A connection between the random pinning model and random walks in sparse random environments

The purpose of this short note is to establish a connection between a one-dimensional random walk in a random sparse environment and the random pinning model. We show that the grand canonical partition function of the pinning model coincides with the mean number of returns to the origin for a random walk in a random sparse environment averaged on the randomness location. We obtain thereof some information on the integrability of the number of return times in the annealed and partially annealed setups.