Abstract
Meinardus proved a general theorem about the asymptotics of the number of weighted partitions, when the Dirichlet generating function for weights has a single pole on the positive real axis. Continuing (Granovsky et al., Adv. Appl. Math. 41:307–328, 2008), we derive asymptotics for the numbers of three basic types of decomposable combinatorial structures (or, equivalently, ideal gas models in statistical mechanics) of size n, when their Dirichlet generating functions have multiple simple poles on the positive real axis. Examples to which our theorem applies include ones related to vector partitions and quantum field theory. Our asymptotic formula for the number of weighted partitions disproves the belief accepted in the physics literature that the main term in the asymptotics is determined by the rightmost pole.
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Benvenuti S., Feng B., Hanany A., He Y.: Counting BPS Operators in Gauge Theories - Quivers, Syzygies and Plethystics. J. High Energy Physics 11, 050 (2007)
Freiman G., Granovsky B.: Asymptotic formula for a partition function of reversible coagulation-fragmentation processes. Isr. J. Math. 130, 259–279 (2002)
Granovsky B., Stark D., Erlihson M.: Meinardus’ theorem on weighted partitions: Extensions and a probabilistic proof. Adv. Appl. Math. 41, 307–328 (2008)
Korevaar, J.: Tauberian Theory. Berlin-Heidelberg-New York: Springer, 2004
Khinchin, A. I.: Mathematical foundations of quantum statistics. Albany, NY: Graylock Press, 1960
Lucietti J., Rangamani M.: Asymptotic counting of BPS operators in superconformal field theories. J. Math. Phys. 49(8), 082301 (2008)
Meinardus G.: Asymptotische Aussagen über Partitionen. Math. Z. 59, 388–398 (1954)
Madritsch M., Wagner S.: A central limit theorem for integer partitions. Monatsh. Math. 161, 85–114 (2010)
Tate T.: A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions. Asymptotic Analysis 67(1–2), 101–123 (2010)
Pitman, J.: Combinatorial stochastic processes. Lecture Notes in Mathematics, 1875, Berlin-Heidelberg-New York: Springer, 2006
Vershik A.: Statistical mechanics of combinatorial partitions and their limit configurations. Funct. Anal. Appl. 30, 90–105 (1996)
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Communicated by S. Zelditch
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Granovsky, B.L., Stark, D. A Meinardus Theorem with Multiple Singularities. Commun. Math. Phys. 314, 329–350 (2012). https://doi.org/10.1007/s00220-012-1526-8
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DOI: https://doi.org/10.1007/s00220-012-1526-8