Excluded volume effect in statistics of self-avoiding walks

The main results of investigations into the self-avoiding walk problem are reviewed. Different approaches to solving this problem are briefly discussed. Attention is paid to the asymptotic solution of the exact equation obtained for the distribution density W_n(R), where R is the vector connecting the end points of an N-step self-avoiding walk. On the basis of the single-functional approach the problem of the diffusion of tracers in a random-velocity field is discussed in the general state. Also, approximate methods allowing one to obtain approximation equations together with the conditions of their validity are discussed. The case of plane parallel average flow is considered in detail and some peculiarities of statistical solutions are discussed for the simplest problem.