MARK KAC SEMINAR

The symmetric inclusion process is a particle system where particles perform random walks and interact with each other by an attractive interaction (inclusion). A crucial property of the process is self-duality which enables to give a fairly complete ergodic theory, i.e., complete characterization of the set of invariant measures. The crucial tool here is to provide a successful coupling of a finite number of inclusion walkers.

The symmetric inclusion process has several related processes including an interacting diffusion process (the Brownian energy process) and a mass distribution model (of KMP type). We show how all these processes can be united via an “abstract generator” which is the co-product of the Casimir operator of the algebra SU(1,1). Once this becomes clear, it is also clear how to define the “correct” asymmetric inclusion process via a q-deformation.

We then focus on a weak asymmetry limit, which naturally leads to a new truly asymmetric process, called the asymmetric Brownian energy process, which has a symmetric dual. This also provides the “correct” asymmetric version of the well-known KMP mass redistribution process. Along the way, we will explain several new developments in the theory of duality of Markov processes such as generating duality functions and orthogonal duality functions.

We provide a new proof of exponential decay for the connection probabilities of subcritical Bernoulli percolation, using the theory of randomized algorithms. This proof does not rely on the domain Markov property or the BK inequality. In particular, it extends to FK percolation and continuum percolation models such as Boolean and Voronoi percolation in arbitrary dimension. This provides the first proof of sharpness of the phase transition for these models.

This presentation is based on a joint work with Hugo Duminil-Copin and Aran Raoufi.