Avalanche frontiers in the dissipative Abelian sandpile model and off-critical Schramm-Loewner evolution

Abstract

Avalanche frontiers in Abelian sandpile model (ASM) are random simple curves whose continuum limit is known to be a Schramm-Loewner evolution with diffusivity parameter κ=2. In this paper we consider the dissipative ASM and study the statistics of the avalanche and wave frontiers for various rates of dissipation. We examine the scaling behavior of a number of functions, such as the correlation length, the exponent of distribution function of loop lengths, and the gyration radius defined for waves and avalanches. We find that they do scale with the rate of dissipation. Two significant length scales are observed. For length scales much smaller than the correlation length, these curves show properties close to the critical curves, and the corresponding diffusivity parameter is nearly the same as the critical limit. We interpret this as the ultraviolet limit where κ=2 corresponding to c=-2. For length scales much larger than the correlation length, we find that the avalanche frontiers tend to self-avoiding walk, and the corresponding driving function is proportional to the Brownian motion with the diffusivity parameter κ=8/3 corresponding to a field theory with c=0. We interpret this to be the infrared limit of the theory or at least a crossover.

Statistical investigation of the cross sections of wave clusters in the three-dimensional Bak-Tang-Wiesenfeld model

“…It is easy to see that in any wave, the set of toppled sites forms a connected cluster with no voids (untoppled sites fully surrounded by toppled sites), and no site topples more than once in one wave. Geometrical aspects of this model have been the subject of intense studies recently [17,23,24]. One example is the exterior perimeter of an avalanche or a wave that is numerically shown to be loop-erased random walk (LERW) in two dimensions [17,23].…”

“…In this theory, the parameter κ identifies the local properties of the model in hand as above, and the parameter ρ has to do with the boundary conditions (bc) imposed. The case of interest in this paper is that the curve starts from the origin and ends on some point on the real axis x ∞ in which it has been shown that ρ = ρ c ≡ κ − 6 [18,23,25,26]. In this case the stochastic equation governing g t (z) is the same as Eq.…”

Statistical investigation of avalanches of three-dimensional small-world networks and their boundary and bulk cross-sections

“…The important aspect of this model is that the system organizes itself in the critical state. The geometrical aspects of the pure two-dimensional regular BTW (which corresponds to c = −2 conformal field theory (CFT)) has been the subject of intense studies [28][29][30]. One example is the exterior perimeter of an avalanche which is numerically shown to be loop-erased random walk (LERW) in two dimensions [28,30].…”

Continuous transforming the BTW to the Manna model

“…There are many examples of this approach. As an example one can mention the massive Abelian sand pile model (ASM) in which by defining intermediate amounts of dissipation, it has been shown that at large spatial scales, it corresponds to self avoiding walk (SAW) [30]. The Ising model out of critical temperature [31][32][33] and off-critical percolation theory [34] can be regarded as the other important examples of this approach.…”