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On minimizing curves in a Brownian potential
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DOI：10.1007/s00440-026-01468-y.png)
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Abstract
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We study a (1+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+1)$$\end{document}-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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d+1)$$\end{document}-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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p=3)$$\end{document}-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.
Keywords:
Minimizing curves
Brownian potential
Random variational problem
Homogenization
Dirichlet energy
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