PHASE-TRANSITION AND LAW OF LARGE NUMBERS FOR A NONSYMMETRICAL ONE-DIMENSIONAL RANDOM-WALK WITH SELF-INTERACTIONS

We study a not necessarily symmetric random walk with interactions on Z, which is an extension of the one-dimensional discrete version of th e sausage Wiener path measure. We prove the existence of a repulsion/a ttraction phase transition for the critical value lambda(c) = -mu of t he repulsion coefficient lambda, where mu is a drift parameter. In the self-repellent case, we determine the escape speed, as a function of lambda and mu, and we prove a law of large numbers for the end-point.