Jacob van den Berg

The expected number of critical percolation clusters intersecting a line segment

We study critical percolation on a regular planar lattice. Let EG(n) be the expected number of open clusters intersecting or hitting the line segment [0; n]. (For the subscript G we either take H, when we restrict to the upper halfplane, or C, when we consider the full lattice). Cardy [2] (see also Yu, Saleur and Haas [11]) derived heuristically that EH(n) = An + p 3 4 log(n) + o(log(n)), where A is some constant. Recently Kovacs, Igloi and Cardy derived in [5] heuristically (as a special case of a more general formula) that a similar result holds for EC(n) with the constant p 3 4 replaced by 5 p 3 32 . In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of EH(n) above, and a rigorous upper bound for the prefactor of the logarithm in the formula of EC(n).

Sublinearity of the travel-time variance for dependent first passage percolation

Let E be the set of edges of the d-dimensional cubic lattice ℤd, with d ≥ 2, and let t(e), e ∈ E, be nonnegative values. The passage time from a vertex v to a vertex w is defined as infπ : v→w ∑e∈π t(e), where the infimum is over all paths π from v to w, and the sum is over all edges e of π. Benjamini, Kalai and Schramm [2] proved that if the t(e)’s are i.i.d. two-valued positive random variables, the variance of the passage time from the vertex 0 to a vertex v is sublinear in the distance from 0 to v. This result was extended to a large class of independent, continuously distributed t-variables by Benaim and Rossignol [1]. We extend the result by Benjamini, Kalai and Schramm in a very different direction, namely to a large class of models where the t(e)’s are dependent. This class includes, among other interesting cases, a model studied by Higuchi and Zhang [9], where the passage time corresponds with the minimal number of sign changes in a subcritical “Ising landscape.”