Katori Makoto, Komatsuda Naoaki
Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 May;67(5 Pt 1):051110. doi: 10.1103/PhysRevE.67.051110. Epub 2003 May 27.
A system of Brownian motions in one dimension all started from the origin and conditioned never to collide with each other in a given finite time interval (0,T] is studied. The spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue statistics of random Hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian orthogonal ensemble (GOE) statistics as the time t goes on from 0 to T. In the present paper, we characterize this GUE-to-GOE transition by presenting the graphical expansion formula for the moments of positions of vicious walkers. In the GUE limit t-->0, only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus. Following the time evolution of the vicious walkers, however, the graphs with twisted ribbons, called Möbius graphs, increase their contribution to our expansion formula, and we have to deal with the topology of nonorientable surfaces. Application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the Möbius expansion using the Stirling numbers of the first kind.
研究了一维布朗运动系统，所有运动均从原点出发，且在给定的有限时间区间(0,T]内条件是永不相互碰撞。这种“恶行者”的空间分布可以用随机厄米矩阵的排斥特征值统计来描述，并且已经表明，随着时间t从0到T的推移，当前的“恶行者”模型呈现出从高斯酉系综(GUE)统计到高斯正交系综(GOE)统计的转变。在本文中，我们通过给出“恶行者”位置矩的图形展开公式来刻画这种从GUE到GOE的转变。在GUE极限t→0时，只有带状图起作用，问题简化为按亏格对可定向曲面进行分类。然而，随着“恶行者”的时间演化，带有扭曲带的图，即所谓的莫比乌斯图，对我们的展开公式的贡献增加，我们必须处理不可定向曲面的拓扑结构。利用动力学相关函数的最新精确结果，使用第一类斯特林数得到了莫比乌斯展开中系数的封闭表达式。
