Francesco Caravenna

We consider disordered systems which are so-called marginally relevant, focusing on the model of directed polymer in random environment in dimension (2+1). As we established in previous work, the partition functions of this model undergo a phase transition, in a suitable weak disorder and continuum limit. In a critical window around the transition point, these partition functions are expected to converge to a generalized random field on R^2 with logarithmically diverging covariances.

We present sharp moment bounds which provide a quantitative step toward proving this convergence. In particular, we show that any subsequential limit is non-trivial and has a unique explicit covariance kernel. Close connections with stochastic PDEs, namely with the two-dimensional Stochastic Heat Equation, will be described.

(Joint work with R. Sun and N. Zygouras)