This paper considers a bivariate random walk model on a rectangular lattice for a particle injected into a fluid flowing in a tank. The numbers of jumps of the particle in the x and y directions in this particular model are correlated. It is shown that when the random walk forms a bivariate Markov chain in continuous time, it is possible to obtain the state probabilities pxy(t) through their Laplace transforms. Two exit rules are considered and results for both of them derived.

The hypercubic lattice with bonds of random electrical resistance affords a model for the specific conductivity of microscopically irregular material. The resulting stochastic process lies between a subadditive and a superadditive process: mesoadditive processes of this type provide several unsolved problems for pure mathematicians.

Let Γn(p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where . Denote by γ n(p) the total number of cycles in the graph Γn(p). The main object of this paper is to prove that the limit distribution of γ n(λ /n) is a Poisson distribution with expectation if n → ∞and 0 < λ< 1.