Branching Brownian motion is a classical process in probability theory belonging to the class of “Log-correlated random fields”. We study the structure of extreme level sets of this process, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida.(joint work with A. Cortines, O Louidor)

In this talk we study the rate of growth of the expectation of the number of holes (the rank of the first homology group) in a polyomino with uniform and percolation distributions. We prove the existence of linear bounds for the expected number of holes of a polyomino with respect to both the uniform and percolation distributions. Furthermore, we exhibit particular constants for the upper and lower bounds in the uniform distribution case. This results can be extend, using the same techniques, to other polyforms and higher dimensions.

Let \(S_n\) denote the set of permutations of length \(n\). For a permutation \(\tau \in S_n\) we say \(\tau\) contains a pattern \(\sigma\in S_k\) if there is a subsequence \(i_1 < \cdots < i_k\) such that \(\tau_{i_1} \cdots \tau_{i_k}\) has the the same relative order of \(\sigma\). If \(\tau\) contains no pattern \(\sigma\), we say that \(\tau\) avoids \(\sigma\). We denote the set of \(\sigma\)-avoiding permutations of length \(n\) by \(S_n(\sigma)\). Recently, there has been a number of results that help describe the geometric properties of a uniformly random element in \(S_n(\sigma)\). Many of these geometric properties are related to well-studied random objects that appear in other settings. For example, if \(\sigma \in S_3\), we have that a permutation chosen uniformly in \(S_n(\sigma)\) converges, in some appropriate sense, to Brownian excursion. Furthermore for \(\sigma = 123,312\) or\( 231\), we can describe properties like the number and location of fixed points in terms of Brownian excursion. Larger patterns are much more difficult to understand. Currently even the simplest question, enumeration, is unknown for the pattern \(\sigma = 4231\). However, for the monotone decreasing pattern \(\sigma= (d+1)d\cdots 21\), \(S_n(\sigma)\) can be coupled with a random walk in a cone that, in some appropriate sense, converges to a traceless Dyson Brownian motion.

In this talk, I will discuss a generalized Moran process from the evolutionary game theory. The generalization incorporates arrangement of by graphs and games among individuals. For these additional features, there has been consistent interest in using general spatial structure as a way to explain the ubiquitous game behavior in biological evolutions; the introduction of games leads to technical complications as basic as nonlinearity and asymmetry in the model. The talk will be centered around a seminal finding in the evolutionary game theory that was obtained more than a decade ago. By an advanced mean-field method, it reduces the infinite-dimensional problem of solving for the game fixation probabilities to a one-dimensional diffusion problem in the limit of a large population. The recent mathematical results and some related mathematical methods will be explained.

The goal of this talk is to further a joint project involving Carla Staver, Simon Levin, Rick Durrett, and Ruibo Ma. The puzzle is: why can savannah and forest display stable coexistence when this is not possible in a spatially homogeneous system.

I will discuss a class of particle systems that serve as models for supercooling in physics, neuron firing in neuroscience and systemic risk in finance. The interaction between the particles falls into the mean-field framework pioneered by McKean and Vlasov in the late 1960s, but many new phenomena arise due to the singularity of the interaction. The most striking of them is the loss of regularity of the particle density caused by the the self-excitation of the system. In particular, while initially the evolution of the system can be captured by a suitable Stefan problem, the following irregular behavior necessitates a more robust probabilistic approach. Based on joint work with Sergey Nadtochiy.

We study the long time behaviour of the heat kernel on Abelian covers of compact Riemannian manifolds. For manifolds without boundary work of Lott and Kotani-Sunada establishes precise long time asymptotics. Extending these results to manifolds with boundary reduces to a cute eigenvalue minimization problem, which we resolve for a Dirichlet and Neumann boundary conditions. We will show how these results can be applied to studying the ``winding'' / ``entanglement'' of Brownian trajectories in Riemannian manifolds.

