虚置换的酉扩张
A unitary extension of virtual permutations
摘要：
类似于虚拟置换的空间，我们定义了等距的投射极限：这些序列的酉算子是自然的，在这个意义上，他们最小化连续矩阵之间的秩范数的大小增加。这样构造的虚等距空间可以看作是Kerov，Olshanski和Vershik的虚置换空间以及Neretin的虚等距空间的自然扩张。然后，我们推导出与纯粹的概率方法的几乎必然收敛这些随机矩阵下的哈尔措施：对于一个相干的哈尔措施的虚拟等距，最小的归一化特征角几乎必然收敛到一个点过程的相关函数给出的正弦核。这几乎必然收敛实际上持有一个更大的一类措施，证明了Borodin和Olshanski。我们给出了一个不同的证明，概率论的意义上，它利用鞅参数，并显示如何交织的特征角时，从n维到n+1。我们的方法还证明了，对于某些普适常数收敛速率>0，当维数n趋于无穷大时，几乎必然由n^{-n}控制。
英文摘要：
Analogously to the space of virtual permutations, we define projective limits of isometries: these sequences of unitary operators are natural in the sense that they minimize the rank norm between successive matrices of increasing sizes. The space of virtual isometries we construct this way may be viewed as a natural extension of the space of virtual permutations of Kerov, Olshanski and Vershik as well as the space of virtual isometries of Neretin. We then derive with purely probabilistic methods an almost sure convergence for these random matrices under the Haar measure: for a coherent Haar measure on virtual isometries, the smallest normalized eigenangles converge almost surely to a point process whose correlation function is given by the sine kernel. This almost sure convergence actually holds for a larger class of measures as is proved by Borodin and Olshanski. We give a different proof, probabilist in the sense that it makes use of martingale arguments and shows how the eigenangles interlace when going from dimension n to n+1. Our method also proves that for some universal constant epsilon>0, the rate of convergence is almost surely dominated by n^{-epsilon} when the dimension n goes to infinity.
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