The spectral gap and the dynamical critical exponent of an exact solvable probabilistic cellular automaton

Abstract

We obtained the exact solution of a probabilistic cellular automaton related to the diagonal-to-diagonal transfer matrix of the six-vertex model on a square lattice. The model describes the flow of ants (or particles), traveling on a one-dimensional lattice whose sites are small craters containing sleeping or awake ants (two kinds of particles). We found the Bethe ansatz equations and the spectral gap for the time-evolution operator of the cellular automaton. From the spectral gap we show that in the asymmetric case it belongs to the Kardar-Parisi-Zhang (KPZ) universality class, exhibiting a dynamical critical exponent value $z=\frac{3}{2}$. This result is also obtained from a direct Monte Carlo simulation, by evaluating the lattice-size dependence of the decay time to the stationary state.