On minimizing curves in a Brownian potential

Abstract

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 [12] and [13, Section 5]; our analysis is reminiscent of the multi-scale approach of the latter work and of [20]. 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 [9] 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.