Pfaff $\tau$ -functions
Copyright policy )Pfaff $\tau$ -functions
Consider the evolution $$ \frac{\pl m_\iy}{\pl t_n}=\Lb^n m_\iy, \frac{\pl m_\iy}{\pl s_n}=-m_\iy(\Lb^\top)^n, $$ 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.
42 pages
- Université Catholique de Louvain Belgium
- Kyoto University Japan
- Brandeis University United States
High Energy Physics - Theory, Nonlinear Sciences - Exactly Solvable and Integrable Systems, Toda system, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), FOS: Physical sciences, Virasoro symmetries, Nonlinear Sciences - Adaptation and Self-Organizing Systems, identities, quasiperiodic solutions, High Energy Physics - Theory (hep-th), Pfaff lattice, \(\tau\)-funciton, Exactly Solvable and Integrable Systems (nlin.SI), Adaptation and Self-Organizing Systems (nlin.AO)
High Energy Physics - Theory, Nonlinear Sciences - Exactly Solvable and Integrable Systems, Toda system, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), FOS: Physical sciences, Virasoro symmetries, Nonlinear Sciences - Adaptation and Self-Organizing Systems, identities, quasiperiodic solutions, High Energy Physics - Theory (hep-th), Pfaff lattice, \(\tau\)-funciton, Exactly Solvable and Integrable Systems (nlin.SI), Adaptation and Self-Organizing Systems (nlin.AO)
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