Talk given at SEPC 2021 in honor of Elizabeth Meckes

We investigate an example of noise-induced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell,Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the addition of additive noise in the vertical direction leads to a unique invariant probability measure to which the system converges at a uniform, exponential rate. These facts are established primarily through the construction of a Lyapunov function which we generate as the solution to a sequence of Poisson equations. Unlike a number of other works, however, our Lyapunov function is constructed in a systematic way, and we present a meta-algorithm we hope will be applicable to other problems. We conclude by proving positivity properties of the transition density by using Malliavin calculus via some unusually explicit calculations. arXiv:1111.175v1 [math.PR]

The multi-population replicator dynamics (RD) can be considered a dynamic approach to the study of multi-player games, where it was shown to be related to Cross-learning, as well as of systems of co-evolving populations. However, not all of its equilibria are Nash equilibria (NE) of the underlying game, and neither convergence to an NE nor convergence in general are guaranteed. Although interior equilibria are guaranteed to be NE, no interior equilibrium can be asymptotically stable in the multi-population RD, resulting, e.g., in cyclic orbits around a single interior NE. We report on our investigation of a new notion of equilibria of RD, called mutation limits, which is based on the inclusion of a naturally arising, simple form of mutation, but is invariant under the specific choice of mutation parameters. We prove the existence of such mutation limits for a large range of games, and consider an interesting subclass, that of attracting mutation limits. Attracting mutation limits are approximated by asymptotically stable equilibria of the (mutation-)perturbed RD, and hence, offer an approximate dynamic solution of the underlying game, especially if the original dynamic has no asymptotically stable equilibria. Therefore, the presence of mutation will indeed stabilise the system in certain cases and make attracting mutation limits near-attainable. Furthermore, the relevance of attracting mutation limits as a game theoretic equilibrium concept is emphasised by the relation of (mutation-)perturbed RD to the Q-learning algorithm in the context of multi-agent reinforcement learning. However, in contrast to the guaranteed existence of mutation limits, attracting mutation limits do not exist in all games, raising the question of their characterization.

The Ising-Kac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighbourhood of radius $\ga^{-1}$ for $\ga \ll1$ around its base point. We study the Glauber dynamics for this model on a discrete two-dimensional torus $\Z^2/ (2N+1)\Z^2$, for a system size $N \gg \ga^{-1}$ and for an inverse temperature close to the critical value of the mean field model. We show that the suitably rescaled coarse-grained spin field converges in distribution to the solution of a non-linear stochastic partial differential equation. This equation is the dynamic version of the $\Phi^4_2$ quantum field theory, which is formally given by a reaction diffusion equation driven by an additive space-time white noise. It is well-known that in two spatial dimensions, such equations are distribution valued and a \textit{Wick renormalisation} has to be performed in order to define the non-linear term. Formally, this renormalisation corresponds to adding an infinite mass term to the equation. We show that this need for renormalisation for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value. This is a joint work with J.C. Mourrat (Lyon).

Consider a Galton-Watson tree. Pick two individuals at random by simple random sampling from the nth generation and trace heir lines of descent back in time till they meet. Call that generation X_n. In this talk we will discuss the probability distribution of X_n and its limits for the four cases m <1, m=1, m greater than 1 but finite, and m infinite, where m is the mean offspring size.

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.

The now ubiquitous Tracy-Widom laws were first discovered in the context of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles (G{O/U/S}E) of random matrix theory. The latter may be viewed as logarithmic gases with quadratic (Gaussian) potential at three special inverses temperatures (beta=1,2,4). A few years back, Jose Ramirez, Balint Virag, and I showed that that one obtains generalizations of the Tracy-Widom laws at all inverse temperatures (beta>0), though still for quadratic potentials. I'll explain how similar ideas (and considerably more labor) extends the result to general potential, general temperature log-gases. This is joint work with Manjunath Krishnapur and Balint Virag.

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.

Dvoretzky's theorem tells us that if we put an arbitrary norm on n-dimensional Euclidean space, no matter what that normed space is like, if we pass to subspaces of dimension about log(n), the space looks pretty much Euclidean. A related measure-theoretic phenomenon has long been observed: the (one-dimensional) marginals of many natural high-dimensional probability distributions look about Gaussian. A question which had received little attention until recently is whether this phenomenon persists for k-dimensional marginals for k growing with n, and if so, for how large a k? In this talk I will discuss recent work showing that the phenomenon does indeed persist if k less than 2log(n)/log(log(n)), and that this bound is sharp (even the 2!).

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.

In this talk we present a recent result on the smoothness of the density for the solution of a semilinear heat equation with multiplicative space-time Gaussian white noise. We assume that the coefficients are smooth and the diffusion coefficient is not identically zero at the initial time. The proof of this result is based on the techniques of the Malliavin calculus, and the existence of negative moments for the solution of a linear heat equation with multiplicative space-time white noise.

