THE NEAREST UNVISITED VERTEX WALK ON RANDOM GRAPHS

We revisit an old topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and ball-covering (metric entropy) measures. For some familiar models of random graphs, this connection allows the order of magnitude of the cover time to be deduced from first passage percolation estimates. Establishing sharper results seems a challenging problem.

Regularity of the time constant for a supercritical Bernoulli percolation

We consider an i.i.d. supercritical bond percolation on Z^d, every edge is open with a probability p > p_c(d), where p_c(d) denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite open cluster C_p [11]. We are interested in the regularity properties of the chemical distance for supercritical Bernoulli percolation. The chemical distance between two points x, y ∈ C_p corresponds to the length of the shortest path in C_p joining the two points. The chemical distance between 0 and nx grows asymptotically like nµ_p(x). We aim to study the regularity properties of the map p → µ_p in the supercritical regime. This may be seen as a special case of first passage percolation where the distribution of the passage time is G_p = pδ_1 + (1 − p)δ_∞, p > p_c(d). It is already known that the map p → µ_p is continuous (see [10]).

First passage percolation in the mean field limit

This dissertation deals with two classical problems in statistical mechanics: the first passage percolation on Euclidean spaces, FPP for short, in both directed and undirected settings.