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.

58 pages, 17 figures (v3: final version, with fewer pages than v2 to comply with journal requirements. In particular Theorems 2.2 and 2.5 are now stated under slightly more restrictive assumptions to simplify presentation. Added Remarks 2.11, 2.12, 5.14 and 6.9 about the transition from pfaffian to determinantal in the bulk.)

References in corpus (6)

• Fluctuations of a one-dimensional polynuclear growth model in a half space
• Stochastic six-vertex model in a half-quadrant and half-line open ASEP
• The free boundary Schur process and applications I
• The periodic Schur process and free fermions at finite temperature
• Facilitated exclusion process
• Correlation Functions of the Pfaffian Schur Process Using Macdonald Difference Operators

Cited by in corpus (19)

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• The free boundary Schur process and applications I
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• Stationary half-space last passage percolation
• Correlation Functions of the Pfaffian Schur Process Using Macdonald Difference Operators
• Boundary current fluctuations for the half space ASEP and six vertex model
• Random walk on nonnegative integers in beta distributed random environment
• Lozenge tilings of hexagons with cuts and asymptotic fluctuations: a new universality class
• Markov duality and Bethe ansatz formula for half-line open ASEP
• Planar maps and random partitions
• Double Interlacing in Random Tiling Models
• New edge asymptotics of skew Young diagrams via free boundaries
• On the combinatorics of last passage percolation in a quarter square and ${\mathrm{GOE}}^2$ fluctuations
• Discrete and continuous Muttalib--Borodin processes I: the hard edge
• Identity between Restricted Cauchy Sums for the $q$-Whittaker and Skew Schur Polynomials
• Correlations in totally symmetric self-complementary plane partitions
• Determinantal point processes from symplectic and orthogonal characters and applications
• Skew doubled shifted plane partitions: calculus and asymptotics
• The half-space Airy stat process