A class of random fields associated with multiple points of a random walk in the plane is studied. It is proved that these fields converge in distribution to analogous fields measuring self-intersections of the planar Brownian motion. The concluding section contains a survey of literature on intersection local times and their renormalizations. A brief look through the first pages of this section could provide the reader with additional motivation for the present work.
E. B. Dynkin. "Self-Intersection Gauge for Random Walks and for Brownian Motion." Ann. Probab. 16 (1) 1 - 57, January, 1988. https://doi.org/10.1214/aop/1176991884