Numerical Study of the Correspondence Between the Dissipative and Fixed Energy Abelian Sandpile Models

We consider the Abelian sandpile model (ASM) on the large square lattice with a single dissipative site (sink). Particles are added by one per unit time at random sites and the resulting density of particles is calculated as a function of time. We observe different scenarios of evolution depending on the value of initial uniform density (height) $h_0=0,1,2,3$. During the first stage of the evolution, the density of particles increases linearly. Reaching a critical density $\rho_c(h_0)$, the system changes its behavior sharply and relaxes exponentially to the stationary state of the ASM with $\rho_s=25/8$. We found numerically that $\rho_c(0)=\rho_s$ and $\rho_c(h_0>0) \neq \rho_s$. Our observations suggest that the equality $\rho_c=\rho_s$ holds for more general initial conditions with non-positive heights. In parallel with the ASM, we consider the conservative fixed-energy Abelian sandpile model (FES). The extensive Monte-Carlo simulations for $h_0=0,1,2,3$ have confirmed that in the limit of large lattices $\rho_c(h_0)$ coincides with the threshold density $\rho_{th}(h_0)$ of FES. Therefore, $\rho_{th}(h_0)$ can be identified with $\rho_s$ if the FES starts its evolution with non-positive uniform height $h_0 \leq 0$.Comment: 6 pages, 8 figure

Discrepancy and Signed Domination in Graphs and Hypergraphs

For a graph G, a signed domination function of G is a two-colouring of the vertices of G with colours +1 and -1 such that the closed neighbourhood of every vertex contains more +1's than -1's. This concept is closely related to combinatorial discrepancy theory as shown by Fueredi and Mubayi [J. Combin. Theory, Ser. B 76 (1999) 223-239]. The signed domination number of G is the minimum of the sum of colours for all vertices, taken over all signed domination functions of G. In this paper, we present new upper and lower bounds for the signed domination number. These new bounds improve a number of known results.Comment: 12 page

The problem of predecessors on spanning trees

We consider the equiprobable distribution of spanning trees on the square lattice. All bonds of each tree can be oriented uniquely with respect to an arbitrary chosen site called the root. The problem of predecessors is finding the probability that a path along the oriented bonds passes sequentially fixed sites $i$ and $j$. The conformal field theory for the Potts model predicts the fractal dimension of the path to be 5/4. Using this result, we show that the probability in the predecessors problem for two sites separated by large distance $r$ decreases as $P(r) \sim r^{-3/4}$. If sites $i$ and $j$ are nearest neighbors on the square lattice, the probability $P(1)=5/16$ can be found from the analytical theory developed for the sandpile model. The known equivalence between the loop erased random walk (LERW) and the directed path on the spanning tree says that $P(1)$ is the probability for the LERW started at $i$ to reach the neighboring site $j$. By analogy with the self-avoiding walk, $P(1)$ can be called the return probability. Extensive Monte-Carlo simulations confirm the theoretical predictions.Comment: 7 pages, 2 figure

Jamming probabilities for a vacancy in the dimer model

Following the recent proposal made by Bouttier et al [Phys. Rev. E 76, 041140 (2007)], we study analytically the mobility properties of a single vacancy in the close-packed dimer model on the square lattice. Using the spanning web representation, we find determinantal expressions for various observable quantities. In the limiting case of large lattices, they can be reduced to the calculation of Toeplitz determinants and minors thereof. The probability for the vacancy to be strictly jammed and other diffusion characteristics are computed exactly.Comment: 19 pages, 6 figure