The boosted loop-erased random walk
Abstract
We consider a near-critical version of loop-erased random walk (LERW) by boosting the weight of the underlying random walk according to its length before loop-erasure. We construct the scaling limit in natural parametrization in the two-dimensional unit disk and the three-dimensional unit ball. The boosted scaling limit undergoes a phase transition in mutual absolute continuity with the classical LERW scaling limit; in the disk the transition occurs at boost rate m2=π2/4. As the boost increases, the curve becomes macroscopically space-filling, in the sense that unvisited cubes have side length O(m−1), while its Hausdorff dimension at each finite boost is unchanged. These results are motivated by Makarov--Smirnov's questions on near-critical lattice models. In the planar chordal setting, we also construct a continuum loop tilt below the spectral threshold of the domain and calculate its Loewner driving function.
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BibTeX
@misc{cai2026boosted,
title = {{The boosted loop-erased random walk}},
author = {Gefei Cai and Xinyi Li and Daisuke Shiraishi and Aoteng Xia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32821},
eprint = {2610.32821},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32821},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The boosted loop-erased random walk
AU - Gefei Cai
AU - Xinyi Li
AU - Daisuke Shiraishi
AU - Aoteng Xia
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32821
UR - https://arxiv.org/abs/2610.32821
AB - We consider a near-critical version of loop-erased random walk (LERW) by boosting the weight of the underlying random walk according to its length before loop-erasure. We construct the scaling limit in natural parametrization in the two-dimensional unit disk and the three-dimensional unit ball. The boosted scaling limit undergoes a phase transition in mutual absolute continuity with the classical LERW scaling limit; in the disk the transition occurs at boost rate m^2=π^2/4. As the boost increases, the curve becomes macroscopically space-filling, in the sense that unvisited cubes have side length O(m^−1), while its Hausdorff dimension at each finite boost is unchanged. These results are motivated by Makarov--Smirnov's questions on near-critical lattice models. In the planar chordal setting, we also construct a continuum loop tilt below the spectral threshold of the domain and calculate its Loewner driving function.
ER -