Abstract:

Pólya urns are urns where at each unit of time a ball is drawn
uniformly at random and is replaced by some other balls according
to its colour. We introduce a more general model: The replacement
rule depends on the colour of the drawn ball AND the value of the
time mod p.

Our key tool are generating functions, which encode all possible
urn compositions after a certain number of steps. The evolution of
the urn is then translated into a system of differential equations
and we prove that the moment generating functions are D-finite in
one variable. From these we derive asymptotic forms of the moments.
When the time tends to infinity, we show that these periodic Pólya
urns follow a rich variety of behaviours: Their asymptotic
fluctuations are described by a family of distributions, the
generalized Gamma distributions, which can also be seen as powers
of Gamma distributions.

Furthermore, we establish some enumerative links with other
combinatorial objects yielding a new result on the asymptotics
of Young tableaux: We prove that the law of the lower right
corner in a triangular Young tableau follows asymptotically a
product of generalized Gamma distributions.

This is joint work with Cyril Banderier and Philippe Marchal.

Zoom-Meeting beitreten

https://zoom.us/j/93778343609?pwd=YngwNmNLZWgxczRnYXNxdHZTSVdSdz09

Meeting-ID: 937 7834 3609
Kenncode: BCfC2J