## Measure-Theoretic Dvoretzky Theorem and Applications to Data Science

- Probability,Uploaded Videos ( 1451 Views )SEPC 2021 in honor of Elizabeth Meckes. Slides from the talks and more information are available <a href="https://services.math.duke.edu/~rtd/SEPC2021/SEPC2021.html">at this link (here).</a>

## Anirban Basak : Ferromagnetic Ising Measures on Large Locally Tree-Like Graphs

- Probability ( 204 Views )We consider the ferromagnetic Ising model on a sequence of graphs $G_n$ converging locally weakly to a rooted random tree. Generalizing [Montanari, Mossel, and Sly (2012)], under an appropriate ?continuity" property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with + and - boundary conditions on that tree. Under the extra assumptions that $G_n$ are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization, is the Ising measure with + boundary condition on the limiting tree. The ?continuity" property holds except possibly for countably many choices of $\beta$, which for limiting trees of minimum degree at least three, are all within certain explicitly specified compact interval. We further show the edge-expander property for (most of) the configuration model graphs corresponding to limiting (multi-type) Galton Watson trees. This talk is based on a joint work with Amir Dembo.

## Elizabeth Meckes : Projections of probability distributions: a measure-theoretic Dvoretzky theorem

- Probability ( 196 Views )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!).

## Swee Hong Chan : Random walks with local memory

- Probability ( 195 Views )In this talk we consider this question for a family of random walks on the square lattice. When the randomness is turned to the maximum, we have the symmetric random walk, which is known to scale to a planar Brownian motion. When the randomness is turned to zero, we have the rotor walk, for which its scaling limit is an open problem. This talk is about random walks that lie in between these two extreme cases and for which we can prove their scaling limit. This is a joint work with Lila Greco, Lionel Levine, and Boyao Li.

## Leonid Koralov : An Inverse Problem for Gibbs Fields

- Probability ( 168 Views )It is well known that for a regular stable potential of pair interaction and a small value of activity one can define the corresponding Gibbs field (a measure on the space of configurations of points in $\mathbb{Z}^d$ or $\mathbb{R}^d$). We consider a converse problem. Namely, we show that for a sufficiently small constant $\overline{\rho}_1$ and a sufficiently small function $\overline{\rho}_2(x)$, $x \in \mathbb{Z}^d$ or $\mathbb{R}^d$, there exist a hard core pair potential, and a value of activity, such that $\overline{\rho}_1$ is the density and $\overline{\rho}_2$ is the pair correlation function of the corresponding Gibbs field.

## David Nualart : Regularity of the density of the stochastic heat equation

- Probability ( 165 Views )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.

## Jeremy Quastel : The effect of noise on KPP traveling fronts

- Probability ( 156 Views )It was noticed experimentally in the late 90's that the speeds of traveling fronts in microscopic systems approximating the KPP equation converge unusually slowly to their continuum values. Brunet and Derrida made a very precise conjecture for the basic model equation, which is the KPP equation perturbed by white noise. We will explain the conjecture and sketch the main ideas of the proof. This is joint work with Carl Mueller and Leonid Mytnik.

## Kevin McGoff : Gibbs posteriors and thermodynamics, Gibbs posterior convergence and the thermodynamic formalism

- Probability ( 145 Views )We consider a Bayesian framework for making inferences about dynamical systems from ergodic observations. The proposed Bayesian procedure is based on the Gibbs posterior, a decision theoretic generalization of standard Bayesian inference. We place a prior over a model class consisting of a parametrized family of Gibbs measures on a mixing shift of finite type. This model class generalizes (hidden) Markov chain models by allowing for long range dependencies, including Markov chains of arbitrarily large orders. We characterize the asymptotic behavior of the Gibbs posterior distribution on the parameter space as the number of observations tends to infinity. In particular, we define a limiting variational problem over the space of joinings of the model system with the observed system, and we show that the Gibbs posterior distributions concentrate around the solution set of this variational problem. In the case of properly specified models our convergence results may be used to establish posterior consistency. This work establishes tight connections between Gibbs posterior inference and the thermodynamic formalism, which may inspire new proof techniques in the study of Bayesian posterior consistency for dependent processes.

## Roberto I. Oliveira : Estimating graph parameters via multiple random walks

- Probability ( 142 Views )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.

## Leonid Bogachev : Gaussian fluctuations for Plancherel partitions

- Probability ( 128 Views )The limit shape of Young diagrams under the Plancherel measure was found by Vershik & Kerov (1977) and Logan & Shepp (1977). We obtain a central limit theorem for fluctuations of Young diagrams in the bulk of the partition 'spectrum'. More specifically, under a suitable (logarithmic) normalization, the corresponding random process converges (in the FDD sense) to a Gaussian process with independent values. We also discuss a link with an earlier result by Kerov (1993) on the convergence to a generalized Gaussian process. The proof is based on poissonization of the Plancherel measure and an application of a general central limit theorem for determinantal point processes. (Joint work with Zhonggen Su.) (see more details hear.

## Eric Foxall : The compulsive gambler with allowances

- Probability ( 127 Views )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.

