Quantum operators in classical probability theory: I. “Quantum spin” techniques and the exclusion model of diffusion

Abstract

The exclusion process is an interacting particle system in which particles perform random walks on a lattice except that they may not move to a position already occupied. In this paper we show how techniques derived from quantum mechanics may be used to achieve asymptotic results for an exclusion process on a complete graph. In particular, an abrupt approach to stationarity is demonstrated. The similarity between the transition matrix for the exclusion process and the Hamiltonian for the Heisenberg ferromagnet is not well understood. However, it allows not only quantum operator techniques to be carried over to a problem in stochastic processes, but also concepts such as the “mean field”.