In this talk I will revisit the idea of viewing the Bayesian update as a variational problem. I will show how the variational interpretation is helpful in establishing the convergence of Bayesian models, and in defining and analysing diffusion processes that have the posterior as invariant measure. I will illustrate the former by proving a consistency result for graph-based Bayesian semi-supervised learning in the large unlabelled data-set regime, and the latter by suggesting new optimality criteria for the choice of metric in Riemannian MCMC.

Quantum field theory is the theoretical framework for studying fundamental interactions. "Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. The recent years witnessed interesting progress in understanding solutions of these stochastic PDEs, one of the remarkable examples being Hairer and Mourrat-Weber's results on the Phi^4_3 equation.

In this talk we will discuss stochastic quantization of quantum field theory with gauge symmetries, with focus on an Abelian example but also provide prospects of non-Abelian Yang-Mills theories. We address issues regarding Wilson’s lattice regularization, dynamical gauge fixing, renormalization, Ward identities, and construction of dynamical loop and string observables.

The classical central limit theorem (CLT) states that for sums of a large number of i.i.d. random variables with finite variance, the distribution of the rescaled sum is approximately Gaussian. However, the statement of the central limit theorem doesn't give any quantitative error estimates for this approximation. Under slightly stronger moment assumptions, quantitative bounds for the CLT are given by the Berry-Esseen estimates. In this talk we will consider similar questions for CLTs for random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE. This talk is based on joint works with Sungwon Ahn and Xiaoqin Guo.

I will discuss a version of stochastic Loewner evolution with branching introduced in my student Vivian Olsiewski Healey's 2017 thesis. Our main motivation was to find natural conformal processes that embed Aldous' continuum random tree in the upper half plane. Unlike previous attempts that rely on lattice models or conformal welding, our model relies on a careful choice of driving measure in the Loewner evolution and the theory of continuous state branching processes. The most important feature of our model is that it has a very nice scaling limit, where the driving measure is a superprocess.

Mean field game theory describes continuum limits of symmetric large-population games. These games can often be seen as competitive extensions of classical models of interacting particle systems, where the particles are now "controlled state process" (with application-specific interpretation, such as position, income, wealth, etc.). The coupled optimization problems faced by each process are typically resolved by Nash equilibrium, and there is a large and growing literature on solvability problems (both theoretical and computational). On the other hand, relatively little is known on how to rigorously pass from a finite population to a continuum, especially for dynamic stochastic games. The basic question is: Given for each N a Nash equilibrium for the N-player game, do the equilibria (more precisely, the empirical distributions of state processes) converge as N tends to infinity? This talk is an overview of the known probabilistic limit theorems in this context (law of large numbers, fluctuations, and large deviations), the ideas behind them, and some open problems.

The efficient delivery of proteins and other molecular products to their correct location within a cell (intracellular transport) is of fundamental importance to normal cellular function and development. Moreover, the breakdown of intracellular transport is a major contributing factor to many degenerative diseases. There are two major types of transport. (I) Passive diffusion within the cytosol or the surrounding plasma membrane of the cell. Since the aqueous environment (cytosol) of a cell is highly viscous at the length and velocity scales of macromolecules (low Reynolds number), a diffusing particle can be treated as an overdamped Brownian particle where inertial effects are ignored. (II) Active motor-driven transport along polymerized filaments such as microtubules and F-actin that comprise the cytoskeleton. At appropriate length and time scales, active transport can either be modeled as a velocity-jump process or as an advection-diffusion process. In this talk I present various PDE models of active and passive transport within cells. The bulk of the talk will focus on three examples: synaptic democracy and vesicular transport in axons and dendrites; stochastically gated diffusion in bounded domains; cytoneme-based transport of morphogens during embryogenesis. (A cytoneme is a thin actin-rich filament that forms direct contacts between cells and is thought to provide an alternative to diffusion-based morphogen gradient formation.) Other applications include cellular length control, cell polarization, and synaptogenesis in C. elegans.

The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.

We give results on the properties of Markov kernels that approximate another Markov kernel. The basic idea is that when the original kernel satisfies a contraction condition in some metric, the long-time dynamics of the two chains -- as well as the invariant measures, when they exist -- will be close in that metric, so long as the approximating kernel satisfies a suitable approximation error condition. We focus on weighted total variation and Wasserstein metrics, and motivate the results with applications to scalable Markov chain Monte Carlo algorithms. This is joint work with Jonathan Mattingly.

