Voltage Distribution in Growing Conducting Networks

We investigate by random-walk simulations and a mean-field theory how growth by biased addition of nodes affects flow of the current through the emergent conducting graph, representing a digital circuit. In the interior of a large network the voltage varies with the addition time $s<t$ of the node as $V(s)\sim \ln (s)/s^\theta$ when constant current enters the network at last added node $t$ and leaves at the root of the graph which is grounded. The topological closeness of the conduction path and shortest path through a node suggests that the charged random walk determines these global graph properties by using only {\it local} search algorithms. The results agree with mean-field theory on tree structures, while the numerical method is applicable to graphs of any complexity

Boundary monomers in the dimer model

The correlation functions of an arbitrary number of boundary monomers in the system of close-packed dimers on the square lattice are computed exactly in the scaling limit. The equivalence of the 2n-point correlation functions with those of a complex free fermion is proved, thereby reinforcing the description of the monomer-dimer model by a conformal free field theory with central charge c=1.Comment: 15 pages, 2 figure

Non-Local Finite-Size Effects in the Dimer Model

We study the finite-size corrections of the dimer model on $\infty \times N$ square lattice with two different boundary conditions: free and periodic. We find that the finite-size corrections depend in a crucial way on the parity of $N$, and show that, because of certain non-local features present in the model, a change of parity of $N$ induces a change of boundary condition. Taking a careful account of this, these unusual finite-size behaviours can be fully explained in the framework of the $c=-2$ logarithmic conformal field theory.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Scaling of avalanche queues in directed dissipative sandpiles

We simulate queues of activity in a directed sandpile automaton in 1+1 dimensions by adding grains at the top row with driving rate $0 < r \leq 1$. The duration of elementary avalanches is exactly described by the distribution $P_1(t) \sim t^{-3/2}\exp{(-1/L_c)}$, limited either by the system size or by dissipation at defects $L_c= \min (L,\xi)$. Recognizing the probability $P_1$ as a distribution of service time of jobs arriving at a server with frequency $r$, the model represents a new example of the $$ server queue in the queue theory. We study numerically and analytically the tail behavior of the distributions of busy periods and energy dissipated in the queue and the probability of an infinite queue as a function of driving rate.Comment: 11 pages, 9 figures; To appear in Phys. Rev.