The Periodic Schur Process and Free Fermions at Finite Temperature

  • Dan BeteaEmail author
  • Jérémie Bouttier


We revisit the periodic Schur process introduced by Borodin in 2007. Our contribution is threefold. First, we provide a new simpler derivation of its correlation functions via the free fermion formalism. In particular, we shall see that the process becomes determinantal by passing to the grand canonical ensemble, which gives a physical explanation to Borodin’s “shift-mixing” trick. Second, we consider the edge scaling limit in the simplest nontrivial case, corresponding to a deformation of the poissonized Plancherel measure on partitions. We show that the edge behavior is described by the universal finite-temperature Airy kernel, which was previously encountered by Johansson and Le Doussal et al. in other models, and whose extreme value statistics interpolates between the Tracy–Widom GUE and the Gumbel distributions. We also define and prove convergence for a stationary extension of our model. Finally, we compute the correlation functions for a variant of the periodic Schur process involving strict partitions, Schur’s P and Q functions, and neutral fermions.


Schur process Free fermions Determinantal point processes Integrable probability Random integer partitions 

Mathematics Subject Classification (2010)

82C23 60K35 05E05 



The authors had illuminating conversations related on the subject of this note with many people, including J. Baik, P. Biane, A. Borodin, S. Corteel, P. Di Francesco, P. Ferrari, T. Imamura, L. Hodgkinson, K. Johansson, C. Krattenthaler, G. Lambert, P. Le Doussal, S. Majumdar, G. Miermont, M. Mucciconi, P. Nejjar, E. Rains, N. Reshetikhin, T. Sasamoto, G. Schehr, M. Schlosser, M. Vuletić and M. Wheeler.

This work was initiated while the authors were at the Département de mathématiques et applications, École normale supérieure, Paris, and continued during several visits D.B. paid to J.B. at the ENS de Lyon. It was finalized while the authors were visiting the Matrix Institute on the University of Melbourne campus in Creswick, Australia. We wish to thank all institutions for their hospitality.

We acknowledge financial support from the “Combinatoire à Paris” project funded by the City of Paris (D.B. and J.B.), from the Laboratoire International Franco-Québécois de Recherche en Combinatoire (J.B.), and from the Agence Nationale de la Recherche via the grants ANR 12-JS02-001-01 “Cartaplus” and ANR-14-CE25-0014 “GRAAL” (J.B.).


