We study the fluctuation problem for the multiple point range of random walks in the two dimensional integer lattice with mean 0 and finite variance. The $p$-multiple point range means the number of distinct sites with multiplicity $p$ of random walk paths before time $n$. The suitably normalized multiple point range is proved to converge to a constant, which is independent of the multiplicity, multiple of the renormalized self-intersection local time of a planar Brownian motion.
"The fluctuation result for the multiple point range of two-dimensional recurrent random walks." Ann. Probab. 25 (2) 598 - 639, April 1997. https://doi.org/10.1214/aop/1024404413