An Ising model having permutation spin motivated by a permutation complexity measure

TL;DR

This paper introduces a novel permutation-based Ising model inspired by a complexity measure for declarative systems, extending classical models to permutations and analyzing elementary properties in one dimension.

Contribution

It defines a permutation-based Ising model motivated by complexity measures, extending classical models to include permutation spins and analyzing its properties.

Findings

Defined a permutation-based Ising model with a new energy function.

Proved elementary properties for the 1D case with length-3 permutations.

Connected the model to classical Ising when permutations are of length 2.

Abstract

In this paper we define a variant of the Ising model in which spins are replaced with permutations. The energy between two spins is a function of the relative disorder of one spin, a permutation, to the other. This model is motivated by a complexity measure for declarative systems. For such systems a state is a permutation and the permutation sorting complexity measures the average sequential disorder of neighbouring states. To measure the relative disorder between two spins we use a symmetrized version of the descent permutation statistic that has appeared in the works of Chatterjee \& Diaconis and Petersen. The classical Ising model corresponds to the length-2 permutation case of this new model. We consider and prove some elementary properties for the 1D case of this model in which spins are length-3 permutations.