ON A LIKELY SHAPE OF THE RANDOM FERRERS DIAGRAM

We study the random partitions of a large integer n, under the assumpt ion that all such partitions are equally likely. We use Fristedt's con ditioning device which connects the parts (summands) distribution to t he one of a g-sequence, that is, a sequence of independent random vari ables, each distributed geometrically with a size-dependent parameter. Confirming a conjecture made by Arratia and Tavare, we prove that the joint distribution of counts of parts with size at most s(n) much les s than n(1/2) (at least s(n) much greater than n(1/2), resp.) is close -in terms of the total variation distance-to the distribution of the f irst s(n) components of the g-sequence (of the g-sequence minus the fi rst s(n) - 1 components, resp.). We supplement these results with the estimates for the middle-sized parts distribution, using the analytica l tools revolving around the Hardy-Ramanujan formula for the partition function. Taken together, the estimates lead to an asymptotic descrip tion of the random Ferrers diagram, close to the one obtained earlier by Szalay and Turan. As an application, we simplify considerably and s trengthen the Szalay-Turan formula for the likely degree of an irreduc ible representation of the symmetric group S-n. We show further that b oth the size of a random conjugacy class and the size of the centralis er for every element from the class are doubly exponentially distribut ed in the limit. We prove that a continuous time process that describe s the random fluctuations of the diagram boundary from the determinist ic approximation converges to a Gaussian (non-Markov) process with con tinuous sample path. Convergence is such that it implies weak converge nce of every integral functional from a broad class. To demonstrate ap plicability of this general result, we prove that the eigenvalue distr ibution for the Diaconis-Shahshahani card-shuffling Markov chain is as ymptotically Gaussian with zero mean, and variance of order n(-3/2). ( C) 1997 Academic Press.