酉积分、组合学和Toeplitz格的递归关系
Recursion relations for Unitary integrals, Combinatorics and the Toeplitz Lattice
摘要：
与符号$exp{t(z+z^{-1})}$或$(1-ξ{z})^α (1-ξ{z^{-1}})^β$相关联的Toeplitz行列式（大小增加）满足递归关系，因此将所有Toeplitz行列式表示为前几个行列式的有理函数。A.鲍罗丁发现这些关系使用黎曼-希尔伯特方法。性质鲍罗丁的关系指向Toeplitz格和它的维拉索罗代数，作为发达国家的作者。在本文中，我们采取的Toeplitz和Virasoro的方法相当大的一类符号，导致系统和简单的方式产生这样的递归关系。后者是非常自然地表示的$L$ -矩阵出现在拉克斯对的Toeplitz格点方程。令人惊讶的是，我们发现，相比鲍罗丁的，一组不同的关系，除了与符号$ e^{t(z+z^{-1})}$相关的3步关系。此外，这些递归关系定义了Toeplitz格的不变流形。这导致了有理递归关系的“离散Painlevé性质”（奇点限制），作为Toeplitz格的经典“连续Painlevé性质”的结果。
英文摘要：
The Toeplitz determinants (of increasing size) associated with the symbols $exp{t(z+z^{-1})}$ or $(1-ξ{z})^α (1-ξ{z^{-1}})^β$ satisfy recursion relations, thus expressing all the Toeplitz determinants as a rational function of the first few determinants. A. Borodin found these relations using Riemann-Hilbert methods. The nature of Borodin's relations pointed towards the Toeplitz lattice and its Virasoro algebra, as developed by the authors. In this paper, we take the Toeplitz and Virasoro approach for a fairly large class of symbols, leading to a systematic and simple way of generating such recursion relations. The latter are very naturally expressed in terms of the $L$-matrices appearing in the Lax pair for the Toeplitz lattice equations. As a surprise, we find, compared to Borodin's, a different set of relations, except for the 3-step relations associated with the symbol $ e^{t(z+z^{-1})}$. Moreover, these recursion relations define an invariant manifold for the Toeplitz lattice. This leads to a "discrete Painlevé property" (singularity confinement) for the rational recursion relations, as a consequence of the classical ``continuous Painlevé property" for the Toeplitz lattice.
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