Fluctuation of maximal particle energy of quantum ideal gas and random partitions

We investigate the limiting distribution of the fluctuations of the maximal summand in a random partition of a large integer with respect to a multiplicative statistics. We show that for a big family of Gibbs measures on partitions (so called generalized Bose--Einstein statistics) this distribution is the well-known Gumbel distribution which usually appears in the context of indepedent random variables. In particular, it means that the (properly rescaled) maximal energy of an individual particle in the grand canonical ensemble of the $d$-dimensional quantum ideal gas has the Gumbel distribution in the limit. We also apply our result to find the fluctuations of the height of a random 3D Young diagram (plane partition) and investigate the order statistics of random partitions under generalized Bose--Einstein statistics.