Dimers with layered disorder

TL;DR

This paper investigates the effects of layered quenched disorder on the square grid dimer model, revealing complex phase behavior, essential singularities, and modified critical exponents due to disorder.

Contribution

It introduces a layered disorder model for dimers, showing non-trivial effects like essential singularities and continuous variation of critical exponents.

Findings

Disorder causes an essential singularity in free energy.

Correlation decay changes from polynomial to exponential.

Critical exponent varies continuously between 3/2 and infinity.

Abstract

We study the dimer model on the square grid, with quenched 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 $e^{-\sqrt{{\rm distance}}}$ 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 $1/({\rm distance})^2$. Moreover, at a point where the homogeneous model has a transition between a massive (gaseous) and massless (liquid) phase, the critical exponent 3/2 (Pokrovsky-Talapov law), characteristic of the transition between the two regimes, is modified by disorder into an exponent that ranges continuously between 3/2 and infinity.