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

TL;DR

This paper presents an exact solution for a probabilistic cellular automaton modeling ant flow, revealing its belonging to the KPZ universality class with a dynamical critical exponent of 1.5, confirmed through analytical and simulation methods.

Contribution

It provides the first exact solution for this automaton, including Bethe ansatz equations and spectral gap analysis, linking it to KPZ universality.

Findings

Spectral gap analysis shows KPZ universality class.

Dynamical critical exponent z=3/2 confirmed.

Monte Carlo simulations support analytical results.

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.