论文标题
平面正交多项式的强渐近学：高斯重量受到有限数量电荷的扰动
Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges
论文作者
论文摘要
我们考虑平面正交多项式$ p_ {n}（z）$相对于在整个复杂平面上支持的度量$$ {\ rm e}^{ - n | z |^2} \ z |^2} \ prod_ {j = 1} d} a $是飞机的lebesgue度量，$ n $是一个正常数，$ \ {c_1，\ cdots，c_ν\} $非零实际数量大于$ -1 $和$ -1 $和$ \ {a_1，\ cdots，a_ν\ \ cdots，a_ν\\} 磁盘。在缩放限制时，当$ n/n = 1 $和$ n \ to \ infty $时，我们获得了多项式$ p_n（z）$的强渐近差。我们表明，根的支持会收敛到我们所谓的“多个szego曲线”，这是一条具有$ν+1 $组件的连接曲线。我们将非线性陡峭下降方法应用于矩阵riemann-hilbert的大小$（ν+1）\ times（ν+1）$的问题。
We consider the planar orthogonal polynomial $p_{n}(z)$ with respect to the measure supported on the whole complex plane $${\rm e}^{-N|z|^2} \prod_{j=1}^ν|z-a_j|^{2c_j}\,{\rm d} A(z)$$ where ${\rm d} A$ is the Lebesgue measure of the plane, $N$ is a positive constant, $\{c_1,\cdots,c_ν\}$ are nonzero real numbers greater than $-1$ and $\{a_1,\cdots,a_ν\}\subset{\mathbb D}\setminus\{0\}$ are distinct points inside the unit disk. In the scaling limit when $n/N = 1$ and $n\to \infty$ we obtain the strong asymptotics of the polynomial $p_n(z)$. We show that the support of the roots converges to what we call the "multiple Szego curve," a certain connected curve having $ν+1$ components in its complement. We apply the nonlinear steepest descent method on the matrix Riemann-Hilbert problem of size $(ν+1)\times(ν+1)$.
