Trees, paths and avalanches on random networks

The investigation of equilibrium and non-equilibrium processes in disordered systems and particularly the relation between them is a complex problem that deserves attention. We concentrate on analyzing several relations of this type and appropriate numerical solutions.; Invasion percolation (IP) model was motivated by the problem of fluid displacement in disordered media but in principle it could be applied to any invasion process which evolves along the minimum resistance path. Finding the invasion paths is a global optimization problem where the front advances by occupying the least resistant bond. Once the invasion is finished, the union of all the invasion paths on the lattice forms a minimum energy spanning tree (MST). We show that the geometry of a MST on random graphs is universal. Due to this geometric universality, we are able to characterize the energy of this optimal tree for any type of disorder using a scaling distribution found using uniform disorder. Therefore we expect the hopping transport in random media to have universal behavior.; Kinetic interfaces is an important branch of statistical mechanics, fueled by application such as fluid-fluid displacement, imbibition in porous media, flame fronts, tumors, etc. These processes can be unified via Kardar-Parisi-Zhang (KPZ) equation, which is mapped exactly to an equilibrium problem (DPRM). We are able to characterize both using Dijkstra's algorithm, which is known to generate shortest path tree in a random network. We found that while obtaining the polymers the algorithm develops a KPZ type interface. We have extracted the interface exponents for both 2d square lattice and 3 d cubic lattice, being in agreement with previously recorded results for KPZ.; The IP and KPZ classes are known to be very different: while the first one generates a distinct self-similar (fractal) interface, the second one has a self-similar invasion front. Though they are different we are able to construct a generalized algorithm that interpolates between these two universality classes. We discuss the relationship with the IP, the directed polymer in a random media; and the implications for the broader issue of universality in disordered systems.; Random Field Ising Model (RFIM) is one of the most important models of phase transitions in disordered systems. We present exact results for the critical behavior of the RFIM on complete graphs and trees, both at equilibrium and away from equilibrium, i.e., models for hysteresis and Barkhausen noise. We show that for stretched exponential and powerlaw distributions of random fields the behavior on complete graphs is non-universal, while the behavior on Cayley trees is universal even in the limit of large coordination.; Until recently, the evolution of WWW, Internet, etc., was thought to be highly complex. The model proposed by Barabási and Albert shows that such networks can be modeled with the help of “preferential attachment”, i.e. a highly connected vertex has a higher chance to get further links compared with a weakly connected vertex. We find that the random network constructed from a self-organized critical mechanism, (IP), falls in the same class without imposing any “preferential” growth rule. The network obtained has a connectivity exponent γ $\approxeq$ 2.45, close to the WWW outgoing-links exponent.