Mark Kac Seminar

October 3, 2008






Location: Utrecht, KNG80, room 130


speaker: Bernard Nienhuis (Amsterdam)

title: Exact correlations for percolation on arbitrary rhombus tilings



In the past it was found that certain correlation functions in percolation models, at the critical point, could be extrapolated exactly from finite size values. This resulted in a number of exact correlations, mainly for bond percolation on the square lattice, but on few occasions also for site percolation on the triangular lattice. These results have been obtained by the study of the Perron-Frobenius eigenvector of the transfer matrix for the cylinder or strip.

In this presentation this approach is generalized, both for site percolation on an arbitrary tiling of the cylinder with arbitrary rhombuses, and for bond percolation on a sublattice of such (bipartite) tiling. The weights on each rhombus are chosen such that they satisfy the Yang-Baxter equation. As a result, the eigenvectors of the (generalized) transfer matrix satisfy recursion relations that are known as q-Knizhnik-Zamolodchikov equations. These equations can be solved for small-diameter cylinders, from which correlation functions can be calculated straightforwardly. The results contain so much structure, and are so regular that an extrapolation to arbitrary sizes can be guessed.


speaker: Sasha Gnedin (Utrecht)

title: Asymptotics in the occupancy scheme with infinitely many boxes


We consider the classical occupancy scheme in which n balls are thrown independently in infinitely many boxes, with positive probability p_j of hitting box j=1,2,... The problem is to describe the asymptotic behaviour, as n tends to infinity, of the number of occupied boxes K_n, and of the number X_{n,r} of boxes occupied by exactly r balls, r=1,2,...

In the first part of the talk we shall review known results about K_n and discuss the multivariate normal approximation to X_{n,r}. In particular, we show that the joint convergence only holds when the probabilities (p_j) satisfy a condition of regular variation. Examples of oscillatory behaviour will be also given.

In the second part of the talk we shall discuss the occupancy problem with random probabilities (p_j), generated by a multiplicative renewal process. This is applied to the asymptotics of exchangeable coalescents with multiple collisions.

Based on joint work with Andrew Barbour and Alex Iksanov.


Mark Kac Seminar 2008-2009


last updated: 29 sep 2008 by Markus