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list Friday, August 24. No SeminarFriday, August 31. Stochastics Seminar. 3-4 PM. LCB 225

Todd Kemp, University of California - San Diego
"Random Matrices with Structured or Unstructured Correlations"

Random matrix theory began with the study, by Wigner in the 1950s, of high-dimensional matrices with i.i.d. entries (up to symmetry). The empirical law of eigenvalues demonstrates two key phenomena: bulk universality (the limit empirical law of eigenvalues doesn't depend on the laws of the entries) and concentration (the convergence is robust and fast).

Several papers over the last decade (initiated by Bryc, Dembo, and Jiang in 2006) have studied certain special random matrix ensembles with structured correlations between some entries. The limit laws are different from the Wigner i.i.d. case, but each of these models still demonstrates bulk universality and concentration.

In this lecture, I will talk about very recent results of mine and my students on these general phenomena:

Bulk universality holds true whenever there are constant-width independent bands, regardless of the correlations within each band. (Interestingly, the same is not true for independent rows or columns, where universality fails.) I will show several examples of such correlated band matrices generalizing earlier known special cases, demonstrating how the empirical law of eigenvalues depends on the structure of the correlations.

At the same time, I will show that concentration is a more general phenomenon, depending not on the the structure of the correlations but only on the sizes of correlated partition blocks. Under some regularity assumptions, we find that Gaussian concentration occurs in NxN ensembles so long as the correlated blocks have size smaller than $N^2/\log N$.

Friday, September 7. No seminar.Friday, September 14. No seminar.Friday, September 21. No seminar.Friday, September 28. Stochastics Seminar.3-4 PM. LCB 225

Yuri Bakhtin, New York University
"Ergodic theory of the stochastic Burgers equation"

The stochastic Burgers equation is one of the basic evolutionary SPDEs related to fluid dynamics and KPZ, among other things. The ergodic properties of the system in the compact space case were understood in 2000's. With my coauthors, Eric Cator, Kostya Khanin, Liying Li, I have been studying the noncompact case. The one force - one solution principle has been proved for positive and zero viscosity. The analysis is based on long-term properties of action minimizers and polymer measures. The latest addition to the program is the convergence of infinite volume polymer measures to Lagrangian one-sided minimizers in the limit of vanishing viscosity (or, temperature) which results in the convergence of the associated global solutions and invariant measures.

Christopher Janjigian, University of Utah
"Busemann functions and Gibbs measures in directed polymer models on $\mathbb Z^2$"

We consider nearest-neighbor random walks on the planar square lattice in a general iid
space-time random potential, also known as directed polymers in a random environment.
We prove results on existence, uniqueness (and non-uniqueness), and the LLN for
semi-infinite path measures. Our main tool is Busemann functions, which are stochastic
processes obtained through limits of ratios of partition functions.

Friday, October 12. No seminar. Fall break.Friday, October 19. No Seminar.Friday, October 26. Stochastics Seminar.3-4 PM. LCB 225

Li-Cheng Tsai, Columbia University
"Lower-tail large deviations of the KPZ equation"

Regarding time as a scaling parameter, we prove the one-point, lower tail Large Deviation Principle (LDP) of the KPZ equation, with an explicit rate function. This result confirms existing physics predictions. We utilize a formula from [Borodin Gorin 16] to convert LDP of the KPZ equation to calculating an exponential moment of the Airy point process, and analyze the latter via stochastic Airy operator and Riccati transform.

Friday, November 2. No Seminar.Friday, November 9. Stochastics Seminar. 3-4 PM. LCB 225

Firas Rassoul-Agha, University of Utah
"Shifted weights and restricted-length paths in first-passage percolation"

We study standard first-passage percolation via related optimization
problems that restrict path length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of geodesic length due to Hammersley, Smythe and Wierman, and Kesten. We study the regularity of the time constant as a function of the shift of weights. For unbounded weights, this function is strictly concave and in case of two or more atoms it has a dense set of singularities. For any weight distribution with an atom at the origin there is a singularity at zero, generalizing a result of Steele and Zhang for Bernoulli FPP. The regularity results are proved by the van den Berg-Kesten modification argument. This is joint work with Arjun Krishnan and Timo Seppalainen.

Leonid Petrov, University of Virginia
"Nonequilibrium particle systems in inhomogeneous space"

I will discuss stochastic interacting particle systems in the KPZ universality class evolving in one-dimensional inhomogeneous space. The inhomogeneity means that the speed of a particle depends on its location. I will focus on integrable examples of such systems, i.e., for which certain observables can be written in exact form suitable for asymptotic analysis. Examples include a continuous-space version of TASEP (totally asymmetric simple exclusion process), and the pushTASEP (=long-range TASEP). For integrable systems, density limit shapes can be described in an explicit way. We obtain asymptotics of fluctuations, in particular, around slow bonds and infinite traffic lights.

