Limit shape of minimal difference partitions and fractional statistics

TL;DR

This paper studies the limit shape of minimal difference partitions with variable gaps, extending previous models to include fractional statistics and providing a unified framework with new asymptotic results.

Contribution

It introduces a generalized model for minimal difference partitions with variable gaps, confirming the limit shape for all gap values and deriving asymptotics for the number of parts.

Findings

Confirmed the limit shape for all gap parameters q.

Derived asymptotics for the number of parts in the partitions.

Unified the fractional statistics interpretation with a generalized model.

Abstract

The class of minimal difference partitions MDP($q$) (with gap $q$) is defined by the condition that successive parts in an integer partition differ from one another by at least $q\ge 0$. In a recent series of papers by A. Comtet and collaborators, the MDP($q$) ensemble with uniform measure was interpreted as a combinatorial model for quantum systems with fractional statistics, that is, interpolating between the classic Bose-Einstein ($q=0$) and Fermi-Dirac ($q=1$) cases. This was done by formally allowing values $q \in (0,1)$ using an analytic continuation of the limit shape of the corresponding Young diagrams calculated for integer $q$. To justify this "replica-trick", we introduce a more general model based on a variable MDP-type condition encoded by an integer sequence $(q_i)$, whereby the (limiting) gap $q$ is naturally interpreted as the Ces\`aro mean of $(q_i)$. In this model, we find the family of limit shapes parameterized by $q \in [0,\infty)$ confirming the earlier answer, and also obtain the asymptotics of the number of parts.