We study completely asymmetric 2-channel exclusion processes in 1 dimension. It describes a two-way traffic flow with cars moving in opposite directions. The interchannel interaction makes cars slow down in the vicinity of approaching cars in other lane. Particularly, we consider in detail the system with a finite density of cars on one lane and a single car on the other one. When the interchannel interaction reaches a critical value, traffic jam occurs, which turns out to be of first order phase transition. We derive exact expressions for the average velocities, the current, the density profile and the $k$- point density correlation functions. We also obtain the exact probability of two cars in one lane being distance $R$ apart, provided there is a finite density of cars on the other lane, and show the two cars form a weakly bound state in the jammed phase.

We consider the ensemble of N-dimensional random symmetric matrices A that have, in average, p non-zero elements per row. We study the asymptotic behavior of the norm of A in the limit of infinitely increasing N and p. We prove that the value p= log N is the critical one for the norm to be either bounded or not. The arguments are based on the calculus of the tree-type graphs. Asymptotic properties of sparse random matrices essentially depend on the typical degree of a tree vertex that we show to be finite.

We observe that the Laplacian of a random graph G on N vertices represents and explicitly solvable model in the limit of infinitely increasing N. Namely, we derive recurrent relations for the limiting averaged moments of the adjacency matrix of G. These relations allow one to study the corresponding eigenvalue distribution function; we show that its density has an infinite support in contrast to the case of the ordinary discrete Laplacian.

We introduce a probabilistic model for protein sliding motion along DNA during the search of a target sequence. The model accounts for possible effects due to sequence-dependent interaction between the nonspecific DNA and the protein. As an example, we focus on T7 RNA-polymerase and exploit the available information about its interaction at the promoter site in order to investigate the influence of bacteriophage T7 DNA sequence on the dynamics of the sliding process. Hydrogen bonds in the major groove are used as the main sequence-dependent interaction between RNA-polymerase and DNA. The resulting dynamical properties and the possibility of an experimental verification are discussed in details. We show that, while at large times the process reaches a pure diffusive regime, it initially displays a sub-diffusive behavior. The subdiffusive regime can lasts sufficiently long to be of biological interest.