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Minimal surfaces in random environment

Barbara Dembin, Dor Elboim, Daniel Hadas, Ron Peled

TL;DR

We study minimal surfaces in a random environment (MSRE) by minimizing an energy that blends elastic interactions with a random environment term, focusing on the harmonic (no-overhang) case with independent disorder. A central main identity controls energy changes under joint shifts of the surface and disorder, enabling sharp probabilistic and multiscale arguments. The paper establishes a rigorous scaling relation chi = 2 xi + d - 2 between ground energy fluctuations chi and transversal height fluctuations xi, proving delocalization in dimensions d <= 4 and localization in d >= 5, with power-law height fluctuations for d <= 3 and sub-power-law behavior at d = 4. The results connect with physical predictions, discuss disorder universality, and outline open problems and extensions including periodic disorder, scaling limits, and more general elastic operators.

Abstract

A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary conditions. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. We wish to study the geometry of $d$-dimensional minimal surfaces in a $(d+n)$-dimensional random environment. Specializing to a model that we term harmonic MSRE, in an ``independent'' random environment, we rigorously establish bounds on the geometric and energetic fluctuations of the minimal surface, as well as versions of the scaling relation $χ=2ξ+d-2$ that ties together these two types of fluctuations. In particular, we prove, for all values of $n$, that the surfaces are delocalized in dimensions $d\le 4$ and localized in dimensions $d\ge 5$. Moreover, the surface delocalizes with power-law fluctuations when $d\le 3$ and sub-power-law fluctuations when $d=4$. Our localization results apply also to harmonic minimal surfaces in a periodic random environment.

Minimal surfaces in random environment

TL;DR

We study minimal surfaces in a random environment (MSRE) by minimizing an energy that blends elastic interactions with a random environment term, focusing on the harmonic (no-overhang) case with independent disorder. A central main identity controls energy changes under joint shifts of the surface and disorder, enabling sharp probabilistic and multiscale arguments. The paper establishes a rigorous scaling relation chi = 2 xi + d - 2 between ground energy fluctuations chi and transversal height fluctuations xi, proving delocalization in dimensions d <= 4 and localization in d >= 5, with power-law height fluctuations for d <= 3 and sub-power-law behavior at d = 4. The results connect with physical predictions, discuss disorder universality, and outline open problems and extensions including periodic disorder, scaling limits, and more general elastic operators.

Abstract

A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary conditions. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. We wish to study the geometry of -dimensional minimal surfaces in a -dimensional random environment. Specializing to a model that we term harmonic MSRE, in an ``independent'' random environment, we rigorously establish bounds on the geometric and energetic fluctuations of the minimal surface, as well as versions of the scaling relation that ties together these two types of fluctuations. In particular, we prove, for all values of , that the surfaces are delocalized in dimensions and localized in dimensions . Moreover, the surface delocalizes with power-law fluctuations when and sub-power-law fluctuations when . Our localization results apply also to harmonic minimal surfaces in a periodic random environment.
Paper Structure (76 sections, 49 theorems, 317 equations, 3 figures, 2 tables)

This paper contains 76 sections, 49 theorems, 317 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The model
  3. The disorder
  4. Smoothed white noise disorder $\eta^\text{white}$.
  5. Assumptions on the disorder
  6. Effect of $\lambda$
  7. Main results
  8. Transversal fluctuations
  9. Scaling relation
  10. The scaling relation in dimension $d=1$
  11. The scaling relation assuming existence of the exponents
  12. Lower bounds on the ground energy fluctuations
  13. Background and comparison with previous work
  14. Harmonic minimal surfaces in random environment
  15. The transversal fluctuations, scaling relation and ground energy fluctuations in the physics literature
  16. ...and 61 more sections

Key Result

Theorem 1.2

There exist $C,c>0$, depending only on $d,n$ and the parameters $(K,\kappa)$ of eq:tal, such that for each integer $L\ge 1$, each $\lambda>0$ and each $v\in\Lambda_L$, where we write $r_v:=d_\infty(v,\Lambda_L^c)$. Moreover, for $d=1$, and for $d\in\{2,3\}$,

Figures (3)

  • Figure 1: A minimal surface in independent disorder with $d=2$, $n=1$.
  • Figure 2: Simulation of directed first-passage percolation with $d=n=1$.
  • Figure 3: Simulation of directed first-passage percolation with $d=1$, $n=2$.

Theorems & Definitions (93)

  • Remark 1.1
  • Theorem 1.2: Localization. \ref{['as:exiuni']}+\ref{['as:stat']}+\ref{['as:conc']}
  • Theorem 1.3: Delocalization. $\eta = \eta^\text{white}$
  • Theorem 1.4: Version of $\chi\ge2\xi+d-2$. \ref{['as:exiuni']}+\ref{['as:stat']}
  • Theorem 1.5: Version of $\chi\le2\xi+d-2$. \ref{['as:exiuni']}+\ref{['as:stat']}+\ref{['as:indep']}
  • Theorem 1.6: \ref{['as:exiuni']}+\ref{['as:stat']}
  • Theorem 1.7: \ref{['as:exiuni']}+\ref{['as:stat']}
  • Remark 1.8
  • Theorem 1.9: \ref{['as:exiuni']}+\ref{['as:stat']}+\ref{['as:indep']}
  • Remark 1.10
  • ...and 83 more