Abstract
The class of minimal difference partitionstext {MDP}(q) (with gap q) is defined by the condition that successive parts in an integer partition differ from one another by at least qge 0. In a recent series of papers by A. Comtet and collaborators, the text {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) and Fermi–Dirac (q=1) cases. This was done by formally allowing values qin (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), whereby the (limiting) gap q is naturally interpreted as the Cesàro mean of {mathfrak {q}}. In this model, we find the family of limit shapes parameterized by qin [0,infty ) confirming the earlier answer, and also obtain the asymptotics of the number of parts.
Highlights
IntroductionThe special case of the MDP(q) model with a constant gap sequence qi ≡ q ∈ N0 in (1.1) was considered in a series of papers by Comtet et al [8,9,10] in connection with fractional exclusion statistics of quantum particle systems (see [23,26] or [28] for a “physical” introduction to this area)
An integer partition is a decomposition of a given natural number into an unordered sum of integers; for example, 35 = 8 + 6 + 6 + 5 + 4 + 2 + 2 + 1 + 1
The non-zero terms λi ∈ λ are called the parts of the partition λ
Summary
The special case of the MDP(q) model with a constant gap sequence qi ≡ q ∈ N0 in (1.1) was considered in a series of papers by Comtet et al [8,9,10] in connection with fractional exclusion statistics of quantum particle systems (see [23,26] or [28] for a “physical” introduction to this area). These authors obtained the limit shape of MDP(q) using a physical argumentation. The Appendix contains proof of the two technical propositions stated in Sect. 2, which establish the equivalence of ensembles
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