Loop-erased random walks associated with Markov processes

A new class of loop-erased random walks (LERW) on a finite set, defined as functionals from a Markov chain is presented. We propose a scheme in which, in contrast to the general settings of LERW, the loop-erasure is performed on a non-markovian sequence and moreover, not all loops are erased with necessity. We start with a special example of a random walk with loops, the number of which at every moment of time does not exceed a given fixed number. Further we consider loop-erased random walks, for which loops are erased at random moments of time that are hitting times for a Markov chain. The asymptotics of the normalized length of such loop-erased walks is established. We estimate also the speed of convergence of the normalized length of the loop-erased random walk on a finite group to the Rayleigh distribution.