Monthly Archives: January 2014

Harmonic functions of minimal growth With Amir, Benjamini, Duminil-Copin, Meyerovich, Yadin. 1. Motivation Gromov’s polynomial growth theorem. Kleiner’s new proof that uses Lipschitz harmonic functions. Any group has non trivial Lipschitz harmonic functions. Polynomial growth implies that they form a … Continue reading →

Random walks on left-orderable groups I view random walks as a tool to study left-orderable groups. 1. Left-orderable groups This means a group with a left-invariant order. Torsion in an obstruction. Not easy to find firther obstructions, and indeed, many … Continue reading →

Invariant random structures 1. First passage percolation Multiply length of edges of the 2-grid by 1 or 10 with equal probabilities. Known : there is an asymptotic shape. What is it ? UIPT : the limit is not deterministic. Stationary … Continue reading →

Heat kernel on affine buildings Consider finite support probability distributions on an affinebuiding which are spherical : depends only on distance. Plays the role of heat kernel on symmetric spaces. For such a kernel, one has rather sharp asymptotic estimates … Continue reading →

Discrete random walks on SOL Joint with J. Brieussel. 1. Case of Theorem 1 (Furstenberg (1963)) Let be an absolutely continuous probability measure on . Then converges to the circle, with asymptotic distribution equal to harmonic measure. This is not … Continue reading →

The Poisson boundary of lamplighter groups Joint with R. Lyons. 1. Entropy The whole subject started with Kesten in 1949 and Furstenberg in early 50’s. Entropy on the boundary. In 1970, Avez proposed to define entropy directly on the group. … Continue reading →