论文标题
毫无生气的操作员及其Lifshitz尾巴
The umpteen operator and its Lifshitz tails
论文作者
论文摘要
正如克罗夫（Kerov）在1990年代初提出的那样，在随后的作品中阐明了Wigner随机矩阵的许多属性，某些线性地图在对称群体的表示理论中发挥了重要的作用。我们介绍并研究了代表理论来源的运营商，该操作员与在$ d $维晶格上发挥作用的离散随机Schrödinger运算符具有一定的相似性。特别是，我们定义了其综合状态密度，并证明在维度$ d \ geq 2 $中，它具有与安德森模型相似的Lifshitz尾巴。 该结构与十五个难题的无限板版本密切相关，这是十五世纪的流行滑动难题。我们估计，使用新的PEIERLS参数，拼图在$ N $随机移动后返回其初始状态的可能性。 LIFSHITZ尾巴是使用我们随机操作员的识别来推导的，该识别是拼图的邻接矩阵在无限对称组的随机表示中的作用。
As put forth by Kerov in the early 1990s and elucidated in subsequent works, numerous properties of Wigner random matrices are shared by certain linear maps playing an important rôle in the representation theory of the symmetric group. We introduce and study an operator of representation-theoretic origin which bears some similarity to discrete random Schrödinger operators acting on the $d$-dimensional lattice. In particular, we define its integrated density of states and prove that in dimension $d \geq 2$ it boasts Lifshitz tails similar to those of the Anderson model. The construction is closely related to an infinite-board version of the fifteen puzzle, a popular sliding puzzle from the XIX-th century. We estimate, using a new Peierls argument, the probability that the puzzle returns to its initial state after $n$ random moves. The Lifshitz tail is deduced using an identification of our random operator with the action of the adjacency matrix of the puzzle on a randomly chosen representation of the infinite symmetric group.
