Excursions of excited random walks on integers

Several phase transitions for excited random walks on the integers are known to be characterized by a certain drift parameter delta is an element of R. For recurrence/transience the critical threshold is vertical bar delta vertical bar = 1, for ballisticity it is vertical bar delta vertical bar = 2 and for diffusivity vertical bar delta vertical bar = 4. In this paper we establish a phase transition at vertical bar delta vertical bar = 3. We show that the expected return time of the walker to the starting point, conditioned on return, is finite iff vertical bar delta vertical bar > 3. This result follows from an explicit description of the tail behaviour of the return time as a function of delta, which is achieved by diffusion approximation of related branching processes by squared Bessel processes.