Minimal surfaces in strongly correlated random environments
Barbara Dembin, Dor Elboim, Ron Peled
TL;DR
This work rigorously analyzes minimal surfaces in strongly correlated random environments generated by fractional Brownian disorder with $H\in(0,1)$. By deriving a main deterministic identity that links shifts in the disorder to energy changes and employing Green's-function estimates and Gaussian concentration, the authors establish a tripartite dimensional behavior: for $d<4$ the surfaces delocalize with power-law fluctuations, for $d=4$ delocalize with sub-power-law fluctuations, and for $d>4$ localize. The fluctuations of height and minimal energy are governed by exponents $\xi$ and $\chi$ related by two scaling relations, $\chi=2\xi+d-2$ and $\chi=H\xi+\frac{d}{2}$, with explicit $H$-dependent expressions $\xi=\frac{4-d}{4-2H}$ and $\chi=\frac{4-d}{2-H}+d-2$ in the subcritical regime. In the critical case $d=4$, the results imply sub-power-law delocalization with logarithmic-in-$L$ corrections and a scaling relation tying energy fluctuations to height, while in $d>4$ the surfaces localize with $\chi=\frac{d}{2}$. These findings align with and rigorously ground the physics predictions for Flory exponents and dimensional transitions in correlated disordered environments, providing precise tail behavior and concentration bounds for the height and ground-energy distributions.
Abstract
A minimal surface in a random environment (MSRE) is a $d$-dimensional surface in $(d+n)$-dimensional space which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary values. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. In this work, we consider the case of strongly correlated environments, realized by the model of harmonic MSRE in a fractional Brownian environment of Hurst parameter $H\in(0,1)$. This includes the case of Brownian environment ($H=1/2$ and $n=1$), which is commonly used to approximate the domain walls of the $(d+1)$-dimensional random-field Ising model. We prove that surfaces of dimension $d\in\{1,2,3\}$ delocalize with power-law fluctuations, and determine their precise transversal and minimal energy fluctuation exponents, as well as the stretched exponential exponents governing the tail decay of their distributions. These exponents are found to be the same in all codimensions $n$, depending only on $d$ and $H$. The transversal and minimal energy fluctuation exponents are specified by two scaling relations. We further show that surfaces of dimension $d=4$ delocalize with sub-power-law fluctuations, with their height and minimal energy fluctuations tied by a scaling relation. Lastly, we prove that surfaces of dimensions $d\ge 5$ localize. These results put several predictions from the physics literature on mathematically rigorous ground.