Duality, supersymmetry and non-conservative random walks

Abstract

We derive a probabilistic interpretation of the observation that the quantum XY chain is supersymmetric in the sense that the Hamiltonian commutes with the generators of a subalgebra of the universal enveloping algebra of the Lie superalgebra and its deformations. The XY chain is shown to be the generator of a Markov process that describes classical vicious random walkers that annihilate immediately when they arrive on the same site, while new random walkers are created at neighbouring sites. The supersymmetry leads to a probabilistic self-duality relation and a duality between the random walk model with an even and odd number of particles, respectively.