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Anomalous scaling limit of a Brownian particle in a log-correlated potential

  • Scott Armstrong
  • Ahmed Bou-Rabee
  • Tuomo Kuusi
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Abstract

We study a Brownian particle in ℝd, d≥2, with drift given by the gradient of a log-correlated Gaussian potential. At weak disorder, we prove convergence to a scaling limit whose law is singular with respect to Brownian motion and whose invariant measure is given by Gaussian multiplicative chaos. The proof uses a renormalization group induction: at each scale, the elliptic operator is well approximated by a Laplacian with effective diffusivity decaying as a power of scale. We compute this exponent exactly in dimension two.

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Cite

BibTeXRIS
BibTeX
@misc{armstrong2026anomalous,
  title = {{Anomalous scaling limit of a Brownian particle in a log-correlated potential}},
  author = {Scott Armstrong and Ahmed Bou-Rabee and Tuomo Kuusi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12335},
  eprint = {2610.12335},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12335},
}
RIS
TY  - UNPB
TI  - Anomalous scaling limit of a Brownian particle in a log-correlated potential
AU  - Scott Armstrong
AU  - Ahmed Bou-Rabee
AU  - Tuomo Kuusi
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12335
UR  - https://arxiv.org/abs/2610.12335
AB  - We study a Brownian particle in ℝ^d, d≥2, with drift given by the gradient of a log-correlated Gaussian potential. At weak disorder, we prove convergence to a scaling limit whose law is singular with respect to Brownian motion and whose invariant measure is given by Gaussian multiplicative chaos. The proof uses a renormalization group induction: at each scale, the elliptic operator is well approximated by a Laplacian with effective diffusivity decaying as a power of scale. We compute this exponent exactly in dimension two.
ER  -