We consider a directed polymer pinned by one-dimensional quenched randomness, modeled by the space-time trajectories of an underlying Markov chain which encounters a random potential of form u + V_i when it visits a particular site, denoted 0, at time i. The polymer depins from the potential when u goes below a critical value. We consider in particular the case in which the excursion length (from 0) of the underlying Markov chain has power law tails. We show that for certain tail exponents, for small inverse temperature \beta there is a constant D(\beta), approaching 0 with \beta, such that if the increment of u above the annealed critical point is a large multiple of D(\beta) then the quenched and annealed systems have very similar free energies, and are both pinned, but if the increment is a small multiple of D(\beta), the annealed system is pinned while the quenched is not. In other words, the breakdown of the ability of the quenched system to mimic the annealed occurs entirely at order D(\beta) above the annealed critical point.

I will discuss the behavior of N bosons trapped in a potential well subject
to a pairwise interaction. In the mean-field limit -- i.e., when N tends to
infinity while keeping the interaction per particle bounded -- the evolution
of a product state remains, asymptotically, a product state. The single
particle wave-function then evolves according to a non-linear Hartree
equation. Versions of this result have been proved before, e.g., by
Hepp in 1977, Spohn in 1980 or, recently, by Rodnianski and Schlein, but
the proofs are often quite technically involved. I will describe a very simple,
and ideologically correct, proof (for bounded interaction potentials) based
on exchengeability and Stormer's (aka quantum deFinetti) theorem.
Based on recent discussions with Nick Crawford.

We study diffusions in random environment in higher dimensions. We assume that the environment is stationary and obeys finite range dependence. Once the environment is chosen, it remains fixed in time. To restore some stationarity, it is common to average over the environment. One then obtains the so-called annealed measures, that are typically non-Markovian measures.
Our goal is to study the asymptotic behavior of the diffusion in random environment under the annealed measure, with particular emphasis on the ballistic regime
('ballistic' means that a law of large numbers with non-vanishing limiting velocity holds). In the spirit of Sznitman, who treated the discrete setting, we introduce
conditions (T) and (T'), and show that they imply, when d>1, a ballistic law of large numbers and a central limit theorem with non-degenerate covariance matrix.
As an application of our results, we highlight condition (T) as a source of new examples of ballistic diffusions in random environment.