On minimizing curves in a Brownian potential

Abstract We study a (1+1) (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) (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) (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.