Criticality and conformality in the random dimer model

TL;DR

This study investigates how localized perturbations affect the random dimer model in 2D, revealing conformal invariance and different excitation behaviors depending on lattice bipartiteness.

Contribution

It provides the first detailed numerical analysis of excitations in the 2D random dimer model, linking their properties to conformal invariance and SLE processes.

Findings

Excitations have fractal dimensions and scaling exponents.

Non-bipartite lattices exhibit domain wall statistics similar to 2D spin glasses.

Bipartite lattices show loop-erased self-avoiding walk behavior.

Abstract

In critical systems, the effect of a localized perturbation affects points that are arbitrarily far from the perturbation location. In this paper, we study the effect of localized perturbations on the solution of the random dimer problem in $2D$. By means of an accurate numerical analysis, we show that a local perturbation of the optimal covering induces an excitation whose size is extensive with finite probability. We compute the fractal dimension of the excitations and scaling exponents. In particular, excitations in random dimer problems on non-bipartite lattices have the same statistical properties of domain walls in the $2D$ spin glass. Excitations produced in bipartite lattices, instead, are compatible with a loop-erased self-avoiding random walk process. In both cases, we find evidence of conformal invariance of the excitations that is compatible with $\mathrm{SLE}_\kappa$ with parameter $\kappa$ depending on the bipartiteness of the underlying lattice only.