Quenched Central Limit Theorem in a Corner Growth Setting

TL;DR

This paper proves a quenched central limit theorem for the energy of typical paths in a two-dimensional random environment, using concentration measures and combinatorial bounds, as the lattice mesh approaches zero.

Contribution

It establishes a quenched CLT for directed paths in a general independent environment, extending previous results to a broader setting.

Findings

Quenched energy satisfies a CLT in the given setting.

Concentration of measure techniques are effective in this context.

Combinatorial bounds are crucial for the proof.

Abstract

We consider point-to-point directed paths in a random environment on the two-dimensional integer lattice. For a general independent environment under mild assumptions we show that the quenched energy of a typical path satisfies a central limit theorem as the mesh of the lattice goes to zero. Our proofs rely on concentration of measure techniques and some combinatorial bounds on families of paths.