Year: 2026
Type: preprint
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 $m^2=π^2/4$. 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.
Source: arXiv (Cornell University)
Institutions: Peking University, Kyoto University, Central South University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223698501