Ergodicity of multiplicative statistics

For a subfamily of multiplicative measures on integer partitions we give conditions for associated Young diagrams to converge in probability after a proper rescaling to a certain curve named the limit shape of partitions. We provide explicit formulas for the scaling function and the limit shape covering some known and some new examples. Introduction It is now widely known that a random partition of a large integer taken with equal probability among all partitions of that integer has the Young diagram which looks (after rescaling) close to a deterministic object called a limit shape of random partition. The discovery of a phenomenon of limit shape formation for the random partition of a large integer has a long history. First it was mentioned in the paper by H. N. V. Temperley [13] in 1952 with heuristic arguments. Much later but independently, the principal calculations leading to this result was made by M. Szalay and P. Turán [12] in 1977, however they did not state their result in the modern way. It was done by A. Vershik and stated in his joint paper with S. Kerov [14] in 1985. Later a new proof of the same result was found based on the fact that the uniform measures on partitions of n are just the product measures restricted to some linear subspace. (The probabilist would say that a random partition of n is just a sequence of independent random variables with specific distributions conditioned to have some weighted sum equal n.) This technique seems to be first applied to random partitions by B. Fristedt [6]; now it is usually referred to as Fristedt’s conditional device. Later it has been frequently used by various authors in the related problems, see [9, 5, 7], to name just a few references. A. Vershik [15] noted that the similar technique is applicable to a wider range of problems with the same property that the measure is a product measure restricted to a certain affine subset. Vershik called such measures on partitions multiplicative; we give the precise definition in Section 1. Similar “limit shape type” results have appeared in diverse contexts including some probability measures on partitions, both multiplicative and not. One of the first results of this type goes back to a seminal paper by P. Erdős and J. Lehner [8]: it can be read from their paper that rescaled Young diagrams of strict partitions of large integer concentrate around a certain limit shape. This is also a multilplicative case, although Erdős and Lehner does not use the related technique. A. Comtet et al. [4] recently found the limit shape ∗This research was made in part during the author’s stay at the Erwin Schrödinger Institute for Physics and Mathematics, Vienna, during the programme “Combinatorics and Statistical Physics” in Spring 2008.