Critical phases in the raise and peel model

Abstract

The raise and peel model (RPM) is a nonlocal stochastic model describing the space and time fluctuations of an evolving one dimensional interface. Its relevant parameter u is the ratio between the rates of local adsorption and nonlocal desorption processes (avalanches) The model at u = 1 is the first example of a conformally invariant stochastic model. For small values u < u0 the model is known to be noncritical, while for u > u0 it is critical. Although previous studies indicate that u0 = 1, a determination of u0 with a reasonable precision is still missing. By calculating numerically the structure function of the height profiles in the reciprocal space we confirm with good precision that indeed u0 = 1. We establish that at the conformal invariant point u = 1 the RPM has a roughening transition with dynamical and roughness critical exponents z = 1 and , respectively. For u > 1 the model is critical with a u-dependent dynamical critical exponent that tends towards zero as . However at 1/u = 0 the RPM is exactly mapped into the totally asymmetric exclusion problem. This last model is known to be noncritical (critical) for open (periodic) boundary conditions. Our numerical studies indicate that the RPM as , due to its nonlocal dynamical processes, has the same large-distance physics no matter what boundary condition we chose. For u > 1, our numerical analysis shows that in contrast to previous predictions, the region is composed of two distinct critical phases. For the height profiles are rough (), and for the height profiles are flat at large distances (). We also observed that in both critical phases (u > 1) the RPM at short length scales, has an effective behavior in the Kardar–Parisi–Zhang critical universality class, that is not the true behavior of the system at large length scales.