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.).


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© 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

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