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.
@article{0501043,
author = {Vershik, A. and Yakubovich, Yu.},
title = {Fluctuation of maximal particle energy of quantum ideal gas and random
partitions},
journal = {arXiv},
volume = {2005},
number = {0},
year = {2005},
language = {en},
url = {http://dml.mathdoc.fr/item/0501043}
}
Vershik, A.; Yakubovich, Yu. Fluctuation of maximal particle energy of quantum ideal gas and random
partitions. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0501043/