A violation of the Harris-Barghathi-Vojta criterion

In 1974, Harris proposed his celebrated criterion: Continuous phase transitions in $d$-dimensional systems are stable against quenched spatial randomness whenever $d\nu>2$, where $\nu$ is the clean critical exponent of the correlation length. Forty years later, motivated by violations of the Harris criterion for certain lattices such as Voronoi-Delaunay triangulations of random point clouds, Barghathi and Vojta put forth a modified criterion for topologically disordered systems: $a\nu >1$, where $a$ is the disorder decay exponent, which measures how fast coordination number fluctuations decay with increasing length scale. Here we present a topologically disordered lattice with coordination number fluctuations that decay as slowly as those of conventional uncorrelated randomness, but for which the universal behaviour is preserved, hence violating even the modified criterion.