  1. 1.
    Aissen, M., Edrei, A., Schoenberg, I.J., Whitney, A.: On the generating functions of totally positive sequences. Proc. Nat. Acad. Sci. USA 37, 303–307 (1951)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aldous, D., Diaconis, P.: Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Relat. Fields 103(2), 199–213 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Commun. Pure Appl. Math. 64(4), 466–537 (2011). arXiv:1003.0443 [math.PR]MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12(4), 1119–1178 (1999). arXiv:math/9810105 [math.CO]MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Betea, D., Boutillier, C., Bouttier, J., Chapuy, G., Corteel, S., Vuletić, M.: Perfect sampling algorithms for Schur processes. Markov Process. Relat. Fields 24(3), 381–418 (2018). arXiv:1407.3764 [math.PR]MathSciNetzbMATHGoogle Scholar
  6. 6.
    Betea, D., Bouttier, J., Nejjar, P., Vuletić, M.: The free boundary Schur process and applications I. Ann. Henri Poincaré 19(12), 3663–3742 (2018). arXiv:1704.05809v2 [math.PR]ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Betea, D., Bouttier, J., Nejjar, P., Vuletić, M.: The free boundary Schur process and applications II. In preparation (2018)Google Scholar
  8. 8.
    Blaizot, J.-P., Ripka, G.: Quantum Theory of finite systems. MIT, Cambridge (1986)Google Scholar
  9. 9.
    Borodin, A.: Periodic Schur process and cylindric partitions. Duke Math. J. 140(3), 391–468 (2007). arXiv:math/0601019 [math.CO]MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Borodin, A.: Schur dynamics of the Schur processes. Adv. Math. 228(4), 2268–2291 (2011). arXiv:1001.3442 [math.CO]MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Borodin, A., Olshanski, G.: Distributions on partitions, point processes, and the hypergeometric kernel. Commun. Math. Phys. 211(2), 335–358 (2000). arXiv:math/9904010 [math.RT]ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Borodin, A., Olshanski, G.: Stochastic dynamics related to Plancherel measure on partitions. In: Representation theory, dynamical systems, and asymptotic combinatorics, volume 217 of Amer. Math. Soc. Transl. Ser. 2, pp 9–21. arXiv:math-ph/0402064. American Mathematical Society, Providence (2006)
  13. 13.
    Borodin, A., Olshanski, G.: Markov processes on partitions. Probab. Theory Relat. Fields 135(1), 84–152 (2006). arXiv:math-ph/0409075 MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Borodin, A., Okounkov, A., Olshanski, G.: Asymptotics of Plancherel measures for symmetric groups. J. Am. Math. Soc. 13(3), 481–515 (2000). arXiv:math/9905032 [math.CO]MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Borodin, A., Corwin, I., Ferrari, P.: Free energy fluctuations for directed polymers in random media in 1 + 1 dimension. Commun. Pure Appl. Math. 67 (7), 1129–1214 (2014). arXiv:1204.1024 [math.PR]MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Bouttier, J., Chapuy, G., Corteel, S.: From Aztec diamonds to pyramids: steep tilings. Trans. Am. Math. Soc. 369(8), 5921–5959 (2017). arXiv:1407.0665 [math.CO]MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Calabrese, P., Doussal, P.L., Rosso, A.: Free-energy distribution of the directed polymer at high temperature. EPL (Europhys. Lett.) 90(2), 20002 (2010). arXiv:1002.4560 [cond-mat.dis-nn]ADSCrossRefGoogle Scholar
  18. 18.
    Corteel, S.: Particle seas and basic hypergeometric series. Adv. Appl. Math. 31(1), 199–214 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Corwin, I., Ghosal, P.: Lower tail of the KPZ equation, arXiv:1802.03273 [math.PR] (2018)
  20. 20.
    Dean, D.S., Le Doussal, P., Majumdar, S.N., Schehr, G.: Finite-temperature free fermions and the Kardar-Parisi-Zhang equation at finite time. Phys. Rev. Lett. 114, 110402 (2015). arXiv:1412.1590 [cond-mat.stat-mech]ADSCrossRefGoogle Scholar
  21. 21.
    Dean, D.S., Le Doussal, P., Majumdar, S.N., Schehr, G.: Noninteracting fermions at finite temperature in a d-dimensional trap: Universal correlations. Phys. Rev. A 94, 063622 (2016). arXiv:1609.04366 [cond-mat.stat-mech]ADSCrossRefGoogle Scholar
  22. 22.
    Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York (1997)Google Scholar
  23. 23.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions, vol. III. McGraw-Hill Book Company Inc., New York-Toronto-London (1955)zbMATHGoogle Scholar
  24. 24.
    Erdős, P., Lehner, J.: The distribution of the number of summands in the partitions of a positive integer. Duke Math. J. 8, 335–345 (1941)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Frobenius, G.: Ueber die elliptischen Functionen zweiter Art. J. Reine Angew. Math. 93, 53–68 (1882)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Gasper, G., Rahman, M.: Basic hypergeometric series, volume 96 of Encyclopedia of Mathematics and its Applications, 2nd edn. Cambridge University Press, Cambridge (2004). With a foreword by Richard AskeyGoogle Scholar
  27. 27.
    Gaudin, M.: Une démonstration simplifée du théorème de Wick en mécanique statistique. Nuclear Phys. 15, 89–91 (1960)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Gessel, I.M., Krattenthaler, C.: Cylindric partitions. Trans. Am. Math. Soc. 349(2), 429–479 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Imamura, T., Sasamoto, T.: Free energy distribution of the stationary O’Connell–Yor directed random polymer model, 2017, arXiv:1701.06904v2 [math-ph]
  30. 30.
