It is well known that there are close connections between non-intersecting processes in one dimension and random matrices, based on the reflection principle. There is a generalisation of the reflection principle for more general (e.g. planar) processes, due to Fomin, in which the non-intersection condition is replaced by a condition involving loop-erased paths. In the context of independent Brownian motions in suitable planar domains, this also has close connections to random matrices. An example of this was first observed by Sato and Katori (Phys Rev E 83:041127, 2011). We present further examples which give rise to various Cauchy-type ensembles. We also extend Fomin’s identity to the affine setting and show that in this case, by considering independent Brownian motions in an annulus, one obtains a novel interpretation of the circular orthogonal ensemble.

中文翻译：

---

循环擦除游走和随机矩阵

众所周知，基于反射原理，一维非相交过程与随机矩阵之间存在密切联系。由于 Fomin，反射原理对更一般的（例如平面）过程进行了推广，其中非相交条件由涉及循环擦除路径的条件代替。在合适的平面域中的独立布朗运动的背景下，这也与随机矩阵密切相关。Sato 和 Katori (Phys Rev E 83:041127, 2011) 首先观察到了这样的一个例子。我们提出了进一步的例子，这些例子产生了各种柯西型集合。我们还将 Fomin 的恒等式扩展到仿射设置，并表明在这种情况下，通过考虑环中的独立布朗运动，