Geometric structures of late points of a two-dimensional simple random walk

As Dembo (In Lectures on Probability Theory and Statistics (2005) 1-101 Springer, and International Congress of Mathematicians, Vol. III (2006) 535-558, Eur. Math. Soc.) suggested, we consider the problem of late points for a simple random walk in two dimensions. It has been shown that the exponents for the number of pairs of late points coincide with those of favorite points and high points in the Gaussian free field, whose exact values are known. We determine the exponents for the number of j-tuples of late points on average.