Year: 2019
Type: article
Abstract: The class of minimal difference partitions$$\text {MDP}(q)$$ MDP(q) (with gap q) is defined by the condition that successive parts in an integer partition differ from one another by at least $$q\ge 0$$ q≥0. In a recent series of papers by A. Comtet and collaborators, the $$\text {MDP}(q)$$ MDP(q) ensemble with uniform measure was interpreted as a combinatorial model for quantum systems with fractional statistics, that is, interpolating between the classical Bose–Einstein ($$q=0$$ q=0) and Fermi–Dirac ($$q=1$$ q=1) cases. This was done by formally allowing values $$q\in (0,1)$$ q∈(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 $${\mathfrak {q}}=(q_i)$$ q=(qi), whereby the (limiting) gap q is naturally interpreted as the Cesàro mean of $${\mathfrak {q}}$$ q. In this model, we find the family of limit shapes parameterized by $$q\in [0,\infty )$$ q∈[0,∞) confirming the earlier answer, and also obtain the asymptotics of the number of parts.
Authors: Leonid V. Bogachev, Yuri V. Yakubovich
Institutions: University of Leeds, St Petersburg University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,690
Funder: University of Leeds
OpenAlex ID: W2891366985