The free boundary Schur process and applications I

Dan Betea (1) , Jérémie Bouttier (2, 3) , Peter Nejjar (4) , Mirjana Vuletić (5)

1 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
2 IPHT - Institut de Physique Théorique - UMR CNRS 3681
3 Phys-ENS - Laboratoire de Physique de l'ENS Lyon
4 IST Austria - Institute of Science and Technology [Klosterneuburg, Austria]
5 - - Department of Mathematics

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.

Domaines: Probabilités [math.PR]; Physique mathématique [math-ph]; Combinatoire [math.CO]

Dan Betea, Jérémie Bouttier, Peter Nejjar, Mirjana Vuletić. The free boundary Schur process and applications I. Annales Henri Poincaré, 2018, 19 (12), pp.3663 - 3742. ⟨10.1007/s00023-018-0723-1⟩. ⟨hal-01619339v2⟩