关于与虚置换相关的算子流
On a flow of operators associated to virtual permutations
摘要：
Kerov、Olshanski和Vershik引入了所谓的虚置换，定义为在$N$阶对称群中的置换族$(σ_N)_{N \geq 1}$、$σ_N$，使得$σ_N$的循环结构可以简单地通过移除元素$N+1$从$σ_{N+1}$的结构推导出来。虚拟排列，特别是概率措施对相应的空间是不变的共轭，已研究了详细的Tsilevich。在本文中，我们证明了，对于一个大类的这种不变的措施（特别是包含任何参数$θ\geq 0$的Ewens措施），它是可能的关联流$(T^α)_{α\in \mathbb{R}}$的随机运营商在一个合适的功能空间。此外，如果$(σ_N)_{N \geq 1}$是遵循上述类别中的分布的随机虚拟置换，则算子$T^α$可以被解释为置换$σ_N^{α_N}$的极限，在必须精确的意义上，其中$N$趋于无穷大，并且$α_N$等价于$αN$。在这种解释的关系，我们证明了无穷小生成元的$(T^α)_{α\in \mathbb{R}}$的特征值等于与$σ_N$的置换矩阵的重标特征角的限制。
英文摘要：
Kerov, Olshanski and Vershik introduced the so-called virtual permutations, defined as families of permutations $(σ_N)_{N \geq 1}$, $σ_N$ in the symmetric group of order $N$, such that the cycle structure of $σ_N$ can be deduced from the structure of $σ_{N+1}$ simply by removing the element $N+1$. The virtual permutations, and in particular the probability measures on the corresponding space which are invariant by conjugation, have been studied in a details by Tsilevich. In the present article, we prove that for a large class of such invariant measures (containing in particular the Ewens measure of any parameter $θ\geq 0$), it is possible to associate a flow $(T^α)_{α\in \mathbb{R}}$ of random operators on a suitable functional space. Moreover, if $(σ_N)_{N \geq 1}$ is a random virtual permutation following a distribution in the class described above, the operator $T^α$ can be interpreted as the limit, in a sense which has to be made precise, of the permutation $σ_N^{α_N}$, where $N$ goes to infinity and $α_N$ is equivalent to $αN$. In relation with this interpretation, we prove that the eigenvalues of the infinitesimal generator of $(T^α)_{α\in \mathbb{R}}$ are equal to the limit of the rescaled eigenangles of the permutation matrix associated to $σ_N$.
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