Central limit theorem for T-graphs

TL;DR

This paper proves a quenched invariance principle for random walks on T-graphs, showing they converge to Brownian motion with isotropic covariance, despite the graphs' quasi-periodic structure and limited symmetry.

Contribution

It establishes a quenched central limit theorem for random walks on T-graphs, linking the result to lozenge tilings and discrete harmonic functions.

Findings

Random walk converges to isotropic Brownian motion.

Covariance matrix is proportional to the identity.

Results apply to quasi-periodic, aperiodic graphs.

Abstract

In this paper, we establish a quenched invariance principle for the random walk on a certain class of infinite, aperiodic, oriented random planar graphs called "T-graphs" [Kenyon-Sheffield04]. These graphs appear, together with the corresponding random walk, in a work [Kenyon07] about the lozenge tiling model, where they are used to compute correlations between lozenges inside large finite domains. The random walk in question is balanced, i.e. it is automatically a martingale. Our main ideas are inspired by the proof of a quenched central limit theorem in stationary ergodic environment on $\mathbb{Z}^2$ [Lawler82, Sznitman02]. This is somewhat surprising, since the environment is neither defined on $\mathbb{Z}^2$ nor really random: the graph is instead quasi-periodic and all the randomness is encoded in a single random variable {\lambda} that is uniform in the unit circle. We prove that the covariance matrix of the limiting Brownian Motion is proportional to the identity, despite the fact that the graph does not have obvious symmetry properties. This covariance is identified using the knowledge of a specific discrete harmonic function on the graph, which is provided by the link with lozenge tilings.