Graph distance and effective resistance of the four-dimensional random walk trace

Refining previous results, we establish a sharp asymptotic estimate on the expected graph distance between the origin and the terminal point of the trace of the first $n$ steps of the walk. A similar conclusion is drawn for the resistance metric.

💡 Research Summary

The paper studies the geometric properties of the trace graph generated by the first n steps of a simple random walk on the four‑dimensional integer lattice ℤ⁴. Specifically, it focuses on two quantities that capture the shape of this random subgraph: the graph distance Dₙ between the origin and the endpoint Sₙ, and the effective electrical resistance Rₙ between the same two vertices. In dimensions d ≥ 5 these quantities grow linearly in n, but d = 4 is the critical dimension where logarithmic corrections appear. Earlier work (notably