Next Mark Kac seminar

Friday June 8
Utrecht, Janskerkhof 15a, room 001

Speaker: Anton Thalmaier (Luxembourg)
Title  : Brownian motion, Ricci curvature, functional inequalities and
         geometric flows III: Heat equations under geometric flows and
         entropy formulas

Speaker: Reka Szabó (Groningen)
Title  : Inhomogeneous percolation on ladder graphs

Speaker: Federico Sau (Delft)
Title  : Exclusion process in symmetric dynamic environment: quenched


Abstract Anton Thalmaier
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 on the Mark Kac website)


Abstract Reka Szabó
We define an inhomogeneous percolation model on “ladder graphs” obtained
as direct products of an arbitrary graph G = (V,E) 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 (0,1).


Abstract Federico Sau
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.