## Lingjiong Zhu : Self-Exciting Point Processes

- Probability ( 121 Views )Self-exciting point processes are simple point processes that have been widely used in neuroscience, sociology, finance and many other fields. In many contexts, self-exciting point processes can model the complex systems in the real world better than the standard Poisson processes. We will discuss the Hawkes process, the most studied self-exciting point process in the literature. We will talk about the limit theorems and asymptotics in different regimes. Extensions to Hawkes processes and other self-exciting point processes will also be discussed.

## Ivan Matic : Decay and Growth of Randomness

- Probability ( 119 Views )Formation of crystals, spread of infections, and flow of fluids through porous rocks are modeled mathematically as systems consisting of many particles that behave randomly. We will use fluctuations to quantify the randomness, and measure its decay as the number of particles increase. Then we will study the opposite problem: growth of randomness. It turns out that situations exist where it is beneficial to increase chaos. As one example, we will study methods to anonymously distribute files over the internet in such a way that nobody can trace the senders.

## Gerard Letac : Dirichlet curve of a probability in \(R^d\)

- Probability ( 116 Views )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.

## Alessandro Arlotto : Distributional Results for Markov Decision Problems

- Probability ( 115 Views )In this talk, I will discuss several finite-horizon Markov decision problems (MDPs) in which the goal is to gather distributional information regarding the total reward that one obtains when implementing a policy that maximizes total expected rewards. I will begin by studying the optimal policy for the sequential selection of an alternating subsequence from a sequence of n independent observations from a continuous distribution, and I will prove a central limit theorem for the number of selections made by that policy. Then, I will discuss a simple version of a sequential knapsack problem, and I will use its structure to characterize a class of MDPs in which the optimal total reward has variance that can be bounded in terms of its mean. Surprisingly, such characterization turns out to be common in several examples of MDPs from operations research, financial engineering and combinatorial optimization. (The talk is based on joint work with Robert W. Chen, Noah Gans, Larry Shepp, and J. Michael Steele.)

## Sayan Banerjee : Coupling, geometry and hypoellipticity

- Probability ( 114 Views )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.

## Soumik Pal : Markov chains on partitions and their diffusion analogs

- Probability ( 113 Views )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.

## Junchi Li : Axelrods Model

- Probability ( 113 Views )Axelrod's model is a voter model in which individuals have multiple opinions and neighbors interact at a rate proprtional to the fraction of opinions they share. I will describe physuicists predictions about the behavior of this model, recent results of Lanchier in one dimension, and a new result I have proved about the two dimensional case.

## George Tauchen : The Realized Laplace Transform of Volatility

- Probability ( 110 Views )We introduce a new measure constructed from high-frequency financial data which we call the Realized Laplace Transform of volatility. The statistic provides a nonparametric estimate for the empirical Laplace transform of the latent stochastic volatility process over a given interval of time. When a long span of data is used, i.e., under joint long-span and fill-in asymptotics, it is an estimate of the volatility Laplace transform. The asymptotic behavior of the statistic depends on the small scale behavior of the driving martingale. We derive the asymptotics both in the case when the latter is known and when it needs to be inferred from the data. When the underlying process is a jump-diffusion our statistic is robust to jumps and when the process is pure-jump it is robust to presence of less active jumps. We apply our results to simulated and real financial data.

## Georg Menz : A two scale proof of the Eyring-Kramers formula

- Probability ( 109 Views )We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian in the regime of small noise. We give a new proof of the Eyring-Kramers formula for the spectral gap of the associated generator of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg and of the mean-difference estimate introduced by Chafai and Malrieu. The Eyring-Kramers formula follows as a simple corollary from two main ingredients : The first one shows that the Gibbs measures restricted to a domain of attraction has a "good" Poincaré constant mimicking the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of the mean-difference by a new weighted transportation distance. It contains the main contribution of the spectral gap, resulting from exponential long waiting times of jumps between metastable states of the diffusion. This new approach also allows to derive sharp estimates on the log-Sobolev constant.

## Ivan Matic : Deterministic Walks in Random Environments

- Probability ( 107 Views )A deterministic walk in a random environment can be understood as a general finite-range dependent random walk that starts repeating the loop once it reaches a site it has visited before. Such process lacks the Markov property. We will talk about the exponential decay of the probabilities that the walk will reach sites located far away from the origin.

## Jingchen Liu : Rare-event Analysis and Monte Carlo Methods for Gaussian Processes

- Probability ( 107 Views )Gaussian processes are employed to model spatially varying errors in various stochastic systems. In this talk, we consider the analysis of the extreme behaviors and the rare-event simulation problems for such systems. In particular, the topic covers various nonlinear functionals of Gaussian processes including the supremum norm and integral of convex functions. We present the asymptotic results and the efficient simulation algorithms for the associated rare-event probabilities.

