arXiv 1206.21852012-06math.PRcond-mat.stat-mechhep-thmath-phmath.MPnlin.SI

与O 'Connell过程相关的复布朗运动系统

System of Complex Brownian Motions Associated with the O'Connell Process

Makoto Katori

备注：: v3: AMS_LaTeX, 25 pages, no figure, minor corrections made for publication in J. Stat. Phys
发表信息：: J. Stat. Phys. 149 (2012) 411-431
DOI：: 10.1007/s10955-012-0602-y

摘要：

O 'Connell过程是非碰撞布朗运动的一个软化版本（一个参数为$a>0$的几何提升），使得相邻粒子可以在特征长度$a$内改变一维位置的顺序。这个过程不是决定性的。然而，在一个特殊的入口定律下，鲍罗丁和科温给出了一个弗雷德霍姆行列式表达式来表示观测量的期望，这是一个粒子位置指示器的软化。我们将它们的积分核改写成类似于行列式过程的相关核的形式，并证明了，如果粒子数为$N$，弗雷德霍姆行列式矩阵的秩为$N$ .然后利用复布朗运动的$N$ -粒子系统给出了该量的一个表示。给出CBM路径权的行列式表达式的复变函数不是整函数，但在组合极限$a \to 0$下，它成为一个提供共形鞅的整函数，从而恢复了非碰撞布朗运动的CBM表示.

英文摘要：

The O'Connell process is a softened version (a geometric lifting with a parameter $a>0$) of the noncolliding Brownian motion such that neighboring particles can change the order of positions in one dimension within the characteristic length $a$. This process is not determinantal. Under a special entrance law, however, Borodin and Corwin gave a Fredholm determinant expression for the expectation of an observable, which is a softening of an indicator of a particle position. We rewrite their integral kernel to a form similar to the correlation kernels of determinantal processes and show, if the number of particles is $N$, the rank of the matrix of the Fredholm determinant is $N$. Then we give a representation for the quantity by using an $N$-particle system of complex Brownian motions (CBMs). The complex function, which gives the determinantal expression to the weight of CBM paths, is not entire, but in the combinatorial limit $a \to 0$ it becomes an entire function providing conformal martingales and the CBM representation for the noncolliding Brownian motion is recovered.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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