Exact value of the resistance exponent for four dimensional random walk trace

Abstract

Let S be a simple random walk starting at the origin in \({\mathbb{Z}^{4}}\). We consider \({{\mathcal G}=S[0,\infty)}\) to be a random subgraph of the integer lattice and assume that a resistance of unit 1 is put on each edge of the graph \({{\mathcal G}}\). Let \({R_{{\mathcal G}}(0,S_{n})}\) be the effective resistance between the origin and Sn. We derive the exact value of the resistance exponent; more precisely, we prove that \({n^{-1}E(R_{{\mathcal G}}(0,S_{n}))\approx (\log n)^{-\frac{1}{2}}}\). As an application, we obtain sharp heat kernel estimates for random walk on \({\mathcal G}\) at the quenched level. These results give the answer to the problem raised by Burdzy and Lawler (J Phys A Math Gen 23:L23–L28, 1990) in four dimensions.