Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis
Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis
复制标题
DOI：
10.1063/1.4922620
复制
发表时间：
2015-06
影响因子：
1.3
通讯作者：
Min Chen;Yang Chen
中科院分区：
文献类型：
--
作者：
Min Chen;Yang Chen
我们继续研究由奇异摄动拉盖尔权w（x;t，α）= xα e-xe-t/x，x ∈ xj+，α > 0，t > 0生成的Hankel行列式，定义为Dn（t，α）=det ∞xj+kw（x; t，α）dxj，k= 0 n −1，并通过拉盖尔权函数w（x; 0，α）= xαe−x，x ∈ n+，α > 0，通过乘法因子e−t/x。较早的调查作出了有限n方面的这种决定，这出现在陈及其[J.近似。Theory 162，270-297（2010）]。发现Hankel行列式的对数具有关于Painleve III（简称PIII）超越及其t阶导数的积分表示.在本文中，我们证明了在双重标度下，当n，Hankel矩阵的大小趋于∞，t趋于0+时，标度化的，因此，在某种意义上，无穷维的Hankel行列式有一个关于C势的积分表示. C所满足的二阶非线性常微分方程，在…
We continue with the study of the Hankel determinant, defined by Dn(t,α)=det∫0∞xj+kw(x;t,α)dxj,k=0n−1, generated by a singularly perturbed Laguerre weight, w(x; t, α) = xαe−xe−t/x, x ∈ ℝ+, α > 0, t > 0, and obtained through a deformation of the Laguerre weight function, w(x; 0, α) = xαe−x, x ∈ ℝ+, α > 0, via the multiplicative factor e−t/x. An earlier investigation was made on the finite n aspect of such determinants, which appeared in Chen and Its [J. Approx. Theory 162, 270–297 (2010)]. It was found that the logarithm of the Hankel determinant has an integral representation in terms of a particular Painleve III (PIII, for short) transcendent and its t derivatives. In this paper, we show that under a double scaling, where n, the size of the Hankel matrix tends to ∞, and t tends to 0+, the scaled—and therefore, in some sense, infinite dimensional—Hankel determinant has an integral representation in terms of a C potential. The second order non-linear ordinary differential equation satisfied by C, after...