Quasi-stationary states in nonlocal stochastic growth models with infinitely many absorbing states

Abstract

We study a two parameter (u, p) extension of the conformally invariant raise and peel model. The model also represents a nonlocal and biased-asymmetric exclusion process with local and nonlocal jumps of excluded volume particles in the lattice. The model exhibits an unusual and interesting critical phase where, in the bulk limit, there are an infinite number of absorbing states. In spite of these absorbing states the system stays, during a time that increases exponentially with the lattice size, in a critical quasi-stationary state. In this critical phase the critical exponents depend only on one of the parameters defining the model (u). The endpoint of this critical phase, where the system changes from an active to an inactive frozen phase, belongs to a distinct universality class. This new behavior, we believe, is due to the appearance of Jordan cells in the Hamiltonian describing the time evolution. The dimensions of these cells increase with the lattice size. In a special case (u = 0) where the model has no adsorptions we are able to calculate analytically the time evolution of some observables. A polynomial time dependence is obtained thanks to the appearance of Jordan cells structures in the Hamiltonian.

Critical phases in the raise and peel model

“…For u > 1 previous results [2,4] indicate that the RPM is in a critical phase, with selforganized criticality having a dynamical critical exponent z(u) that decreases continuously with the parameter u. On the other hand, as noticed in [6], in the limit 1/u = 0 the RPM is exactly mapped into the TASEP, a model with quite distinct properties, if the boundary conditions are taken to be open (free) or closed (periodic). For open boundary conditions the model is massive and for the periodic ones the model is critical and belongs to the KPZ critical universality with z = 3/2 and α = 1/2.…”

“…Previous evaluations [2] of the dynamical critical exponent z for u ≥ 1 indicate a continuous decrease as u increases, tending to z = 0 as u → ∞. On the other hand, more recently [6] it was observed that the RPM at the limiting case 1/u = 0 recovers exactly the totally asymmetric exclusion problem (TASEP). We can see this correspondence easily from the allowed processes in the particle-vacancy representation of the model.…”

“…In this paper we present an extensive numerical study of the phase diagram of the RPM. The original motivation of these calculations is due to the exact connection [6] of the RPM with no desorption (1/u = 0) with the totally asymmetric exclusion problem (TASEP) [7,8]. TASEP is critical (z = 3/2) or not depending on whether the boundary condition is periodic or not.…”

Quasi-Stationary States in Ionic Liquid-Liquid Crystal Mixtures at the Nematic-Isotropic Phase Transition

“…Internal longrange forces can bring in additional spatiotemporal correlations to such open systems, giving rise to states with a very long lifetime, the so-called "quasistationary states" (QSS) [10][11][12]. Quasistationary states involving spin chains [13], lattices with infinity of absorbing states [14], and in hydrodynamics on a torus [15] have proved the general character of these states.…”