On a Likely Shape of the Random Ferrers Diagram

AbstractWe study the random partitions of a large integern, under the assumption that all such partitions are equally likely. We use Fristedt's conditioning device which connects the parts (summands) distribution to the one of ag-sequence, that is, a sequence of independent random variables, each distributed geometrically with a size-dependent parameter. Confirming a conjecture made by Arratia and Tavaré, we prove that the joint distribution of counts of parts with size at mostsn⪡n1/2(at leastsn⪢n1/2, resp.) is close—in terms of the total variation distance—to the distribution of the firstsncomponents of theg-sequence (of theg-sequence minus the firstsn−1 components, resp.). We supplement these results with the estimates for the middle-sized parts distribution, using the analytical tools revolving around the 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 Turán. As an application, we simplify considerably and strengthen the Szalay–Turán formula for the likely degree of an irreducible representation of the symmetric groupSn. 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 doubly 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 the Diaconis–Shahshahani card-shuffling Markov chain is asymptotically Gaussian with zero mean, and variance of ordern−3/2