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The Free Boundary Schur Process and Applications I

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.

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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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Correspondence to Peter Nejjar.

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P. Nejjar: Supported by ERC Advanced Grant No. 338804 and ERC Starting Grant No. 716117.

Communicated by Vadim Gorin.

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Betea, D., Bouttier, J., Nejjar, P. et al. The Free Boundary Schur Process and Applications I. Ann. Henri Poincaré 19, 3663–3742 (2018). https://doi.org/10.1007/s00023-018-0723-1

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