Abstract:One way to convince ourselves that no cooperation can evolve among defectors is via a simple yet one of the most famous games in all of game theory - the Prisoner’s dilemma (PD) game. The players of this game adopt one of the two strategies: a) a cooperator who pays a cost so that another individual can receive a benefit, or b) a defector who can receive benefits, but it has no cost as it does not deal out any benefits at all. As seen from this formulation, no rational individual would opt to be a cooperator. Yet, we can see cooperation everywhere around us and thus (assuming defectors were here first) there must exist at least one mechanism for its evolution. Nowak (2006, 2012) discusses several of such mechanisms, including the kin selection by which cooperation can spread if the benefits go primarily to genetic relatives. In this talk we will introduce a simple PD-like asymmetric matrix game and show how Hamilton’s rule can easily be recovered. We will also introduce a simple PD-like symmetric matrix game to model the evolution of cooperation via greenbeard mechanism, which can be seen as a special case of kin selection.

In 2011 Witten gave a formulation of the Khovanov homology of knots in terms of a system of nonlinear partial differential equations in five dimensions: the Haydys-Witten equations. We highlight the octonionic nature of these equations. This elucidates the importance of the underlying G2 structure and presents the Haydys-Witten equations as a dimensional reduction of the eight-dimensional Spin(7) instanton of Donaldson and Thomas. We conjecture that solutions of the Haydys-Witten equations are in one-to-one correspondence with octonionic monopoles with specific boundary conditions determined by the knot. Octonionic monopole equation also allows to define more general invariants associated to coassociative sumbanifolds in a G2 manifold.

There has been a flurry of recent activity in studying the topology of point cloud data. However, there is a feeling that we are lacking rigorous null hypotheses to compare with the results. This is one motivation for the following: Take n points, independently and identically distributed in R^d, according to some distribution (for example, a standard normal distribution). Connect them if they are close (within distance epsilon, a function of n), and then build the Cech complex or Rips complex. What can one say about the homology of this complex as n approaches infinity? Or the persistent homology with respect to the radius? Using a variety of techniques, including Poissonization, Stein's method, and discrete Morse theory, we are able to identify phase transitions, and for certain ranges of epsilon prove central limit theorems for the Betti numbers. This is joint work with Gunnar Carlsson and Persi Diaconis.

Reliable information transmission between two sites of a network naturally leads to a percolation problem. When the information to be transmitted is quantum an exciting possibility arises: transform the network performing well chosen measurements to enhance the transmission probability. This idea, introduced recently by Acin, Cirac and Lewenstein is now systematically and successfully applied to a variety of two-dimensional networks, but open questions show that a complete theory is missing. The talk will involve some quanta, some network geometry, some percolation and, hopefully, some fun. No knowledge of quantum theory or percolation theory is assumed. Graduate students are encouraged to attend.

What can one say about a graph from multiple (short) random walk trajectories on it? In this talk we consider algorithms that only "see" walk trajectories and the degrees along the way. We will show that the number of vertices, edges and mixing time can be all estimated with a number of RW steps that is sublinear in the size of the graph and in its mixing or relaxation time. Our bounds on the number of RW steps are optimal up to constant or polylog factors. We also argue that such algorithms cannot "know when to stop", and discuss additional conditions that circumvent this limitation. To analyse our results, we rely on novel bounds for random walk intersections. The lower bounds come from a family of explicit constructions.

Points fall into the unit interval according to a certain rule, splitting it up into fragments. An example rule is the following: at each step, two points are randomly drawn from the unit interval and the one that falls into the smaller (or larger) interval is discarded, while the other one is kept. This process is inspired by the so-called "power of choice" paradigm originating in the computer science literature on balanced load allocation models. The question of interest is how much the rule affects the geometry of the point cloud. With Elliot Paquette [1] we introduced a general version of this interval fragmentation model and showed that the empirical distribution of rescaled interval lengths converges almost surely to a deterministic probability measure. I will report on this work as well as on work in progress [2] where we show that the empirical measure of the points converges almost surely to the uniform distribution. The proofs involve techniques from stochastic approximation, non-linear integro-differential equations, ergodic theory for Markov processes and perturbations of semigroups on L^p spaces, amongst other things. [1] Maillard, P., & Paquette, E. (2016). Choices and intervals. Israel Journal of Mathematics, 212(1), 337–384. [2] Maillard, P., & Paquette, E. (in preparation). Interval fragmentations with choice: equidistribution and the evolution of tagged fragments

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.

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.

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.

