Correlations for symplectic and orthogonal Schur measures

Abstract

We show, using either Fock space techniques or Macdonald difference operators, that certain symplectic and orthogonal analogues of Okounkov's Schur measure are determinantal with kernels given by explicit double contour integrals. We give two applications: one equates certain Toeplitz+Hankel determinants of random matrix theory with appropriate Fredholm determinants and computes Szegő asymptotics for the former; another finds that the simplest examples of said measures exhibit discrete sine kernel asymptotics in the bulk and Airy 2 to 1 kernel---along with a certain dual---asymptotics at the edge. We believe the edge behavior to be universal.

23 pages; minor edits compared to v1: added more details in proof of Thm 4, slightly altered the abstract, changed \tilde and \check notation so the former agrees with the literature, changed awkward indexing of Toeplitz+Hankel determinants, corrected typos and other minor errors

References in corpus (4)

• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
• Fluctuations of a one-dimensional polynuclear growth model in a half space
• Point-to-line polymers and orthogonal Whittaker functions
• Correlation Functions of the Pfaffian Schur Process Using Macdonald Difference Operators

Cited by in corpus (4)

• Correlation Functions of the Pfaffian Schur Process Using Macdonald Difference Operators
• Skew Symplectic and Orthogonal Schur Functions
• Limit shape for infinite rank limit of tensor power decomposition for Lie algebras of series so(2n+1)
• Speed of convergence in the Central Limit Theorem for the determinantal point process with the Bessel kernel