Original title: On the rigidity of sloped height functions in $d\ge 3$ and non-crossing surfaces
Arka Adhikari, Reza Gheissari, Ron Peled · September 2026
Scientists study random stepped surfaces, called height functions, tilted at an angle. In 2003, Sheffield predicted that in dimensions d≥3, these sloped surfaces do not wander wildly. Instead, they stay rigid and localized, keeping their height variations tightly bounded even as the system grows infinitely large. The authors proved this conjecture for cases where the slope vector has zero across two coordinates, under even interactions at low temperature. They showed that these infinite surfaces exist, are unique in a specific sense, and have categorized building blocks called extremal componentsThe most extreme states in the system's possible configurations, showing distinct behaviors.. In these surfaces, large dividing walls push apart through entropic repulsionA statistical force that pushes elements apart due to chance configurations, not direct repulsion., matching a circle-rotation system. The results also describe infinitely many flat, stacked surfaces forced never to cross each other.
Imagine two groups of people walking towards each other on a narrow bridge. If they cannot cross, they naturally keep distance to avoid blocking each other, even if no one is pushing. In this paper, the height functions represent such constrained movements, and the systems exhibit a statistical tendency to keep domain walls maximally apart.
This work resolves a long-standing conjecture about how mathematical models of surfaces behave, which affects our understanding of constrained systems in probability theory and could influence models of interfaces in physics.
The proof only applies when two coordinates have zero slope and the interactions are even and at low temperature, not for all slopes or interactions as originally conjectured.
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On the rigidity of sloped height functions in $d\ge 3$ and non-crossing surfaces — Arka Adhikari, Reza Gheissari, Ron Peled
Abstract licence: CC0
Explanation generated by Synthesis from the abstract — it is not the original paper. This is a preprint, so it hasn't necessarily been peer reviewed.