2013 Charles River Lectures on Probability Theory and Related Topics

Agenda

The event features five lectures by distinguished researchers in the areas of probability and related topics.

• 9:30 a.m - 9:50 a.m. Registration and Coffee

• 9:50 a.m. - 10:00 a.m. Opening Remarks

• 10:00 a.m. - 11:00 a.m. Martin Hairer (Warwick)

  Dynamics near criticality

  Heuristically, one can give arguments why the fluctuations of classical models of statistical mechanics near criticality are typically expected to be described by nonlinear stochastic PDEs. Unfortunately, in most examples of interest, these equations seem to make no sense whatsoever due to the appearance of infinities or of terms that are simply ill-posed.

  I will give an overview of a new theory of "regularity structures" that allows to treat such equations in a unified way, which in turn leads to a number of natural conjectures. One interesting byproduct of the theory is a new (and rigorous) interpretation of "renormalisation group techniques" in this context.

  At the technical level, the main novel idea involves a complete rethinking of the notion of "Taylor expansion" at a point for a function or even a distribution. The resulting structure is useful for encoding "recipes" allowing to multiply distributions that could not normally be multiplied. This provides a robust analytical framework to encode renormalisation procedures.

• 11:10 a.m. - 12:10 p.m. Balint Virag (Toronto)

  Mean quantum percolation

  It is an open problem whether the limiting eigenvalue distribution of percolation in a large box has an absolutely continuous part. Similarly, it is not known whether the analogue of the Wigner semicircle law for a sparse Erdos-Renyi random matrix has an absolutely continuous part.

  In joint work with Charles Bordenve and Arnab Sen we can show that non-atomic part exists in many models. I will review the background and some new methods.

• 12:10 p.m. - 1:40 p.m. Lunch Break

• 1:40 p.m. - 2:40 p.m. Ioannis Karatzas (Columbia)

  Competing diffusive particle systems and models of large equity markets

  We introduce and study stable multidimensional diffusions interacting through their ranks. These interactions give rise to invariant measures which are in broad agreement with stability properties observed in large equity markets over long time-periods.

  The models we develop assign growth rates and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset. Such models are able realistically to capture certain critical features of the observed stability of capital distribution over the past century, all the while being simple enough to allow for rather detailed analytical study.

  The methodologies used in this study touch upon the question of triple points for systems of competing diffusive particles; in particular, some choices of parameters may permit triple (or higher-order) collisions to occur. We show, however, that such multiple collisions have no effect on any of the stability properties of the resulting system. This is accomplished through a detailed analysis of collision local times.

  The models have connections with the analysis of Queueing Networks in heavy traffic, and with competing particle systems in Statistical Mechanics (e.g., Sherrington-Kirkpatrick model for spin-glasses). Their hydrodynamic-limit behavior is governed by generalized porous medium equations with convection, whereas limits of a different kind display phase transitions and are governed by Poisson-Dirichlet distributions. We survey briefly recent progress on some of these fronts, and suggest open problems for further study.

• 2:50 p.m. - 3:50 p.m. Elchanan Mossel (Berkeley)

  On sparse block models

  Block models are classical random graphs (random matrices) models which are studied extensively in statistics and computer science. A conjecture from physics by Decelle et. al predicts an exact formula for the location of the phase transition of the model. I will discuss recent progress towards a proof of the conjecture as well as connections with belief propagation, the reconstruction problems on trees, and zeta function of random graphs. Based on joint works with Joe Neeman and Allan Sly.

• 3:50 p.m. - 4:20 p.m. Afternoon Break

• 4:20 p.m. - 5:20 p.m. Assaf Naor (NYU)

  Super-expanders

  A bounded degree n-vertex graph G = (V, E) is an expander if and only if for every choice of n vectors {x_v}_{v \in V} in R^k the average of the Euclidean distance between x_u and x_v is within a constant factor of the average of the same terms over those pairs {u, v} that form an edge in E. The fact that this property is equivalent to the usual combinatorial notion of graph expansion is very simple to prove, and once stated, it is obvious to ask what happens when R^k is replaced by other metric spaces. It turns out that this is a subtle question that relates to a long line of investigations in analysis and geometry. Graphs that are expanders in the classical sense (including random graphs) may or may not be expanders with respect to certain non-Euclidean geometries of interest. Existence of such "metric" expanders becomes a delicate question due in part to the fact that the existing combinatorial, probabilistic and spectral methods that are used for the purpose of constructing classical expanders are insufficient in the metric setting.

  In this talk we will formulate some of the basic questions in this direction and explain some of the ideas and methods that were introduced in order to address them, with emphasis on probabilistic aspects such as the construction of expanders with respect to random graphs and the role of martingale methods in this context.