On the $p,q$-binomial distribution and the Ising model

TL;DR

This paper introduces a novel approach using $p,q$-binomial coefficients to model the magnetisation distributions of the Ising model across dimensions 1 to 5, showing promising fits especially in 1 and 5 dimensions.

Contribution

It develops a new $p,q$-binomial distribution framework for the Ising model and tests its effectiveness across multiple dimensions, revealing strong fits in certain cases.

Findings

Excellent fit for $d=1$ and $d=5$ dimensions

Good fit for $d=4$, with some deviations

Partial success in $d=2,3$ dimensions

Abstract

A completely new approach to the Ising model in 1 to 5 dimensions is developed. We employ $p,q$-binomial coefficients, a generalisation of the binomial coefficients, to describe the magnetisation distributions of the Ising model. For the complete graph this distribution corresponds exactly to the limit case $p=q$. We take our investigation to the simple $d$-dimensional lattices for $d=1,2,3,4,5$ and fit $p,q$-binomial distributions to our data, some of which are exact but most are sampled. For $d=1$ and $d=5$ the magnetisation distributions are remarkably well-fitted by $p,q$-binomial distributions. For $d=4$ we are only slightly less successful, while for $d=2,3$ we see some deviations (with exceptions!) between the $p,q$-binomial and the Ising distribution. We begin the paper by giving results on the behaviour of the $p,q$-distribution and its moment growth exponents given a certain parameterization of $p,q$. Since the moment exponents are known for the Ising model (or at least approximately for $d=3$) we can predict how $p,q$ should behave and compare this to our measured $p,q$. The results speak in favour of the $p,q$-binomial distribution's correctness regarding their general behaviour in comparison to the Ising model. The full extent to which they correctly model the Ising distribution is not settled though.