Sharp one-point estimates and Minkowski content for the scaling limit of three-dimensional loop-erased random walk

Sarai Hernandez-Torres; Xinyi Li; Daisuke Shiraishi

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arXiv · 2403.07256

Sharp one-point estimates and Minkowski content for the scaling limit of three-dimensional loop-erased random walk

Sarai Hernandez-Torres·Xinyi Li·Daisuke Shiraishi

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Abstract

In this work, we consider the scaling limit of loop-erased random walk (LERW) in three dimensions and prove that the limiting occupation measure is equivalent to its $\beta$-dimensional Minkowski content, where $\beta \in (1, 5/3]$ is its Hausdorff dimension. In doing this we also establish the existence of the two-point function and provide some sharp estimates on one-point function and ball-hitting probabilities for 3D LERW in any scale, which is a considerable improvement of previous results.

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Sarai Hernandez-Torres, Xinyi Li, Daisuke Shiraishi. 2024-03-12. Sharp one-point estimates and Minkowski content for the scaling limit of three-dimensional loop-erased random walk. https://arxiv.org/abs/2403.07256

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