# DIMACS Focus on Discrete Probability Seminar

## Title:

On a likely shape of the random Ferrers diagram

## Speaker:

Boris Pittel
Ohio State University

## Place:

CoRE Building, Room 431
Busch Campus, Rutgers University.

## Time:

3:30 - 4:30 PM - After the DIMACS Tea
Thursday, May 8, 1997
Abstract:

We study the random partitions of a large integer \$n\$, under assumption that all such partitions are equally likely. We use Fristedt's conditioning device which connects the parts (summands) distribution to the one of a \$g\$-sequence, that is a sequence of independent random variables, each distributed geometrically with a size-- dependent parameter. Confirming a conjecture made by Arratia and Tavar\'e, we prove that the joint distribution of counts of parts with size at most \$s_n \ll n^{1/2}\$ (at least \$s_n\gg n^{1/2}\$, resp.) is close---in terms of the total variation distance---to the distribution of the first \$s_n\$ components of the \$g\$- sequence (of the \$g\$ sequence minus the first \$s_n-1\$ components, resp.). We supplement these results with the estimates for the middle--size parts distribution, using the analytical tools revolving around Hardy--Ramanujan formula for the partition function. Taken together, the estimates lead to an asymptotic description of the random Ferrers diagram, close to the one obtained earlier by Szalay and Tur\'an. As an application, we simplify considerably and strengthen Szalay--Tur\'an's formula for the likely degree of an irreducible representation of the symmetric group \$S_n\$. We show further that both the size of a random conjugacy class and the size of the centraliser for every element >from the class are double exponentially distributed in the limit. We prove that a continuous time process that describes the random fluctuations of the diagram boundary from the deterministic approximation converges to a Gaussian (non--Markov) process with continuous sample path. Convergence is such that it implies weak convergence of every integral functional from a broad class. To demonstrate applicability of this general result, we prove that the eigenvalue distribution for Diaconis--Shahshahani's card-shuffling Markov chain is asymptotically Gaussian with zero mean, and variance of order \$n^{-3/2}\$.