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Annales Henri Poincaré

, Volume 19, Issue 12, pp 3663–3742 | Cite as

The Free Boundary Schur Process and Applications I

  • Dan Betea
  • Jérémie Bouttier
  • Peter Nejjar
  • Mirjana Vuletić
Open Access
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Abstract

We investigate the free boundary Schur process, a variant of the Schur process introduced by Okounkov and Reshetikhin, where we allow the first and the last partitions to be arbitrary (instead of empty in the original setting). The pfaffian Schur process, previously studied by several authors, is recovered when just one of the boundary partitions is left free. We compute the correlation functions of the process in all generality via the free fermion formalism, which we extend with the thorough treatment of “free boundary states.” For the case of one free boundary, our approach yields a new proof that the process is pfaffian. For the case of two free boundaries, we find that the process is not pfaffian, but a closely related process is. We also study three different applications of the Schur process with one free boundary: fluctuations of symmetrized last passage percolation models, limit shapes and processes for symmetric plane partitions and for plane overpartitions.

Notes

Acknowledgements

We thank Jinho Baik, Guillaume Barraquand, Philippe Biane, Elia Bisi, Alexei Borodin, Cédric Boutillier, Ivan Corwin, Philippe Di Francesco, Patrik Ferrari, Mikael de la Salle, Ole Warnaar, Michael Wheeler, and Nikos Zygouras for useful conversations. We also thank the anonymous referee for suggesting valuable improvements to the paper. Most of this work was done, while the first three authors were at the Département de mathématiques et applications, École normale supérieure, Paris. We also acknowledge hospitality and support from the Galileo Galilei Institute during the 2015 program on “Statistical Mechanics, Integrability and Combinatorics.” Part of this work was done, while D.B. was visiting the ENS de Lyon and László Erdős at IST Austria, and remerciements are due to both institutions and to László for their hosting. J.B. recently enjoyed the hospitality from the PCMI 2017 Research Program and from the ESI program on “Algorithmic and Enumerative Combinatorics.” We acknowledge financial support from the “Combinatoire à Paris” project funded by the City of Paris (D.B. and J.B.), from the Agence Nationale de la Recherche via the Grants ANR 12-JS02-001-01 “Cartaplus” and ANR-14-CE25-0014 “GRAAL” (J.B.), from Paris Sciences and Lettres and from the ERC Advanced Grant No. 338804 (P.N.).

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Dan Betea
    • 1
  • Jérémie Bouttier
    • 2
    • 3
  • Peter Nejjar
    • 4
  • Mirjana Vuletić
    • 5
  1. 1.IRIFCNRS et Université Paris DiderotParis Cedex 13France
  2. 2.Institut de Physique Théorique Université Paris-Saclay, CEA, CNRSGif-sur-YvetteFrance
  3. 3.Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de PhysiqueLyonFrance
  4. 4.IST AustriaKlosterneuburgAustria
  5. 5.Department of MathematicsUniversity of Massachusetts BostonBostonUSA

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