IS09: Random Partitions

Organizer: Harriet Walsh (University College Dublin)

Discrete \(N\)-particle systems at high temperature through Jack generating functions

Maciej Dołęga (Institute of Mathematics of the Polish Academy of Sciences)

We discuss random discrete \(N\)-particle systems, which can be also interpreted as random partitions, with the deformation (inverse temperature) parameter \(\theta\). We find necessary and sufficient conditions for the Law of Large Numbers as their size \(N\) tends to infinity simultaneously with the inverse temperature going to zero.

We apply the general framework to obtain the LLN for a large class of Markov chains of \(N\) nonintersecting particles with interaction of log-gas type, and the LLN for the multiplication of Jack polynomials, as the inverse temperature tends to zero. We express the answer in terms of novel one-parameter deformations of cumulants, and we discuss their relation with quantized free probability and continuous log-gas systems. Based on joint work with Cesar Cuenca.