Travail joint avec T. Bonis, L. Decreusefond and Z. Zhang.

A natural and well-known way to discover the topology of random structures (such as a random graph G), is to have them explored by random walks. The usual random walk jumps from a vertex of G to a neighboring vertex, providing information on the connected components of the graph G. The number of these connected components is the Betti number beta0. To gather further information on the higher Betti numbers that describe the topology of the graph, we can consider the simplicial complex C associated to the graph G: a k-simplex (edge for k=1, triangle for k=2, tetrahedron for k=3 etc.) belongs to C if all the lower (k-1)-simplices that constitute it also belong to the C. For example, a triangle belongs to C if its three edges are in the graph G. Several random walks have already been propose recently to explore these structures, mostly in Informatics Theory. We propose a new random walk, whose generator is related to a Laplacian of higher order of the graph, and to the Betti number betak. A rescaling of the walk for k=2 (cycle-valued random walk) is also detailed when the random walk visits a regular triangulation of the torus. We embed the space of chains into spaces of currents to establish the limiting theorem. An application to statistical topology is proposed.