Eagle House

We introduce and study ergodic multidimensional diffusion processes interacting through their ranks; these interactions lead to invariant measures which are in broad agreement with stability properties of 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 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 interacting diffusions; 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 intersection local times.

The theory we develop has connections with the analysis of Queueing Networks in heavy traffic, as well as with models of competing particle systems in Statistical Mechanics, such as the Sherrington-Kirkpatrick model for spin-glasses.

Probability measures in infinite dimensional spaces especially that induced by stochastic processes are the main objects of the talk. We discuss the role played by measures on analysis on path spaces, Sobolev inequalities, weak formulations and local versions of such inequalities related to Brownian bridge measures.

We relate the partition function associated with a certain Brownian directed polymer model to a diffusion process which is closely related to a quantum integrable system known as the quantum Toda lattice. This result is based on a `tropical' variant of a combinatorial bijection known as the Robinson-Schensted-Knuth (RSK) correspondence and is completely analogous to the relationship between the length of the longest increasing subsequence in a random permutation and the Plancherel measure on the dual of the symmetric group.

We study a simple heat-bath type dynamic for a simple model of
polymer interacting with an interface. The polymer is a nearest neighbor path in
Z, and the interaction is modelised by energy penalties/bonuses given when the
path touches 0. This dynamic has been studied by D. Wilson for the case without
interaction, then by Caputo et al. for the more general case. When the interface
is repulsive, the dynamic slows down due to the appearance of a bottleneck in the
state space, moreover, the systems exhibits a metastable behavior, and, after time
rescaling, behaves like a two-state Markov chain.

We consider a directed random polymer interacting with an interface
that carries random charges some of which attract while others repel
the polymer. Such a polymer can be in a localized or delocalized
phase, i.e., it stays near the interface or wanders away respectively.
 The phase it chooses depends on the temperature and the average bias
of the disorder. At a given temperature, there is a critical bias
separating the two phases. A question of particular interest, and
which has been studied extensively in the Physics and Mathematics
literature, is whether the quenched critical bias differs from the
annealed critical bias. When it does, we say that the disorder is
relevant.

Using a large deviations result proved recently by Birkner, Greven,
and den Hollander, we derive a variational formula for the quenched

critical bias. This leads to a necessary and sufficient condition for
disorder relevance that implies easily some known results as well as
new ones.

The talk is based on joint work with Frank den  Hollander.

In this talk, we will look at the diffusive scaling limit of a class of

one-dimensional random walks in a random space-time environment. In the

scaling limit, this gives rise to a so-called stochastic flow of kernels as

introduced by Le Jan and Raimond and generalized by Howitt and Warren. We will

prove several new results about these stochastic flows of kernels by making

use of the theory of the Brownian web and net. This is joint work with R. Sun

and E. Schertzer.

Abstract:  An instance of the Potts model is defined by a graph of interactions and a number, q, of  different ``spins''.  A configuration in this model is an assignment of spins to vertices. Each configuration has a weight, which in the ferromagnetic case is greater when more pairs of adjacent spins are alike.  The classical Ising model is the special case

of q=2 spins.  We consider the problem of computing an approximation to the partition function, i.e., weighted sum of configurations, of

an instance of the Potts model.  Through the random cluster formulation it is possible to make sense of the partition function also for non-integer q.  Yet another equivalent formulation is as the Tutte polynomial in the positive quadrant.

About twenty years ago, Jerrum and Sinclair gave an efficient (i.e., polynomial-time) algorithm for approximating the partition function of a ferromagnetic Ising system. Attempts to extend this result to q≠2 have been unsuccessful. At the same time, no convincing evidence has been presented to indicate that such an extension is impossible.  An interesting feature of the random cluster model when q>2 is that it exhibits a first-order phase transition, while for 1≤q≤2 only a second-order phase transition is apparent.  The idea I want to convey in this talk is that this first-order phase transition can be exploited in order to encode apparently hard computational problems within the model.  This provides the first evidence that the partition function of the ferromagnetic Potts model may be hard to compute when q>2.

This is joint work with Leslie Ann Goldberg, University of Liverpool.