The Fortuin-Kasteleyn and Damage Spreading transitions in Random bond Ising lattices

TL;DR

This study investigates the relationship between Fortuin-Kasteleyn and damage spreading transition temperatures in random bond Ising models across multiple dimensions, confirming their near equivalence in three dimensions and noting systematic differences in others.

Contribution

It provides the first detailed comparison of T_{FK}(p) and T_{ds}(p) across dimensions two to five, including exact intersection points and their finite size correction properties.

Findings

In 3D, T_{ds}(p) matches T_{FK}(p) within 0.1%.

In other dimensions, T_{ds}(p) is a few percent lower than T_{FK}(p).

Exact intersection points have no finite size corrections.

Abstract

The Fortuin-Kasteleyn and heat-bath damage spreading temperatures T_{FK}(p) and T_{ds}(p) are studied on random bond Ising models of dimension two to five and as functions of the ferromagnetic interaction probability p; the conjecture that T_{ds}(p) ~ T_{FK}(p) is tested. It follows from a statement by Nishimori that in any such system exact coordinates can be given for the intersection point between the Fortuin-Kasteleyn T_{FK}(p) transition line and the Nishimori line, [p_{NL,FK},T_{NL,FK}]. There are no finite size corrections for this intersection point. In dimension three, at the intersection concentration [p_{NL,FK}] the damage spreading T_{ds}(p) is found to be equal to T_{FK}(p) to within 0.1%. For the other dimensions however T_{ds}(p) is observed to be systematically a few percent lower than T_{FK}(p).