Extreme-value statistics and super-universality in critical percolation?

Abstract

Recently, the number of non-standard percolation models has proliferated. In all these models, there exists a phase transition at which long range connectivity is established, if local connectedness increases through a threshold $p_c$. In ordinary (site or bond) percolation on regular lattices, this is a well understood second-order phase transition with rather precisely known critical exponents, but there are non-standard models where the transitions are in different universality classes (i.e. with different exponents and scaling functions), or even are discontinuous or hybrid. It was recently claimed that certain scaling functions are in all such models given by extreme-value theory and thus independent of the precise universality class. This would lead to super-universality (even encompassing first-order transitions!) and would be a major break-through in the theory of phase transitions. We show that this claim is wrong.