Eric Cator, Radboud University
"From WW II bombs to robust statistics"

In this talk I will highlight some of the statistical problems that I have been working on over the years.

The first project is about unexploded bombs from World War II still in the ground in Amsterdam. There is a lot of historical information on bombardment the Allies carried out on German industry in Amsterdam, and the city would like to use this information to determine the risk of starting new building projects in these areas. We developed a new model for this, and were able to implement a Bayesian statistics approach through the Metropolis-Hastings algorithm. Quantative statements about the risk of specific areas can now be given, and this method is about to be implemented across the Netherlands.

The second project is on the Minimum Covariance Determinant (MCD) estimator that is frequently used in robust statistics. Consider a sample from some multidimensional distribution. In robust statistics, we try to estimate properties of the underlying distribution using only a fraction of the data, so that if part of the data is corrupted, this would not influence our estimators too much. The MCD tries to find that fixed fraction of the data that has a covariance matrix with minimal determinant. It was known, for some special multivariate distributions, that this estimator has better convergence properties than the more intuitive Minimum Volume Estimator, but the asymptotic normality and even consistency for general distributions remained out of reach for more than 10 years. By extending the definition of MCD from a sample to a continuous distribution in a proper way, we were able to prove consistency and asymptotic normality, with explicit covariance structure, under mild conditions on the underlying distribution.

Wednesday, November 28. Stochastics Seminar.2-3 PM. LCB 225 NOTE SPECIAL DAY AND TIME

Eric Cator, Radboud University
"The contact process (or SIS model) on large graphs"

The contact process is a model for the spread in time of information (or disease) on a graph. In this talk I will discuss a new idea to study the meta-stable distribution of the contact process on a graph with edge-dependent infection rates and vertex-dependent healing rates, based on a new approximation of the infection rate matrix. This allows us to approximate not only the expectation but also the covariance structure of the meta-stable distribution, from which we extract a second order correction to the mean field approximation. I will show how our model can be combined with data from a real world airport network to produce accurate predictions of infection spread. Finally, I will discuss a new result for the extinction time of the contact process on finite random graphs, extending a famous result by Chatterjee and Durrett that disproved a long-standing physics conjecture, but using a completely different approach.

Tom Kurtz, University of Wisconsin - Madison
"Population models as partial observations of genealogical models "

Classical models of biological populations, for example, Markov branching processes, typically model population size and possibly the distribution of types and/or locations of individuals in the population. The intuition behind these models usually includes ideas about the relationships among the individuals in the population that cannot be directly recovered from the model. This loss of information is even greater if one employs large population approximations such as the diffusion approximations popular in population genetics. "Lookdown" constructions provide representations of population models in terms of countable systems of particles in which each particle has a "type" which may record both spatial location and genetic type and a "level" which incorporates the lookdown structure which in turn captures the genealogy of the population. The original population model can then be viewed as the result of partial observation of the more complex model. We will exploit ideas from filtering of Markov processes to make the idea of partial observation clear and to justify the lookdown construction.

Elina Robeva, MIT
"Orthogonal Tensor Decomposition"

Tensor decomposition has many applications. However, it is often a hard problem. In this talk I will discuss a family of tensors, called orthogonally decomposable tensors, which retain some of the properties of matrices that general tensors don't. A symmetric tensor is orthogonally decomposable if it can be written as a linear combination of tensor powers of n orthonormal vectors. Such tensors are interesting because their decomposition can be found efficiently. We study their spectral properties and give a formula for all of their eigenvectors. We also give equations defining all real symmetric orthogonally decomposable tensors. Analogously, we study nonsymmetric orthogonally decomposable tensors, describing their singular vector tuples and giving polynomial equations that define them. In an attempt to extend the definition to a larger set of tensors, we define tight-frame decomposable tensors and study their properties. Finally, I will conclude with some open questions and future research directions.

Elina Robeva, MIT
"Maximum likelihood estimation under total positivity"

Nonparametric density estimation is a challenging statistical problem -- in general the maximum likelihood estimate (MLE) does not even exist! Introducing shape constraints allows a path forward. In this talk I will discuss non-parametric density estimation under total positivity (i.e. log-supermodularity). Though they possess very special structure, totally positive random variables are quite common in real world data and exhibit appealing mathematical properties. Given i.i.d. samples from a totally positive distribution, we prove that the MLE exists with probability one if there are at least 3 samples. We characterize the domain of the MLE, and give algorithms to compute it. If the observations are 2-dimensional or binary, we show that the logarithm of the MLE is a piecewise linear function and can be computed via a certain convex program. Finally, I will discuss statistical guarantees for the convergence of the MLE, and will conclude with a variety of further research directions.