Eagle House

We prove that when a sequence of Lévy processes $X(n)$ or a normed sequence of random walks $S(n)$ converges a.s. on the Skorokhod space toward a Lévy process $X$, the sequence $L(n)$ of local times at the supremum of $X(n)$ converges uniformly on compact sets in probability toward the local time at the supremum of $X$. A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and

descending) converges jointly in law towards the ladder processes of $X$. As an application, we show that in general, the sequence $S(n)$ conditioned to stay positive converges weakly, jointly with its local time at the future minimum, towards the corresponding functional for the limiting process $X$. From this we deduce an invariance principle for the meander which extends known results for the case of attraction to a stable law.

Joint work with C. Donati-Martin and D. Feral

Condition supercritical percolation so that the origin is enclosed by a dual circuit whose interior traps an area of n^2.

The Wulff problem concerns the shape of the circuit. We study the circuit's fluctuation. A well-known measure of this fluctuation is maximum local roughness (MLR), which is the greatest distance from a point on the circuit to the boundary of circuit's convex hull. Another is maximum facet length (MFL), the length of the longest line segment of which this convex hull is comprised.

In a forthcoming article, I will prove that

for various models including supercritical percolation, under the conditioned measure,

MLR = \Theta(n^{1/3}\log n)^{2/3}) and MFL = \Theta(n^{2/3}(log n)^{1/3}).

An important tool is a result establishing the profusion of regeneration sites in the circuit boundary. The talk will focus on deriving the main results with this tool