MARK KAC SEMINAR

The Yang-Baxter equation (YBE) plays a central role in the theory of two-dimensional solvable lattice models in statistical mechanics. One of its consequences is the existence of a family of commuting transfer matrices. In models which are made inhomogeneous, without violating solvability, the YBE leads to difference equations for the eigenvectors of the transfer matrix. These equations have become known as the q-Kniznhik-Zamolodchikov (qKZ) equation, from an analogy to equations of that name for correlation functions in conformal field theories.

For certain models and geometries, the qKZ equations have particularly simple solutions for the ground state. This is the case for instance in models for critical percolation, and in models for the self-avoiding walk. In this presentation I will report how the this simplicity can be utilized to find exact expressions for certain non-trivial observables in these models on cylinders of finite, but arbitrary, circumference, and infinite length. The focus of the presentation is on the method of calculation and proof. No prior knowledge of solvable models is assumed.

We shall discuss ballistic RWRE and show that the annealed functional CLT holds in the rough path topology. This yields an interesting phenomenon: the scaling limit of the area process is not solely the Levy area (which is the area process of a Brownian motion), but there is also an additive linear correction called the 'area anomaly'. Moreover, the latter is identified in terms of the walk on a regeneration interval and the asymptotic speed. This is a joint work with Olga Lopusanschi (Paris-Sorbonne). Time permitted, we shall discuss also a joint work in progress with Noam Berger (TU Munich) dealing with the convergence with respect to the quenched law.

One advantage of limit theorems in the rough path topology over the uniform topology is demonstrated in the following property - which might be useful e.g., for simulations. Consider a nice SDE and its analogous difference equation, where the role of white noise is played by a sequence of discrete processes. Assume that this sequence converges to the Brownian motion in the rough path topology with some area anomaly. A result by D. Kelly from 2016 then provides a scaling limit of the solution to the deference equation to the solution to a modification of the SDE where the correction is expressed explicitly in terms of the area anomaly. It was observed by T. Lyons in 1998 in his seminal work on rough paths that whenever the convergence of the processes to the Brownian motion holds only in the uniform topology, the discrete solution might not converge in general.