A Geometrical Interpretation of Hyperscaling Breaking in the Ising Model

TL;DR

This paper explores whether the breakdown of hyperscaling in the Ising model can be explained by an infinite number of percolating clusters, drawing parallels with percolation theory, but finds the scenario to be more complex than initially thought.

Contribution

It offers a geometrical interpretation of hyperscaling breaking in the Ising model through the behavior of Fortuin-Kasteleyn clusters near criticality.

Findings

Percolation variables behave differently on either side of T_c.

The scenario of infinite clusters is more complex than simple percolation analogy.

Preliminary results indicate a nuanced understanding of hyperscaling breakdown.

Abstract

In random percolation one finds that the mean field regime above the upper critical dimension can simply be explained through the coexistence of infinite percolating clusters at the critical point. Because of the mapping between percolation and critical behaviour in the Ising model, one might check whether the breakdown of hyperscaling in the Ising model can also be intepreted as due to an infinite multiplicity of percolating Fortuin-Kasteleyn clusters at the critical temperature T_c. Preliminary results suggest that the scenario is much more involved than expected due to the fact that the percolation variables behave differently on the two sides of T_c.