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The simple harmonic urn

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The simple harmonic urn is a discrete-time stochastic process approximating the phase portrait of the simple harmonic oscillator. This urn is a version of 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 a border 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, and for a good reason, as the embedded urn process is, in fact, a Lamperti-type random walk. We also show links between this model and oriented percolation, as well as a few other interesting processes. This is joint work with E. Crane, N. Georgiou, R. Waters and A. Wade.

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