Abstract: We study geometric and electrical properties of the trace of a simple random walk in four dimensions. Viewing the random walk path as a graph, we focus on the graph distance and the effective resistance between the starting point and the current location. Compared to related models such as loop-erased random walk, much less is known about these quantities. We present results describing their typical growth and fluctuation behavior. In particular, we identify their growth rates and show that they satisfy a weak law of large numbers but not a strong law. We also establish a non-Gaussian limiting fluctuation. A key idea in the analysis is a connection between the graph formed by self-intersections of the random walk and a long-range percolation model. The talk is based on two joint works: one with Satomi Watanabe, and another with Arka Adhikari and Izumi Okada.