Observation of SLE(κ, ρ) on the critical statistical models

Abstract

Schramm-Loewner Evolution (SLE) is a stochastic process that helps classify critical statistical models using one real parameter κ. Numerical study of SLE often involves curves that start and end on the real axis. To reduce numerical errors in studying the critical curves which start from the real axis and end on it, we have used hydrodynamically normalized SLE(κ, ρ) which is a stochastic differential equation that is hypothesized to govern such curves. In this paper we directly verify this hypothesis and numerically apply this formalism to the domain wall curves of the Abelian Sandpile Model (ASM) (κ = 2) and critical percolation (κ = 6). We observe that this method is more reliable for analyzing interface loops.

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

“…To this end we discretize the uniformizing map g t by assuming the driving function to be partially constant, in time intervals δt n = t n − t n−1 , i.e., g t n ,ξ n = g δt n−1 ,ξ n−1 og δt n−2 ,ξ n−2 o...og δt 0 ,ξ 0 in which g δt n ,ξ n (z) is the map that uniformizes an infinitesimal vertical line [24]: One can discretize Eq. (9).…”

“…It has also been shown that SLE κ has correspondence with CFT with central charge c κ = (6−κ)(3κ−8) 2κ and at the point where the SLE curve starts a boundary changing operator with weight h 1;2 = (6 − κ)/(2κ) is sitting [18]. SLE(κ,ρ) is a variant of SLE in which the curve experiences some additional boundary changes [24]. 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.…”

“…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].…”

Numerical determination of the boundary condition changing operators

“…The first simulation has been done in 16 in which it was shown that the best values are κ = 1.95 ± 0.07 and ρ c = 3.5 ± 0.5. Our simulation for our case (which is on cylinder and the curves are rescaled) coincides with this report, i.e.…”

“…It has been shown that this method results in more precise and reliable determination of these parameters 16 . Now consider one additional preferred point on the real axis sitting in x 0 .…”