The PDEs of Biorthogonal Polynomials Arising in the Two-Matrix Model
The PDEs of Biorthogonal Polynomials Arising in the Two-Matrix Model
复制标题
二矩阵模型中双正交多项式的偏微分方程
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
B. Eynard
中科院分区：
文献类型：
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作者：
M. Bertola;B. Eynard
双矩阵模型可以通过引入双正交多项式来求解。当测度中的势为多项式时，双正交多项式的有限序列（称为窗）满足多项式常微分方程以及变形方程（PDE）和有限差分方程（ΔE），它们都是Frobenius相容的，并且定义了非规则常微分方程的离散和连续的等单点变形，如我们以前的工作所示.在一个矩阵模型中，这些系统的系数的明确和简洁的表达式是已知的，它允许将配分函数与超定系统的等单轨道τ函数相关联。在这里，我们将这些表达式推广到双正交多项式的情形，从而使我们能够计算超定方程组ODE + PDE + ΔE的基本解的行列式.
The two-matrix model can be solved by introducing biorthogonal polynomials. In the case the potentials in the measure are polynomials, finite sequences of biorthogonal polynomials (called windows) satisfy polynomial ODEs as well as deformation equations (PDEs) and finite difference equations (ΔE) which are all Frobenius compatible and define discrete and continuous isomonodromic deformations for the irregular ODE, as shown in previous works of ours. In the one matrix model an explicit and concise expression for the coefficients of these systems is known and it allows to relate the partition function with the isomonodromic tau-function of the overdetermined system. Here, we provide the generalization of those expressions to the case of biorthogonal polynomials, which enables us to compute the determinant of the fundamental solution of the overdetermined system of ODE + PDEs + ΔE.