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.