Nicholas Georgiou
Durham

The simple harmonic urn is a generalized P\'{o}lya urn model with two
types of ball.  If the drawn ball is red it is replaced together with
a black ball, but if the drawn ball is black it is replaced and a red
ball is thrown out of the urn. When only black balls remain, the roles
of the colours are swapped and the process restarts. This process can
be viewed as a stochastic process on $\mathbb{Z}^2$, that approximates
the phase-portrait of simple harmonic motion.

We will show that the resulting Markov chain is transient, but only
just so: if we throw out a ball every time the colours swap, then the
process becomes positive-recurrent.  Along the way I will discuss the
connections between the urn process and birth-death processes, a
uniform renewal process and the Eulerian numbers.

This is joint work with Edward Crane, Stanislav Volkov, Andrew Wade
and Robert Waters.