Superconcentration in surface growth

TL;DR

This paper introduces the concept of subroughness and proves its equivalence to superconcentration in surface growth models, providing new insights and proofs for superconcentration in various models.

Contribution

It establishes the equivalence between superconcentration and subroughness, and applies this to prove superconcentration in several surface growth models.

Findings

Superconcentration is equivalent to subroughness in surface growth.

Superconcentration is established in RSOS and ballistic deposition models.

New proofs of superconcentration in last-passage percolation and directed polymers.

Abstract

Height functions of growing random surfaces are often conjectured to be superconcentrated, meaning that their variances grow sublinearly in time. This article introduces a new concept, called subroughness, meaning that there exist two distinct points such that the expected squared difference between the heights at these points grows sublinearly in time. The main result of the paper is that superconcentration is equivalent to subroughness in a class of growing random surfaces. The result is applied to establish superconcentration in a variant of the restricted solid-on-solid (RSOS) model and in a variant of the ballistic deposition model, and give new proofs of superconcentration in directed last-passage percolation and directed polymers.