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The statistical properties of the carrier density profile of graphene in the ground state in the presence particle-particle interaction and random charged impurity in zero gate voltage has been recently obtained by Najafi \textit{et al.} (Phys. Rev E95, 032112 (2017)). The non-zero chemical potential ($\mu$) in gated graphene has non-trivial 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 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 $D_2(r,\mu)\propto \mu^2\exp\left[-\left(-a_D\ln\ln r^{\beta_D}\right)^{\alpha_D} \right]$ ($\alpha_D=1.82$, $\beta_D=0.263$ and $a_D=0.955$). The Fourier power spectrum function behaves like $\ln(S(q))=-\beta_S^{-a_S}\left(\ln q \right)^{a_S}+2\ln \mu$ ($a_S=3.0\pm 0.1$ and $\beta_S=2.08\pm 0.03$) in contrast to the ordinary Gaussian rough surfaces for which $a_S=1$ and $\beta_S=\frac{1}{2}(1+\alpha)^{-1}$, ($\alpha$ being the roughness exponent). The geometrical properties are however similar to the un-gated ($\mu=0$) case, with the exponents that are reported in the text., Comment: arXiv admin note: text overlap with arXiv:1609.07096

The BTW sandpile model is considered on three dimensional percolation lattice which is tunned with the occupation parameter $p$. Along with the three-dimensional avalanches, we study the energy propagation in two-dimensional cross-sections. We use the moment analysis to extract the exponents for two separate cases: the critical ($p=p_c\equiv p_c^{3D}$) and the off-critical ($p_c

In many situations we are interested in the propagation of energy in some portions of a three dimensional system with dilute long-range links. In this paper sandpile model is defined on the three-dimensional small world network with real dissipative boundaries and the energy propagation is studied in three-dimensions as well as the two-dimensional cross sections. Two types of cross section are defined in the system, one in the bulk and another in the system boundary. The motivation of this is to make clear how the statistics of the avalanches in the bulk cross section tend to the statistics of the dissipative avalanches, defined in the boundaries as the concentration of long range links ($\alpha$) increases. This trend is numerically shown to be power law in a manner described in the paper. Two regimes of $\alpha$ are considered in this work. For sufficiently small $\alpha$s the dominant behavior of the system is just like that of the regular BTW, whereas for the intermediate values the behavior is non-trivial with some exponents that are reported in the paper. It is shown that the spatial extent up to which the statistics is similar to the regular BTW model scales with $\alpha$ just like the dissipative BTW model with the dissipation factor (mass in the corresponding ghost model) $m^2\sim \alpha$ for the three-dimensional system as well as its two-dimensional cross-sections.

We report on universality in boundary domain growth in cluster aggregation in the limit of maximum concentration. Maximal concentration means that the diffusivity of the clusters is effectively zero and, instead, clusters merge successively in a percolation process, which leads to a sudden growth of the boundary domains. For two-dimensional square lattices of linear dimension L, independent of the models studied here, we find that the maximum of the boundary interface width, the susceptibility $\chi$, exhibits the scaling $\chi \sim L^{\gamma}$ with the universal exponent $\gamma = 1$. The rapid growth of the boundary domain at the percolation threshold, which is guaranteed to occur for almost {\em any} cluster percolation process, underlies the universal scaling of $\chi$., Comment: 9 pages, 13 figures, to appear in Nature Scientific Reports