Exact solution of diffusion limited aggregation in a narrow cylindrical geometry

The diffusion limited aggregation model (DLA) and the more general dielectric breakdown model (DBM) are solved exactly in a two dimensional cylindrical geometry with periodic boundary conditions of width 2. Our approach follows the exact evolution of the growing interface, using the evolution matrix E, which is a temporal transfer matrix. The eigenvector of this matrix with an eigenvalue of one represents the system's steady state. This yields an estimate of the fractal dimension for DLA, which is in good agreement with simulations. The same technique is used to calculate the fractal dimension for various values of eta in the more general DBM model. Our exact results are very close to the approximate results found by the fixed scale transformation approach.Comment: 18 pages RevTex, 6 eps figure

On the Catalyzing Effect of Randomness on the Per-Flow Throughput in Wireless Networks

This paper investigates the throughput capacity of a flow crossing a multi-hop wireless network, whose geometry is characterized by general randomness laws including Uniform, Poisson, Heavy-Tailed distributions for both the nodes' densities and the number of hops. The key contribution is to demonstrate \textit{how} the \textit{per-flow throughput} depends on the distribution of 1) the number of nodes $N_j$ inside hops' interference sets, 2) the number of hops $K$, and 3) the degree of spatial correlations. The randomness in both $N_j$'s and $K$ is advantageous, i.e., it can yield larger scalings (as large as $\Theta(n)$) than in non-random settings. An interesting consequence is that the per-flow capacity can exhibit the opposite behavior to the network capacity, which was shown to suffer from a logarithmic decrease in the presence of randomness. In turn, spatial correlations along the end-to-end path are detrimental by a logarithmic term