Consider the partition function of a directed polymer in an IID
field. Under some mild assumptions on the field, it is a well-known fact
that the free energy of the polymer is equal to some deterministic constant
for almost every realization of the
field and that the upper tail large deviations is exponential. In this
talk I'll discuss the lower tail large deviations and present a method
for estimating it. As a consequence, I'll show that the lower
tail large deviations exhibits three regimes, determined by the
tail of the negative part of the field. The method applies to other
oriented models and can be adapted to non-oriented models as well. This
work extends the results of a recent paper by Cranston Gautier and
Mountford. A preprint is availabe on www.math.uci.edu/~ibenari

Methods of enumeration of spanning trees in a finite graph and relations to
various areas of mathematics and physics have been investigated for more
than 150 years. We will review the history and applications. Then we will
give new formulas for the asymptotics of the number of spanning trees of a
graph. A special case answers a question of McKay (1983) for regular
graphs. The general answer involves a quantity for infinite graphs that we
call ``tree entropy", which we show is a logarithm of a normalized
[Fuglede-Kadison] determinant of the graph Laplacian for infinite graphs.
Proofs involve new traces and the theory of random walks.