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Strassen's LIL and a Phase transition for the capacity of the random walk under diameter constraints
Arka Adhikari, Izumi Okada · 2025-03-17 · via math.PR updates on arXiv.org

We discuss the relationship between the capacity and the geometry for the range of the random walk for $d=3$. In particular, we consider how efficiently the random walk moves or what shape it forms in order to maximize its capacity. In one of our main results, we show a functional law for the capacity of the random walk. In addition, we find that there is a phase transition for the asymptotics of the capacity of the random walk when we condition the diameter of the random walk.

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