A. M. Povolotsky

On the integrability of zero-range chipping models with factorized steady states

Conditions of integrability of general zero range chipping models with factorized steady state, which were proposed in [Evans, Majumdar, Zia 2004 J. Phys. A 37 L275], are examined. We find a three-parametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle models as limiting cases. The solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations, which is proved as a byproduct. We use the Bethe ansatz to solve an eigenproblem for the transition matrix of the Markov process. On its basis we conjecture an integral formula for the Green function of evolution operator for the model on an infinite lattice and derive the Bethe equations for the spectrum of the model on a ring.

Determinant solution for the totally asymmetric exclusion process with parallel update

We consider the totally asymmetric exclusion process in discrete time with the parallel update. Constructing an appropriate transformation of the evolution operator, we reduce the problem to that solvable by the Bethe ansatz. The nonstationary solution of the master equation for the infinite 1D lattice is obtained in a determinant form. Using a modified combinatorial treatment of the Bethe ansatz, we give an alternative derivation of the resulting determinant expression.

Bethe ansatz solution of zero-range process with nonuniform stationary state.

The eigenfunctions and eigenvalues of the master equation for zero-range process with totally asymmetric dynamics on a ring are found exactly using the Bethe ansatz weighted with the stationary weights of particle configurations. The Bethe ansatz applicability requires the rates of hopping of particles out of a site to be the q numbers [n](q). This is a generalization of the rates of hopping of noninteracting particles equal to the occupation number n of a site of departure. The noninteracting case can be restored in the limit q-->1. The limiting cases of the model for q=0, infinity correspond to the totally asymmetric exclusion process, and the drop-push model, respectively. We analyze the partition function of the model and apply the Bethe ansatz to evaluate the generating function of the total distance traveled by particles at large time in the scaling limit. In case of nonzero interaction, q not equal 1, the generating function has the universal scaling form specific for the Kardar-Parizi-Zhang universality class.

Determinant solution for the Totally Asymmetric Exclusion Process with parallel update II. Ring geometry.

Using the Bethe ansatz we obtain the determinant expression for the time dependent transition probabilities in the totally asymmetric exclusion process with parallel update on a ring. Developing a method of summation over the roots of Bethe equations based on the multidimensional analogue of the Cauchy residue theorem, we construct the resolution of the identity operator, which allows us to calculate the matrix elements of the evolution operator and its powers. Representation of results in the form of an infinite series elucidates connection to other results obtained for the ring geometry. As a by-product we also obtain the generating function of the joint probability distribution of particle configurations and the total distance traveled by the particles.

Bethe Ansatz Solution of Discrete Time Stochastic Processes with Fully Parallel Update

We present the Bethe ansatz solution for the discrete time zero range and asymmetric exclusion processes with fully parallel dynamics. The model depends on two parameters: p, the probability of single particle hopping, and q, the deformation parameter, which in the general case, |q| < 1, is responsible for long range interaction between particles. The particular case q = 0 corresponds to the Nagel-Schreckenberg traffic model with vmax = 1. As a result, we obtain the largest eigenvalue of the equation for the generating function of the distance travelled by particles. For the case q = 0 the result is obtained for arbitrary size of the lattice and number of particles. In the general case we study the model in the scaling limit and obtain the universal form specific for the Kardar-Parisi-Zhang universality class. We describe the phase transition occurring in the limit p→ 1 when q < 0.

DYNAMICS OF EULERIAN WALKERS

We investigate the dynamics of Eulerian walkers as a model of self-organized criticality. The evolution of the system is subdivided into characteristic periods which can be seen as avalanches. The structure of avalanches is described and the critical exponent in the distribution of first avalanches

Organization of complex networks without multiple connections.

\tau=2

Finite size behavior of the asymmetric avalanche process

is determined. We also study a mean square displacement of Eulerian walkers and obtain a simple diffusion law in the critical state. The evolution of underlying medium from a random state to the critical one is also described.

From Vicious Walkers to TASEP

We find a new structural feature of equilibrium complex random networks without multiple and self-connections. We show that if the number of connections is sufficiently high, these networks contain a core of highly interconnected vertices. The number of vertices in this core varies in the range between const x N1/2 and const x N2/3, where is the number of vertices in a network. At the birth point of the core, we obtain the size-dependent cutoff of the distribution of the number of connections and find that its position differs from earlier estimates.

Transition from Kardar-Parisi-Zhang to tilted interface critical behavior in a solvable asymmetric avalanche model

We study the behavior of particle flow in the asymmetric avalanche process with partially asymmetric diffusion below the line separating phases of intermittent and continuous flow. Besides the average velocity of flow, that can be obtained in the limit of infinite system size, we obtain the other quantities, such as the dispersion of flow, that does not survive in the thermodynamic limit. Particularly, the generating function of distance travelled by particles is shown to have universal form, specific for Kardar–Parisi–Zhang universality class. To obtain these quantities we apply the method of calculation of finite size corrections to the infinite system size solution based on the Bethe ansatz solution of master equation.