    Johansson, K.: Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. Second Series 153 (1), 259–296 (2001). arXiv:math/9906120 [math.CO]MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Johansson, K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields 123(2), 225–280 (2002). arXiv:math/0011250 [math.PR]MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Johansson, K.: Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242(1-2), 277–329 (2003). arXiv:math/0206208 [math.PR]ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Johansson, K.: The arctic circle boundary and the Airy process. Ann. Probab. 33(1), 1–30 (2005). arXiv:math/0306216 [math.PR]MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Johansson, K.: Random matrices and determinantal processes. In: Mathematical statistical physics. arXiv:math-ph/0510038, pp 1–55. Elsevier B. V., Amsterdam (2006)
  35. 35.
    Johansson, K.: From Gumbel to Tracy–Widom. Probab. Theory Relat. Fields 138(1), 75–112 (2007). arXiv:math/0510181 [math.PR]MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  37. 37.
    Langer, R.: Enumeration of cylindric plane partitions - part II, 2012, arXiv:1209.1807 [math.CO]
  38. 38.
    Le Doussal, P., Majumdar, S.N., Schehr, G.: Periodic Airy process and equilibrium dynamics of edge fermions in a trap. Ann. Phys. 383, 312–345 (2017). arXiv:1702.06931 [cond-mat.stat-mech]ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Liechty, K., Wang, D.: Asymptotics of free fermions in a quadratic well at finite temperature and the Moshe-Neuberger-Shapiro random matrix model, arXiv:1706.06653 [math-ph] (2017)
  40. 40.
    Logan, B.F., Shepp, L.A.: A variational problem for random Young tableaux. Adv. Math. 26(2), 206–222 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. The Clarendon Press Oxford University Press, New York (1995). With contributions by A. Zelevinsky, Oxford Science PublicationsGoogle Scholar
  42. 42.
    Matsumoto, S.: Correlation functions of the shifted Schur measure. J. Math. Soc. Jpn. 57(3), 619–637 (2005). arXiv:math/0312373 [math.CO]MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Nazarov, M.: Factor-representations of the infinite spin-symmetric group. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 181, 132–145 (1990)zbMATHGoogle Scholar
  44. 44.
    Nazarov, M.: Projective representations of the infinite symmetric group. Adv. Soviet Math. 9, 115–130 (1992)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Okounkov, A.: Infinite wedge and random partitions. Selecta Math. (N.S.) 7 (1), 57–81 (2001). arXiv:math/9907127 [math.RT]MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Okounkov, A.: Symmetric functions and random partitions. In: Symmetric functions 2001: surveys of developments and perspectives, volume 74 of NATO Sci. Ser. II Math. Phys. Chem. arXiv:math/0309074 [math.CO], pp 223–252. Kluwer Academic Publishers , Dordrecht (2002)
  47. 47.
    Okounkov, A., Reshetikhin, N.: Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Am. Math. Soc. 16(3), 581–603 (electronic) (2003). arXiv:math/0107056 [math.CO]MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Okounkov, A., Reshetikhin, N.: The birth of a random matrix. Mosc. Math. J. 6(3), 553–566, 588 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Okounkov, A., Reshetikhin, N.: Random skew plane partitions and the Pearcey process. Comm. Math. Phys. 269(3), 571–609 (2007). arXiv:math/0503508 [math.CO]ADSMathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108(5–6), 1071–1106 (2002). arXiv:math/0105240 [math.PR]MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Romik, D.: The Surprising Mathematics of Longest Increasing Subsequences. Institute of Mathematical Statistics Textbooks. Cambridge University Press, Available online at (2015)zbMATHGoogle Scholar
  52. 52.
    Rosengren, H.: Sums of triangular numbers from the Frobenius determinant. Adv. Math. 208(2), 935–961 (2007). arXiv:math/0504272 [math.NT]MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Rosengren, H.: Sums of squares from elliptic Pfaffians. Int. J. Number Theory 4(6), 873–902 (2008). arXiv:math/0610278 [math.NT]MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Sasamoto, T., Spohn, H.: The 1 + 1-dimensional Kardar-Parisi-Zhang equation and its universality class. J. Stat. Mech: Theory Exp. 2010(11), P11013 (2010). arXiv:1010.2691 [cond-mat.stat-mech]CrossRefGoogle Scholar
  55. 55.
    Thoma, E.: Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe. Math. Z. 85, 40–61 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Vershik, A.M.: Statistical mechanics of combinatorial partitions, and their limit shapes. Funct. Anal. Appl. 30(2), 90–105 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Vershik, A.M., Kerov, S.V.: Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Dokl. Akad. Nauk SSSR 233(6), 1024–1027 (1977)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Vuletić, M.: The shifted Schur process and asymptotics of large random strict plane partitions. Int. Math. Res. Not. IMRN (14):Art. ID rnm043, 53. arXiv:math-ph/0702068 (2007)
  59. 59.
    Vuletić, M.: A generalization of MacMahon’s formula. Trans. Am. Math. Soc. 361(5), 2789–2804 (2009). arXiv:0707.0532 [math.CO]MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Watson, G.: A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge (1922). Available online through the Internet Archive at zbMATHGoogle Scholar
  61. 61.
    Wheeler, M.: Free fermions in classical and quantum integrable models, PhD thesis, arXiv:1110.6703v1 1110.6703v1 [math-ph] (2011)

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute for Applied MathematicsUniversity of BonnBonnGermany
  2. 2.Institut de Physique ThéoriqueUniversité Paris-Saclay, CEA, CNRSGif-sur-YvetteFrance
  3. 3.Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRSLaboratoire de PhysiqueLyonFrance

Personalised recommendations