In r-neighbor bootstrap percolation, the vertices of Z^d are initially occupied independently with probability p and empty otherwise. Occupied vertices remain occupied forever, and empty vertices iteratively become occupied when they have at least r occupied neighbors. It is a classic result of van Enter (r=d=2) and Schonmann (d>2 and r between 2 and d) that every vertex in Z^d eventually becomes occupied for any initial density p>0. In the polluted bootstrap percolation model, vertices of Z^d are initially closed with probability q, occupied with probability p and empty otherwise. The r-neighbor bootstrap rule is the same, but now closed vertices act as obstacles, and remain closed forever. This model was introduced 20 years ago by Gravner and McDonald, who studied the case d=r=2 and proved a phase transition exists for this model as p and q tend to 0. We prove a similar phase transition occurs when d=r=3, and we identify the polynomial scaling between p and q at which this transition occurs for the modified bootstrap percolation model. For one direction, our proof relies on duality methods in Lipschitz percolation to find a blocking structure that prevents occupation of the origin. The other direction follows from a rescaling argument, and the recent results of Holroyd and Gravner for d>r=2. This is joint work with Holroyd and Gravner.

This talk concerns to spectral gap in random regular graphs. We prove that almost all bipartite biregular graphs are almost Ramanujan by providing a tight upper bound for the second eigenvalue of its adjacency operator. The proof relies on a technique introduced recently by Massoullie, which we developed for random regular graphs. The same analysis allow us to recover hidden communities in random networks via spectral algorithms.

The frog model is a type of branching random walk model. Active "frogs" move according to random walks, and if they encounter a sleeping frog on their walk, the sleeping frog becomes active and begins an independent random walk. Over the past 20 years, recurrence properties and asymptotic behavior of this system (and many generalizations) have been studied extensively. One way to generalize this system is to consider the continuous version: Brownian motion frogs moving in R^d. In this talk, we will describe a continuous variant of the problem and show a limiting shape theorem analogous to prior discrete results.

Over the past decade, the theory of tumor evolution has largely focused on the selective sweeps model. According to this theory, tumors evolve by a succession of clonal expansions that are initiated by driver mutations. In a 2015 paper, Sottoriva et al collected genetic data of various types from 349 individual tumor glands were sampled from the opposite sides of 15 colorectal tumors and large adenomas. Based on this the authors proposed an alternative theory of tumor evolution, the so-called {\bf Big Bang model}, in which one or more driver mutations are acquired by the founder gland, and the evolutionary dynamics within the expanding population are predominantly neutral. In this talk we will describe a simple mathematical model that reproduces the observed phenomena and makes quantitative predictions.

Parking functions were introduced by combinatorialists in the 1960s, and have recently been studied by probabilists. When the parking lot is an infinite graph and cars drive around at random, we will look at how many parking spots are needed for every car to eventually find a spot.

Coupling is a way of constructing Markov processes with prescribed laws on the same space. The coupling is called Markovian if the coupled processes are co-adapted to the same filtration. We will first investigate Markovian couplings of elliptic diffusions and demonstrate how the rate of coupling (how fast you can make the coupled processes meet) is intimately connected to the geometry of the underlying space. Next, we will consider couplings of hypoelliptic diffusions (diffusions driven by vector fields whose Lie algebra span the whole tangent space). Constructing successful couplings (where the coupled processes meet almost surely) for these diffusions is a much more subtle question as these require simultaneous successful coupling of the driving Brownian motions as well as a collection of their path functionals. We will construct successful Markovian couplings for a large class of hypoelliptic diffusions. We will also investigate non-Markovian couplings for some hypoelliptic diffusions, namely the Kolmogorov diffusion and Brownian motion on the Heisenberg group, and demonstrate how these couplings yield sharp estimates for the total variation distance between the laws of the coupled diffusions when Markovian couplings fail. Furthermore, we will demonstrate how non-Markovian couplings can be used to furnish purely analytic gradient estimates of harmonic functions on the Heisenberg group by purely probabilistic means, providing yet another strong link between probability and geometric analysis. This talk is based on joint works with Wilfrid Kendall, Maria Gordina and Phanuel Mariano.

