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 sl(1|1) 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.

Similarity revisited: shock random walks in the asymmetric simple exclusion process with open boundaries

“…It is well known that many generators of stochastic interacting particle systems are related by a similarity transformation W = A −1 QA between the intensity matrices W of one process and Q of the other [1]. If at least one of the processes is reversible then similarity is a special case of duality between interacting particle systems [2], which has become a topic of intense research during the last decade; see, e.g., [9][10][11][12][13][14][15][16][17] and references therein for various applications to driven diffusive systems and reaction-diffusion processes, and specifically [18][19][20][21][22][23][24][25] for very recent work on duality for models closely related to the ASEP that is the topic of the present work.…”

Duality from integrability: annihilating random walks with pair deposition

“…expressed in terms of the Pauli matrices introduced in definition A.2. For a detailed derivation, see [14]. Thus with the particle number operator…”

“…In order to highlight a major difference to duality for the DAP confined to a box of size L without periodic boundary conditions [14] we point out that one has the trivial duality relation…”

“…There has been renewed interest in the DAP with focus on the relaxation spectrum [13] and, very recently, on duality arising from supersymmetry [14] which has its origin in a close relation of the generator of the process to the quantum Hamiltonian of the XY-quantum spin chain [15]. Duality between two Markov processes η(t), ξ(t) with respect to a duality function D(η, ξ) allows one to express properties of one process in terms of its dual which is useful when the dual process has some simple properties.…”

Dynamical phase transitions in annihilating random walks with pair deposition

“…The paper is organized as follows. In section 2 we give the precise definition of the DAP following [34,35] and we review conditioning and atypical dynamics along the lines of references [31,[36][37][38][39]. The results are presented in section 3 and some conclusions are drawn in section 4.…”