Critical behavior of the Ising model on a lattice with fractional space dimension

Disorder can have a drastic effect on the critical behavior of a magnetic system. The Harris criterion states that if the critical exponent of the correlation length $\nu$ fulfills the inequality $\nu \ge 2/d$, with $d$ the space dimension, disorder does not affect the universality class of the magnetic systems. A recent study reported a violation of this criterion for a two-dimensional three-state Potts model on a Voronoi lattice. To better understand the effects of disorder on the critical behavior of magnetic systems on disordered lattices, we study the critical behavior of a two-dimensional Ising ferromagnet on the largest component of a percolating cluster on a two-dimensional square lattice. There are two possible scenarios: In the weak universality scenario the disordered structure of the underlying lattice slowly changes the critical exponents, whereas in the strong universality scenario the critical exponents are not affected by the fractional space dimension of the system. Our results suggest a strong universality scenario with weak (logarithmic) corrections.