On minimizing curves in a Brownian potential

TL;DR

This work analyzes a (1+1)-dimensional semi-discrete variational problem with energy $E(h)=D(h)-W(h)$, where $D(h)$ is a discrete Dirichlet energy and $W(h)$ is a Brownian-field term, viewed as a geometrically linearized proxy for the 2D random-field Ising model. The authors develop a multi-scale (dyadic) decomposition of minimizers, proving that at each scale the corresponding Dirichlet energy per unit length is order one in a $p$-Dirichlet sense for all $2\le p<3$, and that aggregate scale contributions grow only like a logarithm. A key outcome is a quenched homogenization result: the minimal energy per unit length converges to a deterministic constant and the leading-order Dirichlet energy and Gaussian supremum scale with $L$ in a precise, scale-consistent way. The paper thus provides fine scale-by-scale control that is robust to lattice cut-offs and yields tightness of minimizers, offering a rigorous justification for the geometric linearization and advancing understanding of correlation-length behavior in random-field models. The methods combine concentration, scale decomposition, and a two-scale construction to bridge micro- and macro-scales, with implications for continuum limits and energy fluctuations in disordered systems.

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.

Figures (3)

• Figure 1: Projection operator with $L = 16, l = 2$: $h$ in black, $h_{\ge l}$ in violet, $h_{\ge 2l}$ in red, $h_l$ dotted in violet.
• Figure 2: Minimizing configuration $\sigma_*$ (left). Map from $\sigma_*$ (up right) to the flat configuration $\sigma_0$ (down right) in the violet flat rectangle $[0,L]\times[-H,H]$.
• Figure 3: Step 3 of the construction (dotted, in blue) in the case $W(T'_{\frac{L}{2}, 1}), W(T'_{\frac{L}{4},1}) > 0$, in green, and $W(T'_{\frac{L}{4},2}) < 0$, in red; see also dw for the more complicated two-dimensional geometric problem.

Theorems & Definitions (24)

• Lemma 1
• Lemma 2
• Lemma 3: dw
• Theorem 1
• Corollary 1
• Theorem 2
• Proposition 1
• Proposition 2
• Lemma 4
• Lemma 5