MARK KAC SEMINAR

(Note: the slides of the first lecture in February are available via this link)

Any renewal processes on N with a polynomial tail, with exponent α∈(0,1), has a non-trivial scaling limit, known as the α-stable regenerative set. We consider Gibbs transformations of such renewal processes in an i.i.d. random environment, called disordered pinning models. We show that in the disorder relevant case where α∈(1/2,1), these models have a universal scaling limit, which we call the continuum disordered pinning model (CDPM). This is a random closed subset of R in a white noise random environment, with subtle features:

• Any fixed a.s. property of the α-stable regenerative set (e.g., its Hausdorff dimension) is also an a.s. property of the CDPM, for almost every realization of the environment.
• Nonetheless, the law of the CDPM is singular with respect to the law of the α-stable regenerative set, for almost every realization of the environment.

Joint work with F. Caravenna and N. Zygouras.

We study continuum limits of disordered systems of directed polymer type, for which disorder is so-called marginally relevant. This includes the disordered pinning model on Z with renewal exponent α=1/2, the heavy-tailed directed polymer model on Z¹⁺¹ with tail exponent α=1, and the short-range directed polymer model on Z²⁺¹. We show that in a suitable weak disorder and infinite volume limit, the partition functions of these different models all converge to the same universal limit, which can be identified explicitly and shown to undergo a phase transition as a disorder strength parameter varies. As a by-product, we show that the solution of a suitably regularized two-dimensional Stochastic Heat Equation (SHE) converges pointwise to the same universal limit. Furthermore, the fluctuation field of the solution converges to a variant of the two-dimensional Gaussian free field.

Joint work with F. Caravenna and N. Zygouras.