# Christophe Sabot

Christophe Sabot is Professor at Université Lyon 1, and head of the team "Probability, Statistics and Mathematical Physics" of the Institut Camille Jordan. He is specialist of Probability, more specifically he works on interacting random walks : random walks that interact wit their past trajectory or with a random environment. |

**"Uniform Spanning Trees:"** A spanning tree of a graph is a subset of the edges that connects all vertices but contains no cycle : in the case of the square grid you can think to a labyrinth. Spanning trees are important both in probability and combinatorics.

Pick a spanning tree at random uniformly among all spanning trees : this is the "Uniform Spanning Tree". Uniform spanning trees have deep connections with electrical networks through a remarkable determinantal formula known as the "Matrix-tree Theorem", and with some random processes such as the random walk of the graph or the "loop-erased random walk". In the limit of the d-dimensional lattice, a transition in the structure of the limit appears at each dimension of the form d=4k+1. The limit in dimension 2 involves the beautiful SLE process, which is central in 2D statistical physics.

If the understanding of UST on infinite graphs involves advanced probabilities, the results on finite graphs can be treated by elementary linear algebra.