MARK KAC SEMINAR

In the first part I will talk about the classical result of Groeneveld, Boel and Kasteleyn that boundary spin correlations functions in Ising models on planar graphs satisfy Pfaffian relations. I will consider the reverse question, and show that any classical ferromagnetic spin model whose correlation functions satisfy Pfaffian relations must be (up to local simplifications of the graph) an Ising model on a planar graph. The main tool is a new (coupled) version of the Edwards—Sokal (Fortuin—Kasteleyn) representation of the Ising model applied to two independent copies of the spin model. Joint work with Diederik van Engelenburg.

In the second part I will discuss a geometric formula for a certain set of complex zeros of the partition function of the planar Ising model recently proposed by Livine and Bonzom. Remarkably, the zeros depend locally on the geometry of an immersion of the graph in the three dimensional Euclidean space (different immersions give rise to different zeros). When restricted to the flat case, the weights become the critical weights on circle patterns. I will rigorously prove the formula by geometrically constructing a null eigenvector of the Kac-Ward matrix whose determinant is the squared partition function. The main ingredient of the proof is the realisation that the associated Kac-Ward transition matrix gives rise to an SU(2) connection on the graph, creating a direct link with rotations in three dimensions. The existence of a null eigenvector turns out to be equivalent to this connection being flat.

We consider the two-type Moran model with $N$ individuals. Each individual is assigned a resampling rate, drawn independently from a probability distribution $\mathbb{P}$ on $\mathbb{R}_+$, and a type, either $\heartsuit$ or $\diamondsuit$. Each individual resamples its type at its assigned rate, by adopting the type of an individual drawn uniformly at random. Let $Y^N(t)$ denote the empirical distribution of the resampling rates of the individuals with type $\heartsuit$ at time $Nt$. We show that if $\mathbb{P}$ has countable support and satisfies certain tail and moment conditions, then in the limit as $N\to\infty$ the process $(Y^N(t))_{t \geq 0}$ converges in law to the process $(S(t)\,\mathbb{P})_{t \geq 0}$, in the so-called Meyer-Zheng topology, where $(S(t))_{t \geq 0}$ is the Fisher-Wright diffusion with diffusion constant $D$ given by $1/D = \int_{\mathbb{R}_+} (1/r)\,\mathbb{P}(\mathrm{d} r)$.