrability results from Liouville conformal field theory. Joint work with Morris Ang and Xin Sun.

We study a family of Markov processes on directed graphs where the values at each vertex are influenced by the values of its inbound neighbors and by independent fluctuations either on the vertices themselves or on the edges connecting them to their inbound neighbors. Typical examples include PageRank, the generalized deGroot model, and other information propagation processes on directed graphs. Assuming a stationary distribution exists for this Markov chain, our goal is to characterize the marginal distribution of a uniformly chosen vertex in the graph. In order to obtain a meaningful characterization, we assume that the underlying graph converges in the local weak sense to a marked Galton-Watson process, e.g., a directed configuration model or any rank-1 inhomogeneous random digraph. We then prove that the stationary distribution we study on the graph converges in a Wasserstein metric to a distribution characterized through a branching distributional fixed-point equation and its endogenous solution.

Exchangeability — in which the distribution of an infinite sequence is invariant to reorderings of its elements — implies the existence of a conditional independence structure; this structure may be leveraged for data analysis in the design of probabilistic models, efficient inference algorithms, and randomization-based testing procedures. For instance, in the case of clustering data, the Kingman paintbox and exchangeable partition probability functions (EPPFs) inspire models and inference algorithms, respectively. We start by extending these constructions (A) to feature and trait allocations, where a data point can belong to multiple groups simultaneously (rather than just a single cluster) and (B) to edge-exchangeable graph models, which can exhibit sparsity unlike traditional vertex-exchangeable graph models. A recurring issue in all of these cases and others, though, is that exact exchangeability may be too strong an idealization; real-world data often cannot reasonably be modeled as strictly invariant to permutations across (e.g.) substantial time or space. To address this issue, we introduce a relaxed notion of "local exchangeability" — where swapping data associated with nearby covariates causes a bounded change in the data distribution. We derive representation theory and statistical tests analogous to the case of exact exchangeability.

In 2016, Lubetzky and Peres proved that on every Ramanujan graph $G$ with $n$ vertices and degree $d$, the simple random walk exhibits cutoff at $d(d-2)^{-1} \log_{d-1} n$. In this talk, we will focus on the non-backtracking random walk on $G$ and prove that it exhibits cutoff at $\log_{d-1} n$ with a bounded window, provided that the girth of $G$ is big. This is joint work with Peter Sarnak.

In a seminal influential paper, Lebowitz, Rose and Speer (1988) constructed probability measures on periodic functions inspired by the Gibbs measures of statistical mechanics and based on Brownian motion. These measures are naturally associated to the focusing mass-critical nonlinear Schroedinger equation. They conjectured that these measures are invariant under the nonlinear flow. This was later proved by Bourgain. Lebowitz-Rose-Speer also proposed a critical mass threshold past which the measure no longer exists, given by the mass of the ground state on the real line.

With T. Oh and L. Tolomeo, we prove the optimality of this critical mass threshold. The proof also applies to the two-dimensional radial problem posed on the unit disc. In this case, this answers a question posed by Bourgain and Bulut (2014) on the optimal mass threshold. Furthermore, in the one-dimensional case, we show that the Gibbs measure is normalizable *at* the optimal mass threshold, answering another posed by Lebowitz, Rose, and Speer (1988). This latter fact is somewhat surprising in view of the minimal mass blowup solution for the focusing quintic nonlinear Schroedinger equation on the one-dimensional torus.

Models of cellular processes are often represented with networks that describe the interactions between the constituent molecules. If the counts of the molecular "species" are low, then these systems are most often modeled stochastically using a continuous-time Markov chain. In this talk, I will begin with an introduction to the basic mathematical model and then provide several results relating the properties of the reaction graph (which are easy to check) with the stationary and/or transient behavior of the associated process. I will also provide a number of open problems.

The standard 2-semester course in probability theory (I use Durrett's book for example) is jam-packed with essential stuff. The best students are able both to learn the math and use it for modeling and word problems. But for everyone else (including all the engineers, computer scientists, geneticists, and the mathematics masters students), how can we teach them how to use probability when they are barely keeping up with the formal side of the course?

In this talk I will discuss my experiences with a course focused on modeling, having the same broad content outline as the usual course. To give away the punchline, we cut corners on content in the area of requiring students to know the proofs of some of the more complicated theorems, but not on exact language, statements or constructions. We also increase learning per hour through various pedagogical tricks, to be described in the talk.

In this talk I will introduce graphical representations of stochastic partial differential equations that allow to approximate Matern Gaussian fields on manifolds and generalize the Matern model to abstract point clouds. Graph-based Matern Gaussian fields have a sparse precision matrix, which is important for computationally efficient inference and sampling. Moreover, we will show that they can give optimal posterior contraction in semi-supervised learning applications. This is joint work with Ruiyi Yang.

Unitary Brownian motions have been extensively studied in random matrix theory. In this talk we will look at block determinant processes of a Unitary Brownian motion, and study limit laws of their polar decompositions. The associated radial processes is closely related to the complex Jacobi ensemble. The angular part, namely the associ- ated winding process can be studied by using a fibration structure of the underlying Stiefel manifold. Such geometric structure allows us to present the winding process in terms of a stochastic area process, and hence obtain its limit distribution.

The talk will address the problem of local limit theorem for dependent structures. A local limit theorem is a fine scale behavior of the sums S_n dealing with the rate of convergence of the probability that a partial sum lies in an interval. This type of limit theorem is a fine scale behavior of the sums S_n. Controlling such probabilities is important for finding recurrence conditions for a random walk. The first class considered is the linear fields of random variables constructed from independent and identically distributed innovations with finite second moment. Then, we consider the local limit theorem for additive functionals of a nonstationary Markov chain with infinite or finite second moment. The method of proof is based on the characteristic function factorization

Optimizing a high-dimensional non-convex function is, in general, computationally hard and many problems of this type are hard to solve even approximately. Complexity theory characterizes the optimal approximation ratios achievable in polynomial time in the worst case. On the other hand, when the objective function is random, worst case approximation ratios are overly pessimistic. Mean field spin glasses are canonical families of random energy functions over the discrete hypercube {−1,+1}^N. The near-optima of these energy landscapes are organized according to an ultrametric tree-like structure, which enjoys a high degree of universality. A similar structure is expected to control the energy landscabe of several well studied combinatorial optimization problems, from random K-satisfiability, to coloring of random graphs.

Recently, a precise connection has begun to emerge between this ultrametric structure and the optimal approximation ratio achievable in polynomial time in the typical case. In particular, problems that exhibit full replica symmetry breaking with no overlap gap appear to have algorithm that allow for optimization within an arbitrary constant factor, in polynomial time. I will describe this connection and the algorithms that achieve this goal. [Based on joint work with Ahmed El Alaoui and Mark Sellke]