Periodic Schur process and cylindric partitions. (English) Zbl 1131.22003

This substantial and technical paper is about the so-called periodic Schur process. This process depends on a natural number \(N\), a parameter \(t\) with \(| t| <1\), and \(2N\) specializations of the algebra \(\Lambda\) of symmetric functions \(a[1],a[2],\dots, a[N]\) and \(b[1],b[2],\dots, b[N]\) (where a specialization of \(\Lambda\) is an algebra homomorphism from \(\Lambda\) to \(\mathbb C\). The process lives on periodic sequences of \(N\) partitions, whose period divides \(N\), namely \(\lambda^{(N)}=\lambda^{(0)}\), \(\lambda^{(1)}\), \(\lambda^{(2)},\dots, \lambda^{(N)}\) etc., and assigns a weight proportional to
\[ \sum_{\mu^{(1)},\mu^{(2)},\dots, \mu^{(N)}} t^{| \lambda^{(0)}|} s_{\lambda^{(0)}} \mu^{(1)}(a[1])s_{\lambda^{(1)}}/ \mu^{(1)}(b[1])\dots s_{\lambda^{(N-1)}}/ \mu^{(N)}(a[N]) s_{\lambda^{(N)}}/ \mu^{(N)}(b[N]) \]
to it. Here the \(s_{\lambda/ \mu}\) are the so-called skew Schur functions
(see, e.g., http://www.facstaff.bucknell.edu/pm040/FPSAC07/slides.pdf for some background on skew Schur functions). We emphasise that these weights are complex numbers or formal series in \(\Lambda^{\otimes 2N}\): in particular, though we shall use the language of probability in what follows, some of the probabilities may be negative or just formal series of symmetric functions. Note that the constant of proportionality is obtained explicitly in the article under review.
It is in fact often convenient to look at the so-called shift-mixed periodic Schur process which is the product measure of the periodic Schur process with a measure on \({\mathbb Z}\) which gives to \(S\subseteq {\mathbb Z}\) the weight proportional to \(z^{S}t^{S^{2}/2}\) for some \(z\in {\mathbb C}\), mapped to the space of point configurations in \(\{1,2,\dots, N\}\times ({\mathbb Z}+1/2)\) via the map
\[ (\lambda,S)\rightarrow \{S+\lambda_{i}^{(1)}-i+\tfrac{1}{2}\}_{i\geq 1} \sqcup \dots \sqcup \{S+\lambda_{i}^{(N)}-i+\tfrac{1}{2}\}_{i\geq 1}. \]
In other words, all points of the random point configuration of the periodic Schur process are shifted by an independent integral-valued random variable distributed as \(S\).
Correlation functions (for the periodic Schur process) are defined to be probabilities (with the above caveat!) that the random point configuration contains a fixed finite set of points: \[ \rho_{n}(\tau_{1}, x_{1},\tau_{2},x_{2},\dots, \tau_{n},x_{n})=P\{x_{j}\in \{\lambda_{i}^{(\tau_{j})}-i+\tfrac{1}{2}\}_{i\geq 1}\mid j=1,\dots, n\}. \]
They are defined similarly for the shift-mixed periodic Schur process.
The first result of the paper under review is that the correlation functions of the shift-mixed periodic Schur process are determinants of suitable matrices depending on a so-called correlation kernel \(K\) which depends on the quantities \(a_{m}[k]:=(1/m)p_{m}(a[k])\) and \(b_{m}[k]:=(1/m)p_{m}(b[k])\) where the \(p_{m}\)s are Newton power sums. This statement can be viewed both as a statement about formal identity of series in \(\Lambda^{\otimes 2N}\) or (under suitable convergence conditions) about numeric identities. The author notes that his proof was inspired by formalisms in the so-called Fock space, a point he hopes to explore further later.
The second main result concerns the ‘bulk-scaling’ limit of the correlation functions of the periodic Schur process, and its shift-mixed version, as \(t\rightarrow 1\) with \(N\) and the specializations \(a[k]\) and \(b[k]\) stay fixed. Again the limit is a determinant, this time defined in terms of a contour integral, under some technical assumptions (see the original paper). Various corollaries of the result are noted.
The author then considers the implications of his results for cylindric partitions – that is, ways of filling the boxes of the square grid in the quarter-plane with nonnegative integers so that the numbers do not increase as we move to infinity in the directions of the axes, and the total number of non-zero entries is also finite. (An alternative perspective on the results in this paper is that they initiate the study of random cylindric partitions.) We do not even attempt to describe these results in detail, referring to Theorem C in the paper, and the discussion before it.


22D15 Group algebras of locally compact groups
22E30 Analysis on real and complex Lie groups
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
46L05 General theory of \(C^*\)-algebras
Full Text: DOI arXiv


