Fabio Toninelli: Dimers with layered disorder

Abstract: This talk addresses the question of the effect of quenched randomness on phase transitions, in the framework of the dimer model. Specifically, we will consider the dimer model on the square grid, with random edge weights. Randomness is chosen to have a layered structure, similar to that of the celebrated McCoy–Wu disordered Ising model. Disorder has a highly non-trivial effect and it produces an essential singularity of the free energy, with \(e^{-\sqrt{\text{distance}}}\) decay of dimer-dimer correlations, at a point of the “liquid” (or “massless”) phase where the homogeneous dimer model has instead a real analytic free energy and correlations decaying like \(\frac{1}{(\text{distance})^2}\). Moreover, at a point where the homogeneous model has a transition between a massive (gaseous) and massless (liquid) phase, the critical exponent \(\frac{3}{2}\) (Pokrovsky–Talapov law), characteristic of the transition between the two regimes, is modified by disorder into an exponent that ranges continuously between \(\frac{3}{2}\) and infinity.

Based on joint work with Quentin Moulard, https://arxiv.org/abs/2507.11964.