The simple harmonic urn is a discrete-time stochastic process on $\mathbb Z^2$ approximating the phase portrait of the harmonic oscillator using very basic transitional probabilities on the lattice, incidentally related to the Eulerian numbers.
This urn which we consider can be viewed as a two-colour generalized Polya urn with negative-positive reinforcements, and in a sense it can be viewed as a “marriage” between the Friedman urn and the OK Corral model, where we restart the process each time it hits the horizontal axes by switching the colours of the balls. We show the transience of the process using various couplings with birth and death processes and renewal processes. It turns out that the simple harmonic urn is just barely transient, as a minor modification of the model makes it recurrent.
We also show links between this model and oriented percolation, as well as some other interesting processes.
This is joint work with E. Crane, N. Georgiou, R. Waters and A. Wade.