Dimer models (random lozenge or domino tilings) on large planar domains exhibit universality behavior: local convergence to translation-invariant Gibbs measures, global fluctuations described by the Gaussian Free Field (GFF), and Airy line ensemble at the edges. In this talk, I discuss two mechanisms that break this universality while preserving some exactly solvable structure. First, applying a strong double-well potential parallel to one of the triangular lattice directions induces a new "waterfall" phase in lozenge tilings, where the 2D Gibbs structure collapses into a new 1D process with an emergent period-two structure. The exact solvability is powered by the q-Racah orthogonal polynomials. Second, randomizing edge weights in Aztec domino tilings (in a diagonally layered manner) deforms limit shapes. Moreover, it leads to non-GFF Brownian motion-like fluctuations living on root-N scale or the same constant scale as the GFF, depending on the variance scaling in the random edge weights. The exact solvability here comes from explicit annealed Schur generating functions.
This is based on joint works with Knizel and Bufetov-Zografos.