Complete graph and Gaussian fixed point asymptotics in the five-dimensional Fortuin-Kasteleyn Ising model with periodic boundaries

TL;DR

This study uses Monte Carlo simulations to analyze five-dimensional Fortuin-Kasteleyn Ising models, revealing that both complete graph and Gaussian fixed point asymptotics are essential for understanding finite-size scaling behavior.

Contribution

It demonstrates that both complete graph and Gaussian fixed point asymptotics are necessary to fully describe the finite-size scaling in five-dimensional FK-Ising models.

Findings

Physical quantities with largest cluster contribution follow complete graph asymptotics.

Quantities excluding the largest cluster are governed by Gaussian fixed point behavior.

Both asymptotics are required for a complete scaling description.

Abstract

We present an extensive Markov-chain Monte Carlo study of the finite-size scaling behavior of the Fortuin-Kasteleyn Ising model on five-dimensional hypercubic lattices with periodic boundary conditions. We observe that physical quantities, which include the contribution of the largest cluster, exhibit complete graph asymptotics. However, for quantities, where the contribution of the largest cluster is removed, we observe that the scaling behavior is mainly controlled by the Gaussian fixed point. Our results therefore suggest that \textit{both} scaling predictions, i.e. the complete graph \textit{and} the Gaussian fixed point asymptotics, are needed to provide a complete description for the five-dimensional finite-size scaling behavior on the torus.