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arXiv 1209.6366math-phmath.CAmath.MP
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正交多项式关于平面上测度的强渐近性

Strong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane

Ferenc Balogh, Marco Bertola, Seung Yeop Lee, Kenneth D. T-R McLaughlin

备注：
52 pages, 13 figures, The version accepted for publication
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摘要：

我们考虑在整个复平面上关于测度$|z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$的正交多项式$\{P_{n}(z)\}$。我们得到了复平面上正交多项式的强渐近性和它们的零点在尺度极限中的位置，其中$n$随$N$增长到无穷大。渐近性描述的三个（概率）措施与问题。第一个测量是多项式的零的计数测量的极限，这是由$g$函数捕获的，非常符合实数线上的普通正交多项式的精神。第二个措施是平衡措施，最大限度地减少了一定的对数势能，支持区域$K$的复平面。第三个测度是极点位于$\infty$的$K^c$的调和测度。这表现为由第$n$正交多项式的平方模乘以正交性度量（即，$|P_n(z)|^2 |z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$）给出的概率度量的极限（直到归一化常数）。紧区域$K$是第二个措施的支持下进行的参数$t=n/N$的变化下的拓扑转变;在一个双标度极限附近的临界点由$t_c=a(a+2\sqrt c)$，我们观察到的黑斯廷斯-麦克劳德解决方案Painlevé\ II的正交多项式的渐近性。

英文摘要：

We consider the orthogonal polynomials $\{P_{n}(z)\}$ with respect to the measure $|z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$ over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where $n$ grows to infinity with $N$. The asymptotics are described in terms of three (probability) measures associated with the problem. The first measure is the limit of the counting measure of zeros of the polynomials, which is captured by the $g$-function much in the spirit of ordinary orthogonal polynomials on the real line. The second measure is the equilibrium measure that minimizes a certain logarithmic potential energy, supported on a region $K$ of the complex plane. The third measure is the harmonic measure of $K^c$ with a pole at $\infty$. This appears as the limit of the probability measure given (up to the normalization constant) by the squared modulus of the $n$-th orthogonal polynomial times the orthogonality measure, i.e. $|P_n(z)|^2 |z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$. The compact region $K$ that is the support of the second measure undergoes a topological transition under the variation of the parameter $t=n/N$; in a double scaling limit near the critical point given by $t_c=a(a+2\sqrt c)$ we observe the Hastings-McLeod solution to Painlevé\ II in the asymptotics of the orthogonal polynomials.

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