Universality phenomena for determinantal point processes and relatives.
After the deep results obtained by many great researchers concerning independent random variables, lot of attention has been recently paid to a certain kind of interacting random variables, arising from several (a priori non related) fields of mathematics, which behaves in a same way as the number of such random variables goes to infinity (appearance of the Sine kernel, Tracy-Widom distribution ...) ; the so-called universality phenomenon. This class of interacting random variables is not yet identified but includes
the eigenvalues of many random matrix models
the lengths of the rows of Young diagrams distributed according to the Plancherel measure
models from statistical physics like (T)ASEP, polynuclear growth models, random tilings of geometric shapes, ...)
the zeros of the Riemann Zeta function, once assumed the RH
and many others.
For further information, see e.g. the nice (although not exhaustive) overview of Deift http://arxiv.org/abs/math-ph/0603038
Because of the diversity of the mathematics involved a huge community, including a few Fields medals, is now working on a better understanding of such a class of random variables.
