  In this paper we consider uniformly random lozenge tilings of arbitrary
domains approximating (after suitable normalization) a closed, simply-connected
subset of $\mathbb{R}^2$ with piecewise smooth, simple boundary. We show that
the local statistics of this model around any point in the liquid region of its
limit shape are given by the infinite-volume, translation-invariant, extremal
Gibbs measure of the appropriate slope, thereby confirming a prediction of
Cohn-Kenyon-Propp from 2001 in the case of lozenge tilings. Our proofs proceed
by locally coupling a uniformly random lozenge tiling with a model of Bernoulli
random walks conditioned to never intersect, whose convergence of local
statistics has been recently understood by the work of Gorin-Petrov. Central to
implementing this procedure is to establish a local law for the random tiling,
which states that the associated height function is approximately linear on any
mesoscopic scale.
