Recently, quantitative stochastic homogenization of operators in divergence form has witnessed important progress. Our goal is to go beyond the error bound to analyze statistical fluctuations around the homogenized limit. We prove a pointwise two-scale expansion and a large scale central limit theorem for the solution. The approach is probabilistic. The main ingredients include the Kipnis-Varadhan method applied to symmetric diffusion in random environment, a quantitative martingale central limit theorem, the Helffer-Sjostrand covariance representation and Stein's method. This is joint work with Jean-Christophe Mourrat.

The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the GUE Tracy-Widom distribution. We study the length of the LIS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. We prove limit theorems for the LIS for different regimes of the parameter of the distribution. I will also describe some recent results on the longest common subsequence of independent Mallows permutations. Relevant background for the talk will be introduced as needed. Based on work with Ron Peled, Riddhi Basu and Ke Jin.

Random Polymers are modeled as a one dimensional random walk (S_n), with excursion length distribution P(S_1 = n) = \phi(n)/n^\alpha, \alpha > 1 and \phi(n) a slowly varying function. The polymer gets a random reward whenever it visits or crosses an interface. The random rewards are realised as a sequence of i.i.d. variables (\omega_n). Depending on the relation between the mean value of the disorder \omega_n and the temperature, the polymer might prefer to stick to the interface (pinnings) or undergo a long excursion away from it (depinning). In this talk we will review some aspects of random polymer models. We will also discuss in more detail the pinning-depinning transition of the `Pinning' model and prove its annealed and quenched critical points are distinct. This is joint work with Ken Alexander.

Random multiplicative functions are probabilistic models for multiplicative arithmetic functions, such as Dirichlet characters or the Liouville function. In this talk, I will first quickly give an overview of the area, and then focus on some of the recent works on proving central limit theorems, connections to additive combinatorics, as well as some other deterministic applications. Part of the talk is based on joint work with Soundararajan, with Harper and Soundararajan (in progress) and with Angelo and Soundararajan (in progress).

We consider the heat equation on an interval with heat being introduced according to a random mechanism. When the random mechanism is space-time white noise, this equation has been much studied. We look at the case where the white noise is modified by a function A(u)(x) of the current temperatures u and where A is H\"older continuous as a function of u. Unlike other work along these lines, A(u)(x) can depend on the temperatures throughout the interval. Our method involves looking at the Fourier coefficients, which leads to an infinite dimensional system of stochastic differential equations. This is joint work with Ed Perkins.

The graph Laplacian has been of interest in statistics, machine learning, and theoretical computer science in areas from manifold learning to analysis of Markov chains. A common uses of the graph Laplacian has been in spectral clustering and dimension reduction. A theoretical motivation for why spectral clustering works is the Cheeger inequality which relates the eigenvalues of the graph Laplacian to how disconnected the graph is, Betti zero for topology. We ask how the Cheeger inequality extends to higher-order Laplacians, operators on simplicial complexes, and what clustering means for these higher-order operators. This is joint work with John Steenbergen Related to the graph Laplacian is the idea of random walks on graphs. We will define a random walk on simplicial complexes with a stationary distribution that is related to the k-dimensional Laplacian. The stationary distribution reveals (co)homology of the geometry of the random walk. We apply this random walk to the problem of semi-supervised learning, given some labeled observations and many unlabeled observations how does one propagate the labels.

I will present a tutorial on the mathematical models utilized in molecular biology. I will begin with an introduction to the usual stochastic and deterministic models, and then introduce terminology and results from chemical reaction network theory. I will end by presenting the “deficiency zero” theorem in both the deterministic and stochastic settings.

Dynamical systems defined on networks have applications in many fields in science and engineering. In particular, it is important to understand when networks exhibit synchronous or other types of coherent collective behaviors. Other questions include whether such coherent behavior is stable with respect to random perturbation, or how to described the detailed structure of this behavior during its evolution. We will examine the mathematical challenges of understanding these questions for networked dynamical systems with a particular focus on the dynamics of the Kuramoto oscillator model.

The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this work we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness, and tightness) of certain perturbations of the original process. This is a joint work with P.Dupuis and V.Maroulas.

Recent experiments show that an overdamped Brownian particle in a diffusion gradient experiences an additional drift. Equivalently, the Langevin equation describing the particle's motion should be interpreted according to the "anti-Ito" definition of stochastic integrals. I will explain this effect mathematically by studying the zero-mass limit of the stochastic Newton's equation modeling the particle's motion and, using a multiscale expansion, extend the analysis to a wide class of equations, including systems with colored noise and delay terms, interpreting recent electrical circuit experiments. The results were obtained in a collaboration with experimental physicists in Stuttgart: Giovanni Volpe, Clemens Bechinger, Laurent Helden and Thomas Brettschneider, as well as with the mathematics graduate students at the University of Arizona: Scott Hottovy and Austin McDaniel.

