article epub_ahead yes Felix Otto author Matteo Palmieri author Christian Wagner author bf0c729b-2619-11f0-8024-9d69bb2b8b20 JuFi department 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. https://creativecommons.org/licenses/by/4.0/ Springer Nature2026 eng 0178-8051 1432-206410.1007/s00440-026-01468-y 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> F. Otto, M. Palmieri, C. Wagner, Probability Theory and Related Fields (2026). Otto F, Palmieri M, Wagner C. 2026. On minimizing curves in a Brownian potential. Probability Theory and Related Fields. 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>. F. Otto, M. Palmieri, and C. Wagner, “On minimizing curves in a Brownian potential,” <i>Probability Theory and Related Fields</i>. Springer Nature, 2026. 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> 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>. 213792026-03-02T10:05:23Z2026-03-02T15:15:13Z