{"oa":1,"department":[{"_id":"JuFi"}],"license":"https://creativecommons.org/licenses/by/4.0/","type":"journal_article","day":"14","OA_place":"publisher","date_updated":"2026-03-02T15:15:13Z","has_accepted_license":"1","status":"public","scopus_import":"1","publisher":"Springer Nature","main_file_link":[{"url":"https://doi.org/10.1007/s00440-026-01468-y","open_access":"1"}],"publication":"Probability Theory and Related Fields","_id":"21379","date_created":"2026-03-02T10:05:23Z","OA_type":"hybrid","publication_status":"epub_ahead","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).","language":[{"iso":"eng"}],"doi":"10.1007/s00440-026-01468-y","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"year":"2026","publication_identifier":{"issn":["0178-8051"],"eissn":["1432-2064"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","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."}],"ddc":["510"],"citation":{"short":"F. Otto, M. Palmieri, C. Wagner, Probability Theory and Related Fields (2026).","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>","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>.","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>","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.","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>."},"article_type":"original","quality_controlled":"1","month":"02","author":[{"first_name":"Felix","full_name":"Otto, Felix","last_name":"Otto"},{"last_name":"Palmieri","first_name":"Matteo","full_name":"Palmieri, Matteo"},{"id":"bf0c729b-2619-11f0-8024-9d69bb2b8b20","last_name":"Wagner","full_name":"Wagner, Christian","first_name":"Christian"}],"article_processing_charge":"Yes (via OA deal)","oa_version":"Published Version","corr_author":"1","date_published":"2026-02-14T00:00:00Z","title":"On minimizing curves in a Brownian potential"}