This talk will be about the convergence problem in mean field control (MFC), i.e. the challenge of rigorously justifying the convergence of certain "symmetric" $N$-particle stochastic control problems towards their mean field counterparts. On the one hand, this convergence problem is already well-understood from a qualitative perspective, thanks to powerful probabilistic techniques based on compactness. On the other hand, quantitative results (i.e. rates of convergence) are more difficult to obtain, in large part because the value function of the mean field problem (which is also the solution to a certain Hamilton-Jacobi equation on the Wasserstein space) may fail to be $C^1$, even if all the data is smooth. After giving an overview of the convergence problem, I will discuss the results of two recent joint works with Cardaliaguet, Daudin, Delarue, and Souganidis, in which we use some ideas from the theory of viscosity solutions to overcome this lack of regularity and obtain rates of convergence of the $N$-particle value functions towards the value function of the corresponding MFC problem.

In this talk, we discuss stochastic homogeneization on the Hopf fibration. Let us consider Berger's metrics on the three sphere, obtained by shrinking the Hopf circle directions by a factor epsilon. So we think of three spheres as two spheres attached at each point a circle. We consider a particle that is moved by two vector fields: a unit speed vector field, with respect to Berger's metrices, along the Hopf circle; and also a non-zero vector field in` \(S^2\) direction' with speed given by a one dimensional Brownian motion. In the limit of epsilon goes to zero, we obtain a Brownian motion on \(S^2\). The effective motion is obtained by moving a particle along a fast rotating horizontal direction.

A link to the paper can be found on her web page. In solid tumors, targeted treatments can lead to dramatic regressions, but responses are often short-lived because resistant cancer cells arise. The major strategy proposed for overcoming resistance is combination therapy. We present a mathematical model describing the evolutionary dynamics of lesions in response to treatment. We first studied 20 melanoma patients receiving vemurafenib. We then applied our model to an independent set of pancreatic, colorectal, and melanoma cancer patients with metastatic disease. We find that dual therapy results in long-term disease control for most patients, if there are no single mutations that cause cross-resistance to both drugs; in patients with large disease burden, triple therapy is needed. We also find that simultaneous therapy with two drugs is much more effective than sequential therapy. Our results provide realistic expectations for the efficacy of new drug combinations and inform the design of trials for new cancer therapeutics.

A Dirichlet random probability \(P_t\) on \(\mathbb{R}^d\) of intensity \(t\) and governed by the probability \(\alpha\) is such that for any partition \( (A_0,\ldots,A_n)\) of \(\mathbb{R}^d\) the random variable \( (P_t(A_0),\ldots,P_t(A_n))\) is Dirichlet distributed with parameters \( (t\alpha(A_0),\ldots,t\alpha(A_n).\) If \(\mu(t\alpha)\) is the distribution of \(X_t=\int xP_t(dx),\) the Dirichlet curve is the map \(t\mapsto \mu(t\alpha)\). Its study raises challenging problems and explicit computations are rare. We prove that if \(\lim_{t\to\infty}\mu(t\alpha)\) exists, it is a Cauchy or Dirac distribution on \(\mathbb{R}^d\). If \(\alpha\) has an expectation we prove that \(t\mapsto \int \psi(x)\mu(t\alpha)(dx)\) is decreasing for any positive convex function \(\psi\) on \(\mathbb{R}^d.\) In other terms the Dirichlet curve decreases in the Strassen order. This is joint work with Mauro Piccioni.

Tempered stable distributions were introduced in Rosinski 2007 as models that look like infinite variance stable distributions in some central region, but they have lighter (i.e. tempered) tails. We extend this class of models to allow for more variety in the tails. While some cases no longer correspond to stable distributions they serve to make the class more flexible and in certain subclasses they have been shown to provide a good fit to data. To characterize the possible tails we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs. We then characterize the weak limits of sequences of tempered stable distributions. We will conclude by discussing a mechanism by which distributions that are stable-like in some central region but with lighter tails show up in applications.

In this talk I will discuss a branching process model developed to study intra-tumor diversity (i.e. the variation amongst the cells within a single tumor). This variation arises due to heritable (epi)genetic alterations which can confer changes in cellular fitness and drug response. In the asymptotic (t-> infinity) regime, we study the growth rates of the population as well as some ecological measures of diversity in the tumor. In the second half of the talk I will discuss applications of this model to studying the emergence of drug resistant populations in Chronic Myeloid Leukemia (CML). (Joint work w/K. Leder, J. Mayberry, R. Durrett, F. Michor)

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.

Consider factor (graphical) models on sparse graph sequences that converge locally to a random tree T. Using a novel interpolation scheme we prove existence of limiting free energy density under uniqueness of relevant Gibbs measures for the factor model on T. We demonstrate this for Potts and independent sets models and further characterize this limit via large-deviations type minimization problem and provide an explicit formula for its solution, as the Bethe free energy for a suitable fixed point of the belief propagation recursions on T (thereby rigorously generalize heuristic calculations by statistical physicists using ``replica'' or ``cavity'' methods). This talk is based on a joint work with Andrea Montanari and Nike Sun.