Transition distributions of Young diagrams under periodically weighted Plancherel measures. (English) Zbl 1204.60011

Summary: S. V. Kerov [Funct. Anal. Appl. 27, No. 2, 104–117 (1993); translation from Funkts. Anal. Prilozh. 27, No. 2, 32–49 (1993; Zbl 0808.05098; Transl., Ser. 2, Am. Math. Soc. 188, 111–130 (1999); translation from Tr. St-Peterbg. Mat. Obshch. 4, 165–192 (1996)Zbl 0929.05090)] proved that Wigner’s semi-circular law in Gaussian unitary ensembles is the transition distribution of the omega curve discovered by A. M. Vershik and S. V. Kerov [Sov. Math., Dokl. 18, 527–531 (1977); translation from Dokl. Akad. Nauk SSSR 233, 1024–1027 (1977; Zbl 0406.05008)] for the limit shape of random partitions under the Plancherel measure. This establishes a close link between random Plancherel partitions and Gaussian unitary ensembles. In this paper, we aim at considering a general problem, namely, to characterize the transition distribution of the limit shape of random Young diagrams under Poissonized Plancherel measures in a periodic potential, which naturally arises in Nekrasov’s partition functions and was further studied by N. A. Nekrasov and A. Okounkov [in: The unity of mathematics. In honor of the ninetieth birthday of I. M. Gelfand. Papers from the conference held in Cambridge, MA, USA, August 31–September 4, 2003. Boston, MA: Birkhäuser. Progress in Mathematics 244, 525–596 (2006; Zbl 1233.14029)] and A. Okounkov [The use of random partitions. arXiv:math.ph/0309015; Random partitions and instanton counting. arXiv:math.ph/0601062]. We also find an associated matrix model for this transition distribution. Our argument is based on a purely geometric analysis on the relation between matrix models and Seiberg-Witten differentials.


60B20 Random matrices (probabilistic aspects)
60E05 Probability distributions: general theory
05A17 Combinatorial aspects of partitions of integers
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