Pfaff tau-functions

1999/09/15 by Mark Adler, Adler, M., Takahiro Shiota +3 · 1 citation
Mathematics · #Adaptation and Self-Organizing Systems (nlin.AO) #Advanced Topics in Algebra #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Dynamics and Fractals #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.solv-int/9909010

openalex publication_date 1999/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the evolution (\pl m_\iy)/(\pl t_n)=\Lbⁿ m_\iy, (\pl m_\iy)/(\pl s_n)=-m_\iy(\Lb^\top)ⁿ, on bi- or semi-infinite matrices m_\iy=m_\iy(t,s), with skew-symmetric initial data m_\iy(0,0). Then, m_\iy(t,-t) is skew-symmetric, and so the determinants of the successive "upper-left corners" vanish or are squares of Pfaffians. In this paper, we investigate the rich nature of these Pfaffians, as functions of t. This problem is motivated by questions concerning the spectrum of symmetric and symplectic random matrix ensembles.

Citations

• String-orthogonal polynomials, string equations, and 2-Toda symmetries

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