The Size of the Largest Part of Random Plane Partitions of Large Integers

We study the asymptotic behavior of the largest part size of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that this characteristic, appropriately normalized, tends weakly, as n → ∞, to a random variable having an extreme value probability distribution with distribution function, equal to e−e −z ,−∞ < z < ∞. The representation of a plane partition as a solid diagram shows that the same limit theorem holds for the numbers of rows and columns of a random plane partition of n.