The dimer model on a bipartite graph \( G \subset \mathbb{Z}^2 \) refers to choosing a random dimer cover of \( G \) (equivalently, a random domino tiling on its dual graph \( G^\star \). Each such dimer cover is associated with a height function defined on the vertices of \( G^\star \). If we let \( \Omega_\delta \subset \delta\mathbb{Z}^2 \) be graphs of mesh size \( \delta \) which approximate a domain ( \Omega \subset \C ), then the Kenyon–Okounkov conjecture states that fluctuations of the height function on \( \Omega_\delta^\star \) converge in distribution as \( \delta \downarrow 0 \) to the sum of the Gaussian Free Field on \( \Omega \) and a random harmonic function.

After describing earlier results which prove the Kenyon–Okounkov conjecture for certain discrete approximations, we will introduce the dimer model on cylindrical domains with Temperleyan boundary components of different colors. In this doubly connected setup, we confirm the Kenyon–Okounkov conjecture from general principles rather than explicit computations. Time permitting, we will discuss the main components of our proof. This talk is based on joint work with Dmitry Chelkak.