R. Schoen has conjectured that an asymptotically flat Riemannian n-manifold (M,g) with non-negative scalar curvature is isometric to Euclidean space if it admits a non-compact area-minimizing hypersurface. This has been confirmed by O. Chodosh and M. Eichmair in the case where n=3. In this talk, I will present recent work with M. Eichmair where we confirm this conjecture in the case where 3<n<8 and the area-minimizing hypersurface arises as the limit of large isoperimetric hypersurfaces. By contrast, we show that a large part of spatial Schwarzschild of dimension 3<n<8 is foliated by non-compact area-minimizing hypersurfaces.

There are several functional encodings of random trees which are commonly used to prove (among other things) scaling limit results.  We consider two of these, the height process and Lukasiewicz path, in the classical setting of a branching process tree with critical offspring distribution of finite variance, conditioned to have n vertices.  These processes converge jointly in distribution after rescaling by n^{-1/2} to constant multiples of the same standard Brownian excursion, as n goes to infinity.  Their difference (taken with the appropriate constants), however, is a nice example of a discrete snake whose displacements are deterministic given the vertex degrees; to quote Marckert, it may be thought of as a “measure of internal complexity of the tree”.  We prove that this discrete snake converges on rescaling by n^{-1/4} to the Brownian snake driven by a Brownian excursion.  We believe that our methods should also extend to prove convergence of a broad family of other “globally centred” discrete snakes which seem not to be susceptible to the methods of proof employed in earlier works of Marckert and Janson.

This is joint work in progress with Louigi Addario-Berry, Serte Donderwinkel and Rivka Mitchell.

The vacant set of the random walk on the torus undergoes a percolation phase transition at Poissonian timescales in dimensions 3 and higher. The talk will review this phenomenon and discuss recent progress regarding the nature of the transition, both for this model and its infinite-volume limit, the vacant set of random interlacements, introduced by Sznitman in Ann. Math., 171 (2010), 2039–2087. The discussion will lead up to recent progress regarding the long purported equality of several critical parameters naturally associated to the transition.