We introduce a new method for estimating the parameters of exponential random
graph models. The method is based on a large-deviations approximation to the
normalizing constant shown to be consistent using theory developed by
Chatterjee and Varadhan. The theory explains a host of difficulties encountered
by applied workers: many distinct models have essentially the same MLE,
rendering the problems "practically" ill-posed. We give the first rigorous
proofs of "degeneracy" observed in these models. Here, almost all graphs have
essentially no edges or are essentially complete.

Using techniques from probability theory, we show that the thermodynamics of
the focusing cubic discrete nonlinear Schrodinger equation (NLS) are exactly
solvable in dimensions three and higher. A number of explicit formulas are
derived. The probabilistic results, combined with dynamical information, prove
the existence and typicality of solutions to the discrete NLS with highly
stable localized modes that are sometimes called discrete breathers.

Large graphs are sometimes studied through their degree sequences (power law
or regular graphs). We study graphs that are uniformly chosen with a given
degree sequence. Under mild conditons, it is shown that sequences of such
graphs have graph limits in the sense of Lovasz and Szegedy with identifiable
limits. This allows simple determination of other features such as the number
of triangles. The argument proceeds by studying a natural exponential model
having the degree sequence as a sufficient statistic.

The Ghirlanda-Guerra identities are one of the most mysterious features of
spin glasses. We prove the GG identities in a large class of models that
includes the Edwards-Anderson model, the random field Ising model, and the
Sherrington-Kirkpatrick model in the presence of a random external field.
Previously, the GG identities were rigorously proved only `on average' over a
range of temperatures or under small perturbations.