Oxford-Man Institute

Joint work with Martin Dyer (Leeds) and Leslie Goldberg (Liverpool).

A spin system may be modelled as a graph, in which edges (bonds) indicate interactions between adjacent vertices (sites). A configuration of the system is an assignment of colours (spins) to the vertices of the graph. The interactions between adjacent spins define a certain distribution, the Boltzmann distribution, on configurations. To sample from this distribution it is usually necessary to simulate one of a number of Markov chains on the space of all configurations. Theoretical analyses of the mixing time of these Markov chains usually assume that spins are updated at single vertices chosen uniformly at random. Actual simulations, in contrast, may make (random) updates according to a deterministic, usually highly structured pattern. We'll explore the relationships between systematic scan and random single-site updates, and also between classical uniqueness conditions from statistical physics and more recent techniques in mixing time analysis.

I consider the Metropolis Markov Chain based on the nearest neighbor random walk on the positive half of the Mean Field Ising Model, i.e., on those vectors from $\{−1, 1\}^N$ which contain more $1$ than $−1$. Using randomly-chosen paths I prove a lower bound for the Spectral Gap of this chain which is of order $N^-2$ and which does not depend on the inverse temperature $\beta$. In conjunction with decomposition results such as those in Jerrum, Son, Tetali and Vigoda (2004) this result may be useful for bounding the spectral gaps of more complex Markov chains on the Mean Field Ising Model which may be decomposed into Metropolis chains. As an example, I apply the result to two Multilevel Markov Chain Monte Carlo algorithms, Swapping and Simulated Tempering. Improving a result by Madras and Zheng (2002), I show that the spectral gaps of both algorithms on the (full) Mean Field Ising Model are bounded from below by the reciprocal of a polynomial in the lattice size $N$ and in the inverse temperature $\beta$.

We discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces. More precisely, we consider a random planar map M(n), which is uniformly distributed over the set of all planar maps with n faces in a certain class. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n to the power 1/4. We then discuss the convergence in distribution of the resulting random metric spaces as n tends to infinity, in the sense of the Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Oded Schramm in his plenary address paper at the 2006 ICM, in the special case of triangulations.

In the case of bipartite planar maps, we first establish a compactness result showing that a limit exists along a suitable subsequence. Furthermore this limit can be written as a quotient space of the Continuum Random Tree (CRT) for an equivalence relation which has a simple definition in terms of Brownian labels attached to the vertices of the CRT. Finally we show that any possible limiting metric space is almost surely homomorphic to the 2-sphere. As a key tool, we use bijections between planar maps and various classes of labelled trees.

We study the dynamic of a very simple chain of three anharmonic oscillators with linear nearest-neighbour couplings. The first and the last oscillator furthermore interact with heat baths through friction and noise terms. If all oscillators in such a system are coupled to heat baths, it is well-known that under relatively weak coercivity assumptions, the system has a spectral gap (even compact resolvent) and returns to equilibrium exponentially fast. It turns out that while it is still possible to show the existence and uniqueness of an invariant measure for our system, it returns to equilibrium much slower than one would at first expect. In particular, it no longer has compact resolvent when the potential of the oscillators is quartic and the spectral gap is destroyed when it grows even faster.