arXiv math-ph/04090822004-09-30math-phmath.MP

随机矩阵模型的配分函数的渐近性

Asymptotics of the partition function of a random matrix model

Pavel Bleher, Alexander Its

备注：: 43 pages, 3 figures

摘要：

本文证明了一系列关于具有多项式势$V(z)$的随机矩阵模型自由能的大$N$渐近性的结果。我们的方法是基于变形$τ_tV(z)$的$V(z)$ $z^2$，$0\le t<\infty$和使用的基础可积结构的矩阵模型。主要结果包括：（1）对于单割正则$V$，存在相关正交多项式的递推系数的$N^{-2}$幂的完全渐近展开式;（2）对于允许单割正则形变$τ_tV$的$V$，存在自由能的$N^{-2}$幂的完全渐近展开式;（3）递归系数和自由能的渐近展开式的系数关于$V$的系数的解析性：（4）单割奇异$V$的递归系数和自由能的单侧解析性;（5）奇异四次多项式$V$自由能的双标度渐近性。

英文摘要：

We prove a number of results concerning the large $N$ asymptotics of the free energy of a random matrix model with a polynomial potential $V(z)$. Our approach is based on a deformation $τ_tV(z)$ of $V(z)$ to $z^2$, $0\le t<\infty$ and on the use of the underlying integrable structures of the matrix model. The main results include (1) the existence of a full asymptotic expansion in powers of $N^{-2}$ of the recurrence coefficients of the related orthogonal polynomials, for a one-cut regular $V$; (2) the existence of a full asymptotic expansion in powers of $N^{-2}$ of the free energy, for a $V$, which admits a one-cut regular deformation $τ_tV$; (3) the analyticity of the coefficients of the asymptotic expansions of the recurrence coefficients and the free energy, with respect to the coefficients of $V$; (4) the one-sided analyticity of the recurrent coefficients and the free energy for a one-cut singular $V$; (5) the double scaling asymptotics of the free energy for a singular quartic polynomial $V$.

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