Universality and critical behaviour in the chiral two-matrix model
Universality and critical behaviour in the chiral two-matrix model
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DOI：
10.1088/0951-7715/26/8/2231
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发表时间：
2013-03
期刊：
影响因子：
1.7
通讯作者：
S. Delvaux;Dries Geudens;Lun Zhang
中科院分区：
文献类型：
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作者：
S. Delvaux;Dries Geudens;Lun Zhang
本文研究了Akemann，Damgaard，奥斯本和Splittorff提出的具有多项式势函数V和W的手征双矩阵模型。我们发现，在这个模型中的每个单独的矩阵的平方奇异值形成一个行列式点过程的相关核确定的矩阵值的Riemann-Hilbert问题。黎曼-希尔伯特矩阵的大小取决于势函数W（或V）的阶数。这样我们得到了Kuijlaars-McLaughlin关于非手征双矩阵模型的一个结果的手征类似物。高斯情况对应于V，W是线性的。对于W（y）= y ~ 2/2 + αy是二次函数的情形，我们利用Deift-Zhou最速下降法得到了Riemann-Hilbert问题的大n-渐近解.这证明了这种情况下的普遍性。分析中的一个重要组成部分是三阶微分方程。最后，我们证明了如果V（x）= x也是线性的，那么核的多临界极限存在，它由与Painlevé II方程q″（x）= xq（x）+2 q3（x）− v − 1/2相关的4 × 4矩阵值Riemann-Hilbert问题描述。这样，我们获得了Duits和第二作者最近结果的手性类似物。
We study the chiral two-matrix model with polynomial potential functions V and W, which was introduced by Akemann, Damgaard, Osborn and Splittorff. We show that the squared singular values of each of the individual matrices in this model form a determinantal point process with correlation kernel determined by a matrix-valued Riemann–Hilbert problem. The size of the Riemann–Hilbert matrix depends on the degree of the potential function W (or V respectively). In this way we obtain the chiral analogue of a result of Kuijlaars–McLaughlin for the non-chiral two-matrix model. The Gaussian case corresponds to V, W being linear. For the case where W(y) = y2/2 + αy is quadratic, we derive the large n-asymptotics of the Riemann–Hilbert problem by means of the Deift–Zhou steepest descent method. This proves universality in this case. An important ingredient in the analysis is a third-order differential equation. Finally we show that if also V(x) = x is linear, then a multi-critical limit of the kernel exists which is described by a 4 × 4 matrix-valued Riemann–Hilbert problem associated with the Painlevé II equation q″(x) = xq (x) + 2q3(x) − ν − 1/2. In this way we obtain the chiral analogue of a recent result by Duits and the second author.