TY - JOUR

T1 - Limit Shape of Minimal Difference Partitions and Fractional Statistics

AU - Bogachev, Leonid V.

AU - Yakubovich, Yuri V.

PY - 2019

Y1 - 2019

N2 - The class of minimal difference partitionsMDP (q) (with gap q) is defined by the condition that successive parts in an integer partition differ from one another by at least q≥ 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 classical Bose–Einstein (q= 0) and Fermi–Dirac (q= 1) cases. This was done by formally allowing values 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 q= (qi) , whereby the (limiting) gap q is naturally interpreted as the Cesàro mean of q. In this model, we find the family of limit shapes parameterized by q∈ [0 , ∞) confirming the earlier answer, and also obtain the asymptotics of the number of parts.

AB - The class of minimal difference partitionsMDP (q) (with gap q) is defined by the condition that successive parts in an integer partition differ from one another by at least q≥ 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 classical Bose–Einstein (q= 0) and Fermi–Dirac (q= 1) cases. This was done by formally allowing values 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 q= (qi) , whereby the (limiting) gap q is naturally interpreted as the Cesàro mean of q. In this model, we find the family of limit shapes parameterized by q∈ [0 , ∞) confirming the earlier answer, and also obtain the asymptotics of the number of parts.

UR - http://www.scopus.com/inward/record.url?scp=85069670684&partnerID=8YFLogxK

U2 - 10.1007/s00220-019-03513-5

DO - 10.1007/s00220-019-03513-5

M3 - Article

AN - SCOPUS:85069670684

VL - 373

SP - 1085

EP - 1131

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

ER -