MSI Weekly Bulletin - Week starting Monday 30 April, 2007

A perfect matching in a graph G is a set of pairwise vertex-disjoint edges in which every vertex of G is an endpoint of exactly one edge. We denote the collection of all sets of k edge-disjoint perfect matchings in K_{2n} by P(k,n). In this talk, two methods of calculating an asymptotic number of sets of k edge-disjoint perfect matchings in K_{2n} will be presented. The first method employs a switching technique to prove that if k=o(n^{1/3}), then |P(k,n)|\sim\frac{1}{k!}\biggl(\frac{(2n)!}{(2^n\,n!)}\biggr)^k\exp\bigl(\tfrac{1}{4}k(1{-}k)\bigr). The second method produces a more accurate result than the first. It relies on a theorem of Godsil's which proves that the number of perfect matchings in the complement of $G$ is \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-x^2/2}\alpha(G,x)\,dx, where \alpha(G,x) is the matchings polynomial of G. By considering the case where G is a k-regular graph with 2n vertices and using the Laplace method to approximate the above integral, it will be shown that |P(k,n)|\sim \frac{1}{k!}\biggl(\frac{(2n)!}{2^{n}n!}\biggr)^{\negmedspace k} \biggl(\frac{(2n)!}{(2n)^k(2n{-}k)!}\biggr)^{\negthickspace n} \biggl(1{-}\frac{k}{2n}\biggr)^{n/2}e^{k/4}, for k=o(n^{5/6}). Both estimates of |P(k,n)| improve upon an existing result of Bollob\'{a}s who solved this problem for constant k.

There is a heuristic in physics for estimating the number of bound states of a quantum system (or in mathematical terms, the number of negative eigenvalues of a self-adjoint operator) by regarding the eigenfunctions as disjoint 'blobs' of phase space, each occupying a fixed volume. In the talk I will investigate the worth of this heuristic in the case of a simple quantum system, that of the Laplacian in R3 plus a potential function. We find very precise asymptotics for the number of bound states in some cases, and see that the heuristic is a very good, but not perfect, guide to the actual situation.

I will give a basic introduction to rings of Witt vectors. I probably won't have time to discuss any geometry, and so I will do that next week.

Since introduced in 1999, SLE has put on a rigorous footing many predictions from statistical physics and shed new light on old problems. This talk will motivate SLE as the continuum limit for a class of random curves and interfaces appearing in lattice models. The Loewner Equation will be introduced in this context, with the equation governing the evolution of an SLE trace the end result of this discussion.

In recent years, a theory of stochastic integration for Banach space-valued stochastic processes with respect to Brownian motion has been developed. Its main ingredients are generalised square functions (connected with the operator ideal of the so-called $\gamma$-radonifing operators) and decoupling inequalities for random sums with values in UMD Banach spaces. In this talk we survey these developments and present some applications to Malliavin calculus.