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.

Consider a connected network and suppose each edge in the network has a random positive edge weight. Understanding the structure and weight of the shortest path between nodes in the network is one of the most fundamental problems studied in modern probability theory and goes under the name first passage percolation. It arises as a fundamental building block in many interacting particle system models such as the spread of epidemics on networks. To a large extent such problems have been only studied in the context of the n-dimensional lattice. In the modern context these problems take on an additional significance with the minimal weight measuring the cost of sending information while the number of edges on the optimal path (hopcount) representing the actual time for messages to get between vertices in the network. Given general models of random graphs with random edge costs, can one develop techniques to analyze asymptotics of functionals of interest which are robust to the model formulation? The aim of this talk is to describe a heuristic based on continuous time branching processes which gives very easily, a wide array of asymptotic results for random network models in terms of the Malthusian rate of growth and the stable age distribution of associated branching process. These techniques allow us to solve not only first passage percolation problems rigorously but also understand functionals such as the degree distribution of shortest path trees, congestion across edges as well as asymptotics for “betweeness centrality” a concept of crucial interest in social networks, in terms of Cox processes and extreme value distributions. These techniques also allow one to exactly solve models of “weak disorder” in the context of the stochastic mean field model of distance, a model of great interest in probabilistic combinatorial optimization.

We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d \ge 3$. Combining this result with properties of the PDE and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first two applications confirm conjectures of Cox and Perkins and Ohtsuki et al.

Pratima Hebbar, Probability Seminar on October 21, 2021

Our primary goal is to describe a strong quantitative form of the Shannon-McMillan-Breiman theorem for log-concave probability measures on linear spaces, even in the absence of stationarity. The main technical result is a concentration of measure inequality for the ``information content'' of certain random vectors. We will also briefly discuss implications. In particular, by combining this concentration result with ideas from information theory and convex geometry, we obtain a reverse entropy power inequality for convex measures that generalizes the reverse Brunn-Minkowski inequality of V. Milman. Along the way, we also develop a new information-theoretic formulation of Bourgain's hyperplane conjecture, as well as some Gaussian comparison inequalities for the entropy of log-concave probability measures. This is joint work with Sergey Bobkov (Minnesota).

In the late 20th century, statistical physicists introduced a chemical reaction model called ballistic annihilation. In it, particles are placed randomly throughout the real line and then proceed to move at independently sampled velocities. Collisions result in mutual annihilation. Many results were inferred by physicists, but it wasn’t until recently that mathematicians joined in. I will describe my trajectory through this model. Expect tantalizing open questions.

In '53, GI Taylor estimated the effective dispersion rate of a solute diffusing in the presence of a laminar flow in a pipe. It turns out that the length scales involved in typical pipes are too short for Taylor's result to apply. The goal of my talk will be to establish a preliminary estimate for the effective dispersion rate in a model problem at time scales much shorter than those required in Taylor's result. Precisely, I will study a diffusive tracer in the presence of a fast cellular flow. The main result (joint with A. Novikov) shows that the variance at intermediate time scales is of order $\sqrt{t}$. This was conjectured by W. Young, and is consistent with an anomalous diffusive behaviour.

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.

In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. In a recent breakthrough work, Sourav Chatterjee proved this conjecture using a strong definition of the exponents. I will discuss work I just completed with Tuca Auffinger, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the relation. One advantage of our argument is that it does not require a non-trivial technical assumption of Chatterjee on the weight distribution.

The goal of this talk is to give a self-contained introduction to some aspects of the thermodynamic formalism in ergodic theory that should be accessible to probabilists. In particular, the talk will focus on equilibrium states and Gibbs measures on the Z^d lattice. We'll present some basic examples in the theory, as well as some recent results that are joint with Christopher Hoffman.

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.

For neutral branching models of two types of populations there are three universality classes of behavior: independent branching, (one-sided) catalytic branching and mutually catalytic branching. Loss of independence in the two latter models generates many new features in the way that the populations evolve. In this talk I will focus on describing the genealogy of a catalytic branching diffusion. This is the many individual fast branching limit of an interacting branching particle model involving two populations, in which one population, the "catalyst", evolves autonomously according to a Galton-Watson process while the other population, the "reactant", evolves according to a branching dynamics that is dependent on the number of catalyst particles. We show that the sequence of suitably rescaled family forests for the catalyst and reactant populations converge in Gromov-Hausdorff topology to limiting real forests. We characterize their distribution via a reflecting diffusion and a collection of point-processes. We compare geometric properties and statistics of the catalytic branching forests with those of the "classical" (independent branching) forest. This is joint work with Andreas Greven and Anita Winter.

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.