Character expansion method for the first order asymptotics of a matrix integral
Character expansion method for the first order asymptotics of a matrix integral
复制标题
矩阵积分一阶渐进的特征展开法
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
Mylène Maïda
中科院分区：
文献类型：
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作者：
A. Guionnet;Mylène Maïda
当矩阵的大小达到无穷大时，各种矩阵积分的估计是由理论物理、几何和自由概率问题所驱动的。在严格的基础上，到目前为止，只能计算一个矩阵或几个具有简单二次相互作用(称为AB相互作用)的矩阵的积分(见[19]、[17]或[9])。在本文中，我们遵循物理文献中广泛发展的一种基于特征扩展的思想来研究更复杂的相互作用。在此背景下，我们推导出了Young场景的经验度量的大偏差原理。然后，我们使用它来研究一个矩阵模型，该模型是在文献[13]中介绍的“双权图模型”的精神下定义的，但具有截断函数，使得矩阵积分及其特征展开收敛。我们证明了该模型的自由能随着矩阵的大小趋于无穷大而收敛，并研究了极限的临界点。
Abstract.The estimation of various matrix integrals as the size of the matrices goes to infinity is motivated by theoretical physics, geometry and free probability questions. On a rigorous ground, only integrals of one matrix or of several matrices with simple quadratic interaction (called AB interaction) could be evaluated so far (see e.g. [19], [17] or [9]). In this article, we follow an idea widely developed in the physics literature, which is based on character expansion, to study more complex interaction. In this context, we derive a large deviation principle for the empirical measure of Young tableaux. We then use it to study a matrix model defined in the spirit of the ‘dually weighted graph model’ introduced in [13], but with a cutoff function such that the matrix integral and its character expansion converge. We prove that the free energy of this model converges as the size of the matrices goes to infinity and study the critical points of the limit.