Phase Transition in a Self-repairing Random Network

We consider a network, bonds of which are being sequentially removed; that is done at random, but conditioned on the system remaining connected (Self-Repairing Bond Percolation SRBP). This model is the simplest representative of a class of random systems for which forming of isolated clusters is forbidden. It qualitatively describes the process of fabrication of artificial porous materials and degradation of strained polymers. We find a phase transition at a finite concentration of bonds $p=p_c$, at which the backbone of the system vanishes; for all $p<p_c$ the network is a dense fractal.Comment: 4 pages, 4 figure

Percolation with excluded small clusters and Coulomb blockade in a granular system

We consider dc-conductivity $\sigma$ of a mixture of small conducting and insulating grains slightly below the percolation threshold, where finite clusters of conducting grains are characterized by a wide spectrum of sizes. The charge transport is controlled by tunneling of carriers between neighboring conducting clusters via short ``links'' consisting of one insulating grain. Upon lowering temperature small clusters (up to some $T$-dependent size) become Coulomb blockaded, and are avoided, if possible, by relevant hopping paths. We introduce a relevant percolational problem of next-nearest-neighbors (NNN) conductivity with excluded small clusters and demonstrate (both numerically and analytically) that $\sigma$ decreases as power law of the size of excluded clusters. As a physical consequence, the conductivity is a power-law function of temperature in a wide intermediate temperature range. We express the corresponding index through known critical indices of the percolation theory and confirm this relation numerically.Comment: 7 pages, 6 figure

Universality and non-universality in behavior of self-repairing random networks

We numerically study one-parameter family of random single-cluster systems. A finite-concentration topological phase transition from the net-like to the tree-like phase (the latter is without a backbone) is present in all models of the class. Correlation radius index $\nu_B$ of the backbone in the net-like phase; graph dimensions -- $d_{\min}$ of the tree-like phase, and $D_{\min}$ of the backbone in the net-like phase appear to be universal within the accuracy of our calculations, while the backbone fractal dimension $D_B$ is not universal: it depends on the parameter of a model.Comment: Published variant; more accurate numerical data and minor corrections. 4 pages, 5 figure