Probability and Statistics Seminar

After introducing essential notions related to Markov chains and their applications to Markov chain Monte Carlo (MCMC) algorithms, we will present a general class of non-reversible Markov chains defined on partially ordered discrete set. This type of Markov chains, which have appeared in several MCMC methods, are often believed to be more efficient (in terms of asymptotic variance) than their reversible equivalent. The main focus of this talk is to present conditions under which this is provably the case. Applications to the sampling of an Ising model and variable selection illustrate our work.

One versatile probabilistic approach to study directed percolation and polymer models is through comparison with their equilibrium versions when the latter are sufficiently tractable and provide a satisfactory approximation for the purposes of the problem at hand. In this talk, we focus on the paradigmatic setting of last-passage percolation with i.i.d. exponential weights on the lattice quadrant. The equilibrium versions of this model are explicitly obtained by placing additional independent exponential weights with suitable rates on the boundary (axes). Then an important aspect of the aforementioned comparison scheme is to control the point where a given geodesic from the origin exits the boundary. The main results to be presented in the talk are sharp upper bounds on the tails of the exit points. While these bounds can be and, in part, have been concurrently established via known tail bounds for the largest eigenvalue of the Laguerre ensemble, our technique is new and relies entirely on the stationarity of the equilibrium models. We also aim to discuss two applications of the exit bounds related to the geometry of geodesics. These results provide upper bounds on the speed of distributional convergence to the Busemann limits and to the limiting direction of the competition interface.

Joint work with C. Janjigian and T. Seppäläinen.

I will discuss some results on extreme eigenvalue distributions of adjacency matrices of random $d$-regular graphs, which are believed to be universal following the Tracy-Widom distribution.

In the first part of the talk I will present the results on random $d$-regular graphs, where $d$ grows with the size $N$ of the graph, we confirmed that on the regime $N^{2/9}<< d<< N^{1/3}$ the extremal eigenvalues after proper rescaling are concentrated at scale $N^{-2/3}$ and their fluctuations are governed by the Tracy-Widom statistics. Thus, in the same regime of $d$, about fifty two percent of all $d$-regular graphs have the second-largest eigenvalue strictly less than $2\sqrt{d-1}$. In the second part of the talk, I will focus on random $d$-regular graphs with fixed $d>=3$, and give a new proof of Alon's second eigenvalue conjecture that with high probability, the second eigenvalue of a random $d$-regular graph is bounded by $2\sqrt{d-1}+o(1)$, where we can show that the error term is polynomially small in the size of the graph. These are based on joint works with Roland Bauerschmids, Antti Knowles and Horng-Tzer Yau.

In 2002, Johansson conjectured that the maximum of the Airy process minus a parabola is almost surely achieved at a unique location. This result was proved a decade later by Corwin and Hammond; Moreno Flores, Quastel and Remenik; and Pimentel. The Airy process is the fixed time spatial marginal of the KPZ fixed point run from narrow wedge initial data. None of these results rules out the possibility that at random times, the KPZ fixed point spatial marginal violates maximizer uniqueness. In terms of directed polymers, these times of non-uniqueness are instants of instability in the zero temperature polymer measure — moments at which the endpoint jumps from one location to another. We showed in our work that, for a broad class of initial data, with positive probability the set of such times is non-empty, and conditionally on this event this set almost surely has Hausdorff dimension two-thirds.

In this talk I will discuss the work in progress on the study of multi-point joint distribution of the so-called discrete time totally asymmetric simple exclusion process with parallel updates over a spatially periodic domain , which is a typical model in the so-called 1+1 Kardar-Parisi-Zhang universality class with finite-volume space. This is an extension of the recent work of Baik and Liu(and independently Prohlac for one-point case) on the study of continuous-time periodic TASEP which can be obtained by taking a certain continuum limit of our results. In particular we obtain a finite-time multi-point joint distribution formula and perform asymptotic analysis under the so-called relaxation time scale. The limiting distribution function agrees with the one obtained by Baik and Liu which interpolates equilibrium dynamics and KPZ dynamics over infinite-volume spaces(so in particular the one-point marginal interpolates gaussian distribution and certain Tracy-Widom type distribution depending on the initial data). This provides evidence for KPZ universality under finite-volume spaces with periodic boundary conditions.

Phylodynamics seeks to extract information on the transmission and evolution of a pathogen from genetic sequences. The problem boils down to that of constructing the likelihood function that links a dynamical system model to the genetic data. The two approaches to the problem currently in use view the pathogen genealogy as a static, historical record of the generating process and rely on questionable assumptions and approximations. By viewing the generative model as a genealogy-valued Markov process, we show that one can derive exact expressions for the likelihood and efficient Monte Carlo algorithms for its calculation.

Gibbs point processes form an important class of models in statistical mechanics, stochastic geometry and spatial statistics. They model randomly distributed points in space. "Points" can be molecules of a gas or trees of a forest. A notorious difficulty is that many quantities cannot be computed explicitly; for example, the intensity measure (density) of a Gibbs point process is a highly non-trivial function of the intensity of the underlying Poisson point process (activity). As a partial way out, physicists and mathematical physicists have long worked with perturbation series, called cluster expansions.

The talk gives a brief introduction to Gibbs point processes and cluster expansions and presents some recent developments.