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.

The self-avoiding walk on the square site-diluted correlated percolation lattice is considered. The Ising model is employed to realize the spatial correlations of the metric space. As a well-accepted result, the (generalized) Flory's mean-field relation is tested to measure the effect of correlation. After exploring a perturbative Fokker-Planck-like equation, we apply an enriched Rosenbluth Monte Carlo method to study the problem. To be more precise, the winding angle analysis is also performed from which the diffusivity parameter of Schramm-Loewner evolution theory (κ) is extracted. We find that at the critical Ising (host) system, the exponents are in agreement with Flory's approximation. For the off-critical Ising system, we find also a behavior for the fractal dimension of the walker trace in terms of the correlation length of the Ising system ξ(T), i.e., D_{F}^{SAW}(T)-D_{F}^{SAW}(T_{c})∼1/sqrt[ξ(T)].

We study the sandpile model on three-dimensional spanning Ising clusters with the temperature T treated as the control parameter. By analyzing the three dimensional avalanches and their twodimensional projections (which show scale-invariant behavior for all temperatures), we uncover two universality classes with different exponents (an ordinary BTW class, and SOCT =∞), along with a tricritical point (at Tc, the critical temperature of the host) between them. The SOCT =∞ universality class is characterized by the exponent of the avalanche size distribution τ T =∞ = 1.27 ± 0.03, consistent with the exponent of the size distribution of the Barkhausen avalanches in amorphous ferromagnets (Phys. Rev. L 84, 4705 (2000)). The tricritical point is characterized by its own critical exponents and also some additional scaling behavior in its vicinity. In addition to the avalanche exponents, some other quantities like the average height, the spanning avalanche probability (SAP) and the average coordination number of the Ising clusters change significantly the behavior at this point, and also exhibit power-law behavior in terms of ≡ T −Tc Tc , defining further critical exponents. Importantly the finite size analysis for the activity (number of topplings) per site shows the scaling behavior with exponents β = 0.19 ± 0.02 and ν = 0.75 ± 0.05. A similar behavior is also seen for the SAP and the average avalanche height. The fractal dimension of the external perimeter of the two-dimensional projections of avalanches is shown to be robust against T with the numerical value D f = 1.25 ± 0.01.In the context of out-of-equilibrium critical phenomena, self-organized critical (SOC) systems have attracted much attention because of their role in a wide range of systems, from finance [1] and biological [2] to granular matter [3], the brain [4] and neural networks in general [5]. SOC systems are characterized by their avalanche dynamics resulting from slow driving of the system. Vortex avalanche dynamics in type II superconductors [6], earthquakes [7], solar flares [8], microfracturing processes [9], fluid flow in porous media [10], phase transition-like behavior of the magnetosphere [11], bursts in filters [12], phase transitions in jammed granular matter [3], and avalanches dynamics in the rat cortex [13] are some natural examples for SOC. This large class of natural systems inspired theoretical models with the aim of capturing the dominant internal dynamics that causes the avalanches.Here we find evidence for a novel non-equilibrium universality class, and propose a new type of outof-equilibrium phase transition between SOC models induced by the geometry of the underlying graph upon which the model is defined. It might be applicable to experiments with spatial flow patterns of transport in heterogeneous porous media [14], which involve the toppling of fluid [15]. Another example is the Barkhausen effect in magnetic systems [16], for which the avalanches have been shown to exhibit scaling behavior with an avalanche size exponent 1.27 ± 0....

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The statistical properties of the carrier density profile of graphene in the ground state in the presence of particle-particle interaction and random charged impurity in zero gate voltage has been recently obtained by Najafi et al. [Phys. Rev. E 95, 032112 (2017)2470-004510.1103/PhysRevE.95.032112]. The nonzero chemical potential (μ) in gated graphene has nontrivial effects on electron-hole puddles, since it generates mass in the Dirac action and destroys the scaling behaviors of the effective Thomas-Fermi-Dirac theory. We provide detailed analysis on the resulting spatially inhomogeneous system in the framework of the Thomas-Fermi-Dirac theory for the Gaussian (white noise) disorder potential. We show that the chemical potential in this system as a random surface destroys the self-similarity, and also the charge field is non-Gaussian. We find that the two-body correlation functions are factorized to two terms: a pure function of the chemical potential and a pure function of the distance. The spatial dependence of these correlation functions is double logarithmic, e.g., the two-point density correlation behaves like D_{2}(r,μ)∝μ^{2}exp[-(-a_{D}lnlnr^{β_{D}})^{α_{D}}] (α_{D}=1.82, β_{D}=0.263, and a_{D}=0.955). The Fourier power spectrum function also behaves like ln[S(q)]=-β_{S}^{-a_{S}}(lnq)^{a_{S}}+2lnμ (a_{S}=3.0±0.1 and β_{S}=2.08±0.03) in contrast to the ordinary Gaussian rough surfaces for which a_{S}=1 and β_{S}=1/2(1+α)^{-1} (α being the roughness exponent). The geometrical properties are, however, similar to the ungated (μ=0) case, with the exponents that are reported in the text.

The Gaussian free field (GFF) is considered in the background of random iso-height islands which is modeled by the site percolation with the occupation probability p. To realize GFF, we consider the Poisson equation in the presence of normal distributed white-noise charges, as the stationary state of the Edwards-Wilkinson (EW) model. The iso-potential (metallic in the terminology of the electrostatic problem) sites are chosen over the lattice according to the percolation problem, giving rise to some metallic islands and some active (not metallic, nor surrounded by a metallic island) area. We see that the dilution of the system by incorporating metallic particles (or equivalently considering the iso-height islands) annihilates the spatial correlations and also the potential fluctuations. Some local and global critical exponents of the problem are reported in this work. The GFF, when simulated on the active area show a cross over between two regimes: small (UV) and large (IR) scales. Importantly, by analyzing the change of exponents (in and out of the critical occupation pc) under changing the system size and the change of the cross-over points, we find two fixed points and propose that GFFp=p c is unstable towards GFFp=1.

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