MARK KAC SEMINAR

In the first talk, I will review (without assuming much background) some of the important fields in (Euclidean) QFT, such as:

• the magnetic field of the Ising model
• the Energy field of the Ising model
• the $\phi^4_d$ field
• the Sine-Gordon field.

Often, these fields are viewed through their $k$-point correlation functions (defined as $f(x_1,...,x_n):=\langle \phi(x_1),\ldots,\phi(x_n)\rangle$. I will focus on the following natural question: in which cases do these celebrated fields correspond to actual random Schwartz distributions in ${\bf R}^d$ ? In the second talk, I will give a tentative answer to this question which will require revisiting the mathematical construction of these fields. This is based on a joint work with Martin Hairer and Antti Kupiainen.

We consider a directed polymer interacting with a linear interface. The monomers carry random charges. Each monomer contributes an energy to the interaction Hamiltonian that depends on its charge as well as its height with respect to the interface, modulated by an interaction potential. The configurations of the polymer are weighted according to the Gibbs measure associated with the interaction Hamiltonian at a given inverse temperature, where the reference measure is given by a recurrent Markov chain. We are interested in both the quenched and the annealed free energy per monomer in the limit as the polymer becomes large. We find that each exhibits a phase transition along a critical curve separating a localized phase (where the polymer stays close to the interface) from a delocalized phase (where the polymer wanders away from the interface). We obtain variational formulas for the critical curves, and find that the quenched phase transition is at least of second order. We obtain upper and lower bounds on the quenched critical curve in terms of the annealed critical curve. In addition, for the special case where the reference measure is given by a Bessel random walk, we identify the weak disorder scaling limit of the annealed free energy and the annealed critical curve in three different regimes for the tail exponent of the interaction potential. Based on joint work with Francesco Caravenna (University of Milano-Bicocca, Italy)