---
OA_place: publisher
OA_type: hybrid
_id: '21379'
abstract:
- lang: eng
  text: We study a (1 + 1)-dimensional semi-discrete random variational problem that
    can be interpreted as the geometrically linearized version of the critical 2-dimensional
    random field Ising model. The scaling of the correlation length of the latter
    was recently characterized in Probab. Duke Math. J. 172(9), 1781–1811 (2023) and
    arXiv:2011.08768v3, (2022); our analysis is reminiscent of the multi-scale approach
    of the latter work and of Combinatorica 9, 161–187 (1989) . We show that at every
    dyadic scale from the system size down to the lattice spacing the minimizer contains
    at most order-one Dirichlet energy per unit length. We also establish a quenched
    homogenization result in the sense that the leading order of the minimal energy
    becomes deterministic as the ratio system size / lattice spacing diverges. To
    this purpose we adapt arguments from arXiv:2401.06768, (2024) on the (d + 1)-dimensional
    version our the model, with a Brownian replacing the white noise potential, to
    obtain the initial large-scale bounds. Based on our estimate of the (p = 3)-Dirichlet
    energy, we give an informal justification of the geometric linearization. Our
    bounds, which are oblivious to the microscopic cut-off scale provided by the lattice
    spacing, yield tightness of the law of minimizers in the space of continuous functions
    as the lattice spacing is sent to zero.
acknowledgement: FO and CW thank Ron Peled for insightful discussions on the white-noise
  multi-dimensional case in the Fall of 2023. CW thanks Barbara Dembin for the discussion
  during a workshop in Spring 2025. The work was done while the authors were affiliated
  with the Max Planck Institute for Mathematics in the Sciences; CW thanks the MPI
  for the support and warm hospitality. Open access funding provided by Institute
  of Science and Technology (IST Austria).
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Felix
  full_name: Otto, Felix
  last_name: Otto
- first_name: Matteo
  full_name: Palmieri, Matteo
  last_name: Palmieri
- first_name: Christian
  full_name: Wagner, Christian
  id: bf0c729b-2619-11f0-8024-9d69bb2b8b20
  last_name: Wagner
citation:
  ama: Otto F, Palmieri M, Wagner C. On minimizing curves in a Brownian potential.
    <i>Probability Theory and Related Fields</i>. 2026. doi:<a href="https://doi.org/10.1007/s00440-026-01468-y">10.1007/s00440-026-01468-y</a>
  apa: Otto, F., Palmieri, M., &#38; Wagner, C. (2026). On minimizing curves in a
    Brownian potential. <i>Probability Theory and Related Fields</i>. Springer Nature.
    <a href="https://doi.org/10.1007/s00440-026-01468-y">https://doi.org/10.1007/s00440-026-01468-y</a>
  chicago: Otto, Felix, Matteo Palmieri, and Christian Wagner. “On Minimizing Curves
    in a Brownian Potential.” <i>Probability Theory and Related Fields</i>. Springer
    Nature, 2026. <a href="https://doi.org/10.1007/s00440-026-01468-y">https://doi.org/10.1007/s00440-026-01468-y</a>.
  ieee: F. Otto, M. Palmieri, and C. Wagner, “On minimizing curves in a Brownian potential,”
    <i>Probability Theory and Related Fields</i>. Springer Nature, 2026.
  ista: Otto F, Palmieri M, Wagner C. 2026. On minimizing curves in a Brownian potential.
    Probability Theory and Related Fields.
  mla: Otto, Felix, et al. “On Minimizing Curves in a Brownian Potential.” <i>Probability
    Theory and Related Fields</i>, Springer Nature, 2026, doi:<a href="https://doi.org/10.1007/s00440-026-01468-y">10.1007/s00440-026-01468-y</a>.
  short: F. Otto, M. Palmieri, C. Wagner, Probability Theory and Related Fields (2026).
corr_author: '1'
date_created: 2026-03-02T10:05:23Z
date_published: 2026-02-14T00:00:00Z
date_updated: 2026-03-02T15:15:13Z
day: '14'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00440-026-01468-y
has_accepted_license: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1007/s00440-026-01468-y
month: '02'
oa: 1
oa_version: Published Version
publication: Probability Theory and Related Fields
publication_identifier:
  eissn:
  - 1432-2064
  issn:
  - 0178-8051
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: On minimizing curves in a Brownian potential
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2026'
...