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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.

The Bak-Tang-Wiesenfeld (BTW) model is considered on the site-diluted square lattice, tuned by the occupancy probability p. Various statistical observables of the avalanches are analyzed in terms of p, e.g. the fractal dimension of their exterior frontiers, gyration radius, loop lengths and Green's function. The model exhibits critical behavior for all amounts of p, and the exponents of the statistical observables are analyzed. We find a distinct universality class at = p p c , which is unstable towards a p = 1 (BTW) fixed point. This universality class displays some common features such as a twodimensional (2D) Ising universality class, e.g. the fractal dimension of loops in the thermodynamic limit is1.38 0.01 F p p c which is compatible with the fractal dimension of geometrical spin clusters of the 2D critical Ising model (with = D F Ising 11 8 ).

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.

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SLE(κ,ρ[over arrow]) is a variant of Schramm-Loewner Evolution (SLE) which describes the curves which are not conformal invariant, but are self-similar due to the presence of some other preferred points on the boundary. In this paper we study the left passage probability (LPP) of SLE(κ,ρ[over arrow]) through field theoretical framework and find the differential equation governing this probability. This equation is numerically solved for the special case κ=2 and h(ρ)=0 in which h(ρ) is the conformal weight of the boundary changing (bcc) operator. It may be referred to loop erased random walk (LERW) and Abelian sandpile model (ASM) with a sink on its boundary. For the curve which starts from ξ(0) and conditioned by a change of boundary conditions at x(0), we find that this probability depends significantly on the factor x(0)-ξ(0). We also present the perturbative general solution for large x(0). As a prototype, we apply this formalism to SLE(κ,κ-6) which governs the curves that start from and end on the real axis.

In this paper, we present an electronic avalanche model for the transport of electrons in the disordered two-dimensional (2D) electron gas which has the potential to describe the 2D metalinsulator transition (MIT) in the zero electron-electron interaction limit. The disorder is considered to be uncorrelated-Coulomb noise with a uniform distribution. In this model we sub-divide the system to some virtual cells each of which has a linear size of the order of phase coherence length of the system. Using Thomas-Fermi-Dirac theory we propose some simple energy functions for the cells and using the thermodynamics of 2DEG we develop some rules for the charge transfer between the cells. A second order transition line arises from our model with some similarities with the experiments. The compressibility of the system also diverges on this line. We characterize this (disorder-driven) phase transition which is between the non-percolating phase and the percolating phase (in which the system shows metallic behavior) and obtain some geometrical critical exponents. The fractal dimension of the exterior frontier of the electronic avalanches on the transition line is compatible with the percolation theory, whereas the other exponents are different. The exponents are robust against disorder in the low disordered 2DEGs and change considerably in the high disordered ones.

The self-organized criticality on the random fractal networks has many motivations, like the movement pattern of fluid in the porous media. In addition to the randomness, introducing correlation between the neighboring portions of the porous media has some nontrivial effects. In this paper, we consider the Ising-like interactions between the active sites as the simplest method to bring correlations in the porous media, and we investigate the statistics of the BTW model in it. These correlations are controlled by the artificial "temperature" T and the sign of the Ising coupling. Based on our numerical results, we propose that at the Ising critical temperature T_{c} the model is compatible with the universality class of two-dimensional (2D) self-avoiding walk (SAW). Especially the fractal dimension of the loops, which are defined as the external frontier of the avalanches, is very close to D_{f}^{SAW}=4/3. Also, the corresponding open curves has conformal invariance with the root-mean-square distance R_{rms}∼t^{3/4} (t being the parametrization of the curve) in accordance with the 2D SAW. In the finite-size study, we observe that at T=T_{c} the model has some aspects compatible with the 2D BTW model (e.g., the 1/log(L)-dependence of the exponents of the distribution functions) and some in accordance with the Ising model (e.g., the 1/L-dependence of the fractal dimensions). The finite-size scaling theory is tested and shown to be fulfilled for all statistical observables in T=T_{c}. In the off-critical temperatures in the close vicinity of T_{c} the exponents show some additional power-law behaviors in terms of T-T_{c} with some exponents that are reported in the text. The spanning cluster probability at the critical temperature also scales with L^{1/2}, which is different from the regular 2D BTW model.