  We consider uniformly random lozenge tilings of simply connected polygons
subject to a technical assumption on their limit shape. We show that the edge
statistics around any point on the arctic boundary, that is not a cusp or
tangency location, converge to the Airy line ensemble. Our proof proceeds by
locally comparing these edge statistics with those for a random tiling of a
hexagon, which are well understood. To realize this comparison, we require a
nearly optimal concentration estimate for the tiling height function, which we
establish by exhibiting a certain Markov chain on the set of all tilings that
preserves such concentration estimates under its dynamics.