## Christopher Hoffman : Geodesics in first passage percolation

- Probability ( 104 Views )First passage percolation is the study of a random metric space generated by replacing each edge in a graph by an edge of a random length. The distance between two vertices u and v is the length of the shortest path connecting u and v. An infinite path P is a geodesic if for any two vertices u and v on P the shortest path between them in the random graph is along P. It is easy to show that in the nearest neighbor graph with vertices Z^2 that there exists at least one (one sided) infinite geodesic starting at any given vertex. It is widely expected that there are infinitely many such one sided infinite geodesics that begin at the origin, with (at least) one in every direction. But it turns out to be very difficult to prove that there are even two with positive probability. We will discuss some recent results which get closer to proving this widely held belief.

## Shankar Bhamidi : Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erd\H{o}s-R\enyi random graph

- Probability ( 103 Views )Over the last few years a wide array of random graph models have been postulated to understand properties of empirically observed networks. Most of these models come with a parameter t (usually related to edge density) and a (model dependent) critical time t_c which specifies when a giant component emerges. There is evidence to support that for a wide class of models, under moment conditions, the nature of this emergence is universal and looks like the classical Erdos-Renyi random graph, in the sense of the critical scaling window and (a) the sizes of the components in this window (all maximal component sizes scaling like n^{2/3}) and (b) the structure of components (rescaled by n^{-1/3}) converge to random fractals related to the continuum random tree. Till date, (a) has been proven for a number of models using different techniques while (b) has been proven for only two models, the classical \erdos random graph and the rank-1 inhomogeneous random graph. The aim of this paper is to develop a general program for proving such results. The program requires three main ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent (ii) scaling exponents of susceptibility functions in the barely subcritical regime are the same as the Erdos-Renyi random graph and (iii) macroscopic averaging of expected distances between random points in the same component in the barely subcritical regime. We show that these apply to a number of fundamental random graph models including the configuration model, inhomogeneous random graphs modulated via a finite kernel and bounded size rules. Thus these models all belong to the domain of attraction of the classical Erdos-Renyi random graph. As a by product we also get the first known results for component sizes at criticality for a general class of inhomogeneous random graphs. This is joint work with Xuan Wang, Sanchayan Sen and Nicolas Broutin.

## Mariana Olvera-Cravioto : Opinion dynamics on complex networks: From mean-field limits to sparse approximations

- Probability ( 98 Views )In a world of polarized opinions on many cultural issues, we propose a model for the evolution of opinions on a large complex network. Our model is akin to the popular Friedkin-Johnsen model, with the added complexity of vertex-dependent media signals and confirmation bias, both of which help explain some of the most important factors leading to polarization. The analysis of the model is done on a directed random graph, capable of replicating highly inhomogeneous real-world networks with various degrees of assortativity and community structure. Our main results give the stationary distribution of opinions on the network, including explicitly computable formulas for the conditional means and variances for the various communities. Our results span the entire range of inhomogeneous random graphs, from the sparse regime, where the expected degrees are bounded, all the way to the dense regime, where a graph having n vertices has order n^2 edges.

## Leonid Petrov : Lax equations for integrable stochastic particle systems

- Probability ( 93 Views )Integrable stochastic particle systems in one space dimension, like the Totally Asymmetric Simple Exclusion Process (TASEP), have been studied for over 50 years (introduced simultaneously in biology and mathematics in 1969-70). They strike a balance between being simple enough to be mathematically tractable and complicated enough to describe many interesting phenomena. Many natural questions about these systems can be generalized by introducing multiple parameters. The interplay between these parameters is powered by the Yang-Baxter equation, which brings new intriguing results to the well-traveled territory. In particular, I will discuss new Lax-type equations for the Markov semigroups of the TASEP and its relatives. Based on a joint work with Axel Saenz.

## Erik Bates : The Busemann process of (1+1)-dimensional directed polymers

- Probability ( 78 Views )Directed polymers are a statistical mechanics model for random growth. Their partition functions are solutions to a discrete stochastic heat equation. This talk will discuss the logarithmic derivatives of the partition functions, which are solutions to a discrete stochastic Burgers equation. Of interest is the success or failure of the ??one force-one solution principle? for this equation. I will reframe this question in the language of polymers, and share some surprising results that follow. Based on joint work with Louis Fan and Timo Seppäläinen.

## Haotian Gu : Universality and Phase Transitions of Holomorphic Multiplicative Chaos

- Probability ( 65 Views )The random distribution Holomorphic multiplicative chaos (HMC) with Gaussian inputs is recently introduced independently by Najnudel, Paquette, and Simm as a limiting object on the unit complex circle of characteristic polynomial of circular beta ensembles, and by Soundararajan and Zaman as an analogue of random multiplicative functions. In this talk, we will explore this rich connection between HMC and random matrix theory, number theory, and Gaussian multiplicative chaos. We will also discuss the regularity of this distribution, alongside the fractional moments and tightness of its Fourier coefficients (also referred to as secular coefficients). Furthermore, we introduce non-Gaussian HMC, and discuss the Gaussian universality and two phase transitions phenomenon in the fractional moments of its secular coefficients. A transition from global to local effect is observed, alongside an analysis of the critical local-global case. As a result, we unveil the regularity of some non-Gaussian HMC and tightness of their secular coefficients. Based on joint work with Zhenyuan Zhang.