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.Comment: 42 page

The Pfaff lattice and skew-orthogonal polynomials

Consider a semi-infinite skew-symmetric moment matrix, m_{\iy} evolving according to the vector fields \pl m / \pl t_k=\Lb^k m+m \Lb^{\top k} , where \Lb is the shift matrix. Then the skew-Borel decomposition m_{\iy}:= Q^{-1} J Q^{\top -1} leads to the so-called Pfaff Lattice, which is integrable, by virtue of the AKS theorem, for a splitting involving the affine symplectic algebra. The tau-functions for the system are shown to be pfaffians and the wave vectors skew-orthogonal polynomials; we give their explicit form in terms of moments. This system plays an important role in symmetric and symplectic matrix models and in the theory of random matrices (beta=1 or 4).Comment: 21 page

Ikehara-type theorem involving boundedness

Consider any Dirichlet series sum a_n/n^z with nonnegative coefficients a_n and finite sum function f(z)=f(x+iy) when x is greater than 1. Denoting the partial sum a_1+...+a_N by s_N, the paper gives the following necessary and sufficient condition in order that (s_N)/N remain bounded as N goes to infinity. For x tending to 1 from above, the quotient q(x+iy)=f(x+iy)/(x+iy) must converge to a pseudomeasure q(1+iy), the distributional Fourier transform of a bounded function. The paper also gives an optimal estimate for (s_N)/N under the "real condition" that (1-x)f(x) remain bounded as x tends to 1 from above.Comment: 6 page