The range of random walks means the number of distinct sites visited at least once by the random walk. In two-or-more-dimensional cases, we established the law of large numbers for the range of simple symmetric random walks under the conditional probability given the event that the last point is the origin. Moreover we studied the large deviations in the upward direction and obtained similar results to the original random walk.
"On the range of pinned random walks." Tohoku Math. J. (2) 58 (3) 329 - 357, 2006. https://doi.org/10.2748/tmj/1163775134