Self-intersections of random walks on lattices (Q1410400)

Let \((X^d_n)_{n\geq 0}\) be a uniform symmetric random walk on \(\mathbb Z^d\). Suppose that \(f:\mathbb N\to \mathbb N\) is an increasing function tending to \(\infty\) as \(n\to \infty\). The aim of the present paper is to investigate the probability of the events \[ E^d_n:=\{X^d_k:0\leq k\leq n\}\cap \{X^d_k:n+f(n)\leq k<\infty\}. \] The author proves lower and upper estimates for \(\mathbb P(E^d_n)\), \(d\geq 3\), in dependence of the function \(f\). These results extend earlier work by \textit{P. Erdős} and \textit{S. J. Taylor} [Acta Math. Acad Sci. Hung. 11, 231-248 (1960; Zbl 0096.33302)] in the case of the simple random walk. Furthermore, if \(d\geq 3\), necessary and sufficient conditions for \(f\) in order that \(\mathbb P(E^d_n)=0 \text{ or } 1\) are proved.