Consider a hexagon overlayed on a regular triangular grid, where the equilateral triangles have sidelength 1. A "lozenge" is a pair of adjacent triangles, and there are three types of lozenges that can be built on this grid. Using these lozenges, one can tile the entire hexagon. Lozenge tilings of a regular hexagon are in bijection with boxed plane partitions (i.e. stacks of boxes in the back of a cubic room) and can therefore be assigned a volume; a fact that is best illustrated by staring at a picture of one such tiling. The q^(Volume) tiling model is a measure on the space of tilings of the hexagon which assigns to each tiling a probability proportional to q^(Volume), where q is a real parameter. In this talk, I will recall the model and basic result about it and propose an approach to studying its statistical properties as the size of the hexagon grows by analyzing a related family of non-Hermitian orthogonal polynomials. The talk is based on ongoing joint work with Maurice Duits.