基于图的低温不相交布朗运动极限分布的平衡问题
A graph-based equilibrium problem for the limiting distribution of non-intersecting Brownian motions at low temperature
摘要：
我们考虑n条不相交的布朗运动路径，p条路径在t=0时具有指定的起始位置，q条路径在t=1时具有指定的终止位置。路径在任意中间时刻的位置都是一个行列式点过程，在p=1的情况下，它等价于具有外源的高斯酉系综的随机矩阵的特征值分布.对于一般的p和q，我们表明，如果一个温度参数是足够小的，然后分布的布朗路径的特点是在大n极限的向量平衡问题的相互作用矩阵，是基于一个二分可平面图。我们的证明是基于一个相关的（p + q）（p +q）矩阵值的黎曼-希尔伯特问题，其解决方案是建立在多个正交多项式的最速下降分析。最速下降分析的一个新特征是系统地打开大量的全局透镜。
英文摘要：
We consider n non-intersecting Brownian motion paths with p prescribed starting positions at time t=0 and q prescribed ending positions at time t=1. The positions of the paths at any intermediate time are a determinantal point process, which in the case p=1 is equivalent to the eigenvalue distribution of a random matrix from the Gaussian unitary ensemble with external source. For general p and q, we show that if a temperature parameter is sufficiently small, then the distribution of the Brownian paths is characterized in the large n limit by a vector equilibrium problem with an interaction matrix that is based on a bipartite planar graph. Our proof is based on a steepest descent analysis of an associated (p+q) by (p+q) matrix valued Riemann-Hilbert problem whose solution is built out of multiple orthogonal polynomials. A new feature of the steepest descent analysis is a systematic opening of a large number of global lenses.
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