A (conjectural) 1/3-phenomenon for the number of rhombus tilings of a hexagon which contain a fixed rhombus

Abstract

We state, discuss, provide evidence for, and prove in special cases the conjecture that the probability that a random tiling by rhombi of a hexagon with side lengths $2n+a,2n+b,2n+c,2n+a,2n+b,2n+c$ contains the (horizontal) rhombus with coordinates $(2n+x,2n+y)$ is equal to $1/3+g_{a,b,c,x,y}(n){\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}^3/\left(\genfrac{}{}{0ex}{}{6n}{3n}\right)$, where $g_{a,b,c,x,y}(n)$ is a rational function in $n$. Several specific instances of this "1/3-phenomenon" are made explicit.

16 pages, AmS-LaTeX, uses TeXDraw; a few typos corrected