MARK KAC SEMINAR

We continue our description of Ricci curvature and Ricci flow in terms of Brownian motion. After a general discussion of heat equations under a geometric flow, we outline a probabilistic approach to entropy formulas. In particular, we define variants of Perelman's entropy functionals, using the Wiener measure as reference measure, and investigate their monotonicity along the flow.

The slides of the first and second lecture are available here and here, respectively.

We define an inhomogeneous percolation model on “ladder graphs” obtained as direct products of an arbitrary graph $G=\left(V,E\right)$ and the set of integers (vertices are thought of as having a “vertical” component indexed by an integer). We make two natural choices for the set of edges, producing an unoriented graph and an oriented graph. These graphs are endowed with percolation configurations in which independently, edges inside a fixed infinite “column” are open with probability $q$, and all other edges are open with probability $p$. We prove that the function that maps $q$ into the corresponding critical percolation threshold is continuous in $\left(0,1\right)$.

For the simple exclusion process evolving in a symmetric dynamic random environment, we derive the hydrodynamic limit from the quenched invariance principle of the corresponding random walk. For instance, if the limiting behavior of a test particle resembles that of Brownian motion on a diffusive scale, the empirical density, in the limit and suitably rescaled, evolves according to the heat equation.

In this talk we make this connection explicit for the simple exclusion process and show how self-duality of the process enters the problem. This allows us to extend the result to other conservative particle systems (e.g. IRW, SIP) which share a similar property.

Work in progress with F. Collet, F. Redig and E. Saada.