Jean-François Le Gall

Uniqueness and universality of the Brownian map

We consider a random planar map Mn which is uniformly distributed over the class of all rooted q-angulations with n faces. We let mn be the vertex set of Mn, which is equipped with the graph distance dgr. Both when q≥4 is an even integer and when q=3, there exists a positive constant cq such that the rescaled metric spaces (mn,cqn−1/4dgr) converge in distribution in the Gromov–Hausdorff sense, toward a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.

The topological structure of scaling limits of large planar maps

We discuss scaling limits of large bipartite planar maps. If p≥2 is a fixed integer, we consider, for every integer n≥2, a random planar map Mn which is uniformly distributed over the set of all rooted 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of Mn, equipped with the graph distance rescaled by the factor n-1/4, converges in distribution as n→∞ towards a limiting random compact metric space, in the sense of the Gromov–Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.We discuss scaling limits of large bipartite planar maps. If p≥2 is a fixed integer, we consider, for every integer n≥2, a random planar map Mn which is uniformly distributed over the set of all rooted 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of Mn, equipped with the graph distance rescaled by the factor n-1/4, converges in distribution as n→∞ towards a limiting random compact metric space, in the sense of the Gromov–Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.

The Brownian snake and solutions of Δu=u2 in a domain

SummaryWe investigate the connections between the path-valued process called the Brownian snake and nonnegative solutions of the partial differential equation Δu=u2 in a domain of ℝd. In particular, we prove two conjectures recently formulated by Dynkin. The first one gives a complete characterization of the boundary polar sets, which correspond to boundary removable singularities for the equation Δu=u2. The second one establishes a one-to-one correspondence between nonnegative solutions that are bounded above by a harmonic function, and finite measures on the boundary that do not charge polar sets. This correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the Brownian snake. Our proofs combine probabilistic and analytic arguments. An important role is played by a new version of the special Markov property, which is of independent interest.

A Path-Valued Markov Process and its Connections with Partial Differential Equations

With every continuous Markov process (ξ t ) having state space E, we associate a Markov process (W s ) with values in the space of stopped paths in E, whose value at time s is a path of the process (ξ t ) stopped at a random time depending on s. In the case when (ξ t ) is standard Brownian motion, our path-valued process is related to the nonlinear equation Δu = u 2 in the same way as Brownian motion is related to the Laplace equation. The values of the process (W s ) form a “tree of paths” in E, and a key tool is the exit measure, which describes the way all these paths exit a given domain Ω. The exit measure plays a role similar to harmonic measure in the classical setting. Our path-valued process is closely related to the measure-valued processes called superprocesses, so that some of our statements are reformulations of results due to Dynkin. As a typical application, we investigate the uniqueness of the nonnegative solution of Δu = u 2 in a domain with infinite boundary conditions, in the spirit of the recent analytic work of Veron and Marcus.

Scaling limits of random planar maps with large faces

We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index

The uniform random tree in a Brownian excursion

\alpha\in(1,2)

Wiener sausage and self-intersection local times

. When the number

A new approach to the single point catalytic super-Brownian motion

n

Enlacements du mouvement brownien autour des courbes de l’espace

of vertices of the map tends to infinity, the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process, which can be constructed from a centered stable process with no negative jumps and index

Mouvement brownien, cônes et processus stables

\alpha