In the range 4 < \kappa < 8, the intersection of the Schramm-Loewner Curve (one of the central objects in the theory of 2-D Conformally Invariant Systems) with the boundary of its domain is a random fractal set. After reviewing some previous results on the dimension and measure of this set, I will describe recent joint work with Ilia Binder and Fredrik Viklund that partitions this set of points according to the generalized "angle" at which the curve hits the boundary, and computes the Hausdorff dimension of each partition set. The Hausdorff dimension as a function of the angle is what we call the dimension spectrum.

 We study the path properties of the random polymer attracted to a defect line by a potential with disorder, and we prove that in the delocalized regime, at any temperature, the number of contacts with the defect line remains in a certain sense ``tight in probability'' as the polymer length varies. On the other hand we show that at sufficiently low temperature, there exists a.s. a subsequence where the number of contacts grows like the log of the length of the polymer.

We consider the Williams Bjerknes model, also known as the biased voter model on the d-regular tree T^d, where d \geq 3. Starting from an initial configuration of ``healthy'' and ``infected'' vertices, infected vertices infect their neighbors at Poisson rate \lambda \geq 1, while healthy vertices heal their neighbors at Poisson rate 1. All vertices act independently. It is well known that starting from a configuration with a positive but finite number of infected vertices, infected vertices will continue to exist at all time with positive probability iff \lambda > 1. We show that there exists a threshold \lambda_c \in (1, \infty) such that if \lambda > \lambda_c then in the above setting with positive probability all vertices will become eventually infected forever, while if \lambda < \lambda_c, all vertices will become eventually healthy with probability 1. In particular, this yields a complete convergence theorem for the model and its dual, a certain branching coalescing random walk on T^d -- above \lambda_c. We also treat the case of initial configurations chosen according to a distribution which is invariant or ergodic with respect to the group of automorphisms of T^d. Joint work with A. Vandenberg-Rodes, R. Tessler.

The aim of this talk is to introduce the subject of spin glasses,
and more generally the statistical mechanics of quenched disorder,
as a problem of general interest to physicists from multiple disciplines and
backgrounds. Despite years of study, the physics of quenched
disorder remains poorly understood, and represents a major gap in our
understanding of the condensed state of matter. While there are many
active areas of investigation in this field, I will narrow the focus of this
talk to our current level of understanding of the low-temperature
equilibrium structure of
realistic (i.e., finite-dimensional) spin glasses.

I will begin with a brief survey of why the subject is of interest not only
to physicists,
but also mathematicians, computer scientists, and scientists working in
other areas. A brief review of the basic features of spin glasses and what
is
known experimentally will follow. I will then turn to the problem of
understanding the nature of the spin glass phase --- if it exists.
The central question to be addressed is the nature of broken symmetry in
these systems. Parisi's replica symmetry breaking approach,
now mostly verified for mean field spin glasses, attracted great excitement
and interest as a novel and exotic form of symmetry breaking. But does it
hold also for real spin glasses in finite dimensions? This has been a
subject of intense controversy, and although the issues surrounding it have
become more sharply defined
in recent years, it remains an open question. I will explore this problem,
introducing new mathematical constructs such as the metastate along the way.
The talk will conclude with an examination of how and in which respects the
statistical mechanics of disordered systems might differ from that of
homogeneous systems.

Abstract: For the site percolation model on the triangular lattice and
certain generalizations for which Cardy’s Formula has been established
we acquire a power law estimate for the rate of convergence of the
crossing probabilities to Cardy’s Formula.