Strassen's LIL and a Phase transition for the capacity of the random walk under diameter constraints

Arka Adhikari; Izumi Okada

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arXiv · 2503.12728

Strassen's LIL and a Phase transition for the capacity of the random walk under diameter constraints

Arka Adhikari·Izumi Okada

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Abstract

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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Arka Adhikari, Izumi Okada. 2025-03-17. Strassen's LIL and a Phase transition for the capacity of the random walk under diameter constraints. https://arxiv.org/abs/2503.12728

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