In this talk I will go over two examples of one-dimensional interacting particle systems: Aldous' up-the-river problem, and a modified Diffusion Limited Growth. I will explain how these systems connect to certain PDE problems with boundaries. For the up-the-river problem this connection helps to solve Aldous’ conjecture regarding an optimal strategy. For the modified DLA, this connection helps to characterize the scaling exponent and scaling limit of the boundary at the critical density. This talk is based on joint work with Amir Dembo and Wenpin Tang.

We will discuss recent progress, jointly with R. Gheissari, on the dynamical phase transition for the critical q-state Potts model on the 2D torus (both single-site dynamics such as Glauber/Metropolis and cluster dynamics such as Swendsen--Wang), where the conjectured behavior was a mixing time that is polynomial in the side-length for $q = 2,3,4$ colors but exponential in it for $q>4$. We will then present a proof from a recent work with R. Gheissari and Y. Peres, that, on the complete graph on $n$ vertices with $q>2$ colors, the Swendsen--Wang dynamics is exponentially slow in $n$, improving on the lower bound of $\exp(c\sqrt{n})$ due to Gore and Jerrum in 1999. If time permits, we will then revisit the model on the 2D lattice, and describe the effect of different boundary conditions on its dynamical behavior at criticality.

We consider a process in which a finite set of n agents continually receive a 1 dollar allowance and gamble their fortunes, all in, with one another at a constant rate. This is a variation on the existing compulsive gambler process; in that process, initial fortunes are prescribed and no further allowances are given out. For our process, we find that after some time the distribution of wealth settles into a pattern in which most people have only a few dollars, a few are very wealthy, and a single person possesses most of the cash currently present in the population. In addition, eventually the only way to attain first rank is by winning a bet against the current champion. Moreover, if agents play a fair game, i.e., the probability of winning a bet is proportional to the players' fortunes, the title of champion is assumed by every player infinitely often, although it changes less and less frequently as time goes on. Finally, by examining the process from both the perspective of typical fortune, and that of large fortune, we can go one step further and obtain two distinct limiting processes as n --> infty, with each one admitting a detailed description of its dynamics.

Consider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.

The dynamics of a set of chemical reactions are usually modeled by mass action kinetics as a set of algebraic ordinary differential equations. This model sees the state space of the system as a continuum, whereas chemical reactions represent interactions of a discrete set of molecules. We study large fluctuations of the stochastic mass action kinetics model through Freidlin-Wentzell theory. The application of such a theory to this framework requires justification, in particular because of the non-uniformily Lipschitz character of the model. We therefore find, using tools of Lyapunov stability theory, a set of sufficient conditions for the applicability of large deviations theory to this framework, and prove that such conditions are satisfied by a large class of chemical reaction networks identified exclusively on the base of their topological structure.

A popular family of models of random partitions is called the Chinese Restaurant Process. We imagine n customers being seated randomly and sequentially at tables of a restaurant according to a fixed stochastic rule. Grouping customers by the tables gives us a partition of n. Consider a Markov chain on such partitions where we remove a randomly chosen customer and reseat her. How can one describe the limit of such a Markov chain as n tends to infinity? We will construct such limits as diffusions on partitions of the unit interval. Examples of such random partitions of the unit interval are given by the complement of the zeros of the Brownian motion or the Brownian bridge. The processes of ranked interval lengths of our partitions are members of a family of diffusions introduced by Ethier and Kurtz (1981) and Petrov (2009) that are stationary with respect to the Poisson-Dirichlet distributions. Our construction is a piece of a more complex diffusion on the space of real trees, stationary with respect to the law of the Brownian Continuum Random Tree, whose existence has been conjectured by David Aldous. Joint work with Noah Forman, Doug Rizzolo, and Matthias Winkel.