Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis
Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis
复制标题
双矩阵模型的普适性：黎曼-希尔伯特最速下降分析
DOI：
--
复制
发表时间：
2008
期刊：
影响因子：
--
通讯作者：
A. Kuijlaars
中科院分区：
文献类型：
--
作者：
M. Duits;A. Kuijlaars
The eigenvalue statistics of a pair (M1, M2) of n × n Hermitian matrices taken randomly with respect to the measure $${1 over Z_{n}} exp left({-n} {
m Tr},(V(M_{1})+ W(M_{2})- au M_{1}M_{2})
ight){
m d}M_{1},{
m d}M_{2}$$ can be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest‐descent analysis of a 4 × 4 matrix‐valued Riemann‐Hilbert problem characterizing one of the families of biorthogonal polynomials in the special case W(y) = y4/4 and V an even polynomial. As a result, we obtain the limiting behavior of the correlation kernel associated to the eigenvalues of M1 (when averaged over M2) in the global and local regime as n → ∞ in the one‐cut regular case. A special feature in the analysis is the introduction of a vector equilibrium problem involving both an external field and an upper constraint. © 2008 Wiley Periodicals, Inc.
The eigenvalue statistics of a pair (M1, M2) of n × n Hermitian matrices taken randomly with respect to the measure $${1 over Z_{n}} exp left({-n} {
m Tr},(V(M_{1})+ W(M_{2})- au M_{1}M_{2})
ight){
m d}M_{1},{
m d}M_{2}$$ can be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest‐descent analysis of a 4 × 4 matrix‐valued Riemann‐Hilbert problem characterizing one of the families of biorthogonal polynomials in the special case W(y) = y4/4 and V an even polynomial. As a result, we obtain the limiting behavior of the correlation kernel associated to the eigenvalues of M1 (when averaged over M2) in the global and local regime as n → ∞ in the one‐cut regular case. A special feature in the analysis is the introduction of a vector equilibrium problem involving both an external field and an upper constraint. © 2008 Wiley Periodicals, Inc.