[1] M. Aissen, A. Edrei, I. J. Schoenberg, and A. Whitney, On the generating functions of totally positive sequences , Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 303–307. · Zbl 0042.29205 · doi:10.1073/pnas.37.5.303
[2] A. Borodin and A. Okounkov, A Fredholm determinant formula for Toeplitz determinants , Integral Equations Operator Theory 37 (2000), 386–396. · Zbl 0970.47014 · doi:10.1007/BF01192827
[3] A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups , J. Amer. Math. Soc. 13 (2000), 481–515. JSTOR: · Zbl 0938.05061 · doi:10.1090/S0894-0347-00-00337-4
[4] A. Borodin and G. Olshanski, Distributions on partitions, point processes, and the hypergeometric kernel , Comm. Math. Phys. 211 (2000), 335–358. · Zbl 0966.60049 · doi:10.1007/s002200050815
[5] -, “Stochastic dynamics related to Plancherel measure on partitions” in Representation Theory, Dynamical Systems, and Asymptotic Combinatorics (St. Petersburg, Russia, 2004) , Amer. Math. Soc. Transl. Ser. 2 217 , Amer. Math. Soc., Providence, 2006, 9–21. · Zbl 1109.60041
[6] A. Borodin and E. M. Rains, Eynard-Mehta theorem, Schur process, and their Pfaffian analogs , J. Stat. Phys. 121 (2005), 291–317. · Zbl 1127.82017 · doi:10.1007/s10955-005-7583-z
[7] R. Cerf and R. Kenyon, The low-temperature expansion of the Wulff crystal in the 3D Ising model , Comm. Math. Phys. 222 (2001), 147–179. · Zbl 1013.82010 · doi:10.1007/s002200100505
[8] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes , Springer Ser. Statist., Springer, New York, 1988. · Zbl 0657.60069
[9] A. Edrei, On the generating functions of totally positive sequences, II , J. Analyse Math. 2 (1952), 104–109. · Zbl 0049.17202 · doi:10.1007/BF02786971
[10] A. ErdéLyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. 3 , McGraw-Hill, New York, 1955. · Zbl 0064.06302
[11] F. G. Frobenius, Über die elliptischen Funktionen zweiter Art , J. Reine Angew. Math. 93 (1882), 53–68.
[12] G. Gasper and M. Rahman, Basic Hypergeometric Series , Encyclopedia Math. Appl. 35 , Cambridge Univ. Press, Cambridge, 1990. · Zbl 0695.33001
[13] I. M. Gessel and C. Krattenthaler, Cylindric partitions , Trans. Amer. Math. Soc. 349 (1997), 429–479. · Zbl 0865.05003 · doi:10.1090/S0002-9947-97-01791-1
[14] K. Johansson, Discrete polynuclear growth and determinantal processes , Comm. Math. Phys. 242 (2003), 277–329. · Zbl 1031.60084 · doi:10.1007/s00220-003-0945-y
[15] -, The arctic circle boundary and the Airy process , Ann. Probab. 33 (2005) 1–30. · Zbl 1096.60039 · doi:10.1214/009117904000000937
[16] S. V. Kerov, Asymptotic Representation Theory of the Symmetric Group and Its Applications in Analysis , Transl. Math. Monogr. 219 , Amer. Math. Soc., Providence, 2003. · Zbl 1031.20007
[17] K. Koike, On the decomposition of tensor products of the representations of the classical groups: By means of the universal characters , Adv. Math. 74 (1989), 57–86. · Zbl 0681.20030 · doi:10.1016/0001-8708(89)90004-2
[18] O. Macchi, “The Fermion process –.-A model of stochastic point process with repulsive points” in Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians, Vol. A (Prague, 1974) , Reidel, Dordrecht, Netherlands, 1977, 391–398. · Zbl 0413.60048
[19] I. G. Macdonald, Symmetric Functions and Hall Polynomials , 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995. · Zbl 0824.05059
[20] N. A. Nekrasov and A. Okounkov, “Seiberg-Witten theory and random partitions” in The Unity of Mathematics (Cambridge, Mass., 2003) , Progr. Math. 244 , Birkhäuser, Boston, 2006, 525–596. · Zbl 1233.14029 · doi:10.1007/0-8176-4467-9_15
[21] A. Okounkov, Infinite wedge and random partitions , Selecta Math. (N.S.) 7 (2001), 57–81. · Zbl 0986.05102 · doi:10.1007/PL00001398
[22] -, “The uses of random partitions” in XIVth International Congress on Mathematical Physics (Lisbon, 2003) , World Sci., Hackensack, N.J., 2005, 379–403. · Zbl 1120.05301
[23] A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles , Ann. of Math. (2) 163 (2006), 517–560. · Zbl 1105.14076 · doi:10.4007/annals.2006.163.517
[24] A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random \(3\)-dimensional Young diagram , J. Amer. Math. Soc. 16 (2003), 581–603. · Zbl 1009.05134 · doi:10.1090/S0894-0347-03-00425-9
[25] -, Random skew plane partitions and the Pearcey process , Comm. Math. Phys. 269 (2007), 571–609. · Zbl 1115.60011 · doi:10.1007/s00220-006-0128-8
[26] M. PräHofer and H. Spohn, Scale invariance of the PNG droplet and the Airy process , J. Statist. Phys. 108 (2002), 1071–1106. · Zbl 1025.82010 · doi:10.1023/A:1019791415147
[27] E. M. Rains, Transformations of elliptic hypergeometric integrals , preprint,\arxivmath/0309252v4[math.QA] · Zbl 1329.14012
[28] E. Thoma, Die unzerlegbaren, positive-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe , Math. Z. 85 (1964), 40–61. · Zbl 0192.12402 · doi:10.1007/BF01114877
[29] T. Tsuda, Universal characters and an extension of the KP hierarchy , Comm. Math. Phys. 248 (2004), 501–526. · Zbl 1233.37042 · doi:10.1007/s00220-004-1098-3
[30] A. M. Vershik, Statistical mechanics of combinatorial partitions, and their limit configurations (in Russian), Funktsional. Anal. i Prilozhen. 30 , no. 2 (1996), 19–39.; English translation in Funct. Anal. Appl. 30 (1996), 90–105.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.