Mark Kac Seminar

November 14, 2008






Location: Utrecht, KNG80, room 130


speaker: Petter Brändén (KTH Stockholm)

title: Correlation inequalities and the geometry of multivariate polynomials



The main topic of the talk is zeros of partition functions and correlation inequalities. We prove that certain conditions on the zero-set of the partition function entail negative association for the corresponding measure. A consequence is that the symmetric exclusion process with product initial distribution is negatively associated at positive times, as conjectured by Liggett and Pemantle.

We also characterize linear operators preserving restrictions on the zero-set and show how the classification provides a uniform platform for proving Lee-Yang type theorems.

This talk is based on joint work with J. Borcea (Stockholm) and T. M. Liggett (UCLA).


speaker: Andras Balint (VU Amsterdam)

title: The high temperature Ising model on the triangular lattice is a critical percolation model


The Ising model at inverse temperature \beta and zero external field can be obtained via the Fortuin-Kasteleyn (FK) random-cluster model with q=2 and density of open edges p=1-e^{-\beta} by assigning spin +1 or -1 to each vertex in such a way that
(1) all the vertices in the same FK cluster get the same spin and
(2) +1 and -1 have equal probability.

We generalize the above procedure by assigning spin +1 with probability r and -1 with probability 1-r, with r \in [0,1], while keeping condition (1). For fixed \beta, this generates a dependent (spin) percolation model with parameter r. We show that, on the triangular lattice and for \beta<\beta_c, this model has a sharp percolation phase transition at r=1/2, corresponding to the Ising model.

Our analysis therefore shows that the high temperature Ising model on the triangular lattice with no external field can be interpreted as a critical percolation model with exponentially decaying correlations between sites, and should therefore be in the same universality class as Bernoulli percolation.

Joint work with Federico Camia and Ronald Meester.


Mark Kac Seminar 2008-2009


last updated: 29 sep 2008 by Markus