Combinatorics, Probability and Computing
European Journal of Combinatorics
On Dimensions of a Random Solid Diagram
Combinatorics, Probability and Computing
Limit shape of a random integer partition with a bounded max-to-min ratio of parts sizes
Journal of Combinatorial Theory Series A
Ergodicity of multiplicative statistics
Journal of Combinatorial Theory Series A
On the connected components of a random permutation graph with a given number of edges
Journal of Combinatorial Theory Series A
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We 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 Tavare, we prove that the joint distribution of counts of parts with size at mosts"n@?n^1^/^2(at leasts"n@?n^1^/^2, resp.) is close-in terms of the total variation distance-to the distribution of the firsts"ncomponents of theg-sequence (of theg-sequence minus the firsts"n-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 Turan. As an application, we simplify considerably and strengthen the Szalay-Turan formula for the likely degree of an irreducible representation of the symmetric groupS"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 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.