We consider a polymer with configuration modeled by the trajectory of a Markov chain, interacting with a potential of form u+V_n when it visits a particular state 0 at time n, with V_n representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned by the potential. Typically the probability of an excursion of length n for the underlying Markov chain is taken to decay as a power of n (called the loop exponent), perhaps with a slowly varying correction. A particular case not covered in a number of previous studies is that of loop exponent one, which includes simple random walk in two dimensions. We show that in this case, at all temperatures, the critical values of u in the quenched and annealed models are equal, in contrast to all other loop exponents, for which these critical values are known to differ at least at low temperatures. The work is joint with N. Zygouras.

The beta ensembles of random matrix theory are natural generalizations of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles, these classical cases corresponding to beta = 1, 2, and 4. We prove that the extremal eigenvalues for the general ensembles have limit laws described by the low lying spectrum of certain raandom Schroedinger operators, as conjectured by Edelman-Sutton. As a corollary, a second characterization of these laws is made the explosion probability of a simple one-dimensional diffusion. A complementary pictures is developed for beta versions of random sample-covariance matrices. (Based on work with J. Ramirez and B. Virag.)

We consider random matrices associated to random walks on the complete
graph with random weights. When the weights have finite second moment we
find Wigner-like behavior for the empirical spectral density. If the
weights have finite fourth moment we prove convergence of extremal
eigenvalues to the edge of the semi-circle law. The case of weights with
infinite second moment is also considered. In this case we prove
convergence of the spectral density on a suitable scale and the limiting
measure is characterized in terms certain Poisson weighted infinite
trees associated to the starting graph. Connections with recent work on
random matrices with i.i.d. heavy-tailed entries and several open
problems are also discussed. This is recent work in collaboration with
D. Chafai and C. Bordenave (from Univ. P.Sabatier, Toulouse - France).

Consider a crystal formed of two types of atoms placed at the nodes of the
integer lattice. The type of each atom is chosen at random, but the crystal
is statistically shift-invariant. Consider next an electron hopping from atom
to atom. This electron performs a random walk on the integer lattice with
randomly chosen transition probabilities (since the configuration seen by
the electron is different at each lattice site). This process is highly
non-Markovian, due to the interaction between the walk and the
environment.

We will present a martingale approach to proving the invariance principle
(i.e. Gaussian fluctuations from the mean) for (irreversible) Markov chains
and show how this can be transferred to a result for the above process
(called random walk in random environment).

This is joint work with Timo Sepp\"al\"ainen.