2020/10/04 by Ohad N. Feldheim, Feldheim, Ohad N., Sasha Sodin +1
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Random Matrices and Applications #math-ph #math.MP #math.PR #math.SP
paper · pdf · doi:10.48550/arxiv.2010.01552
v2: 20 pages, 2 figures. Minor revision. v3: The text is identical to v2. We note that the upper bound in Theorems 1 and 2 can be improved using the results of Erschler and Zheng (Annales de l'Institut Fourier, 2020, arXiv:1708.04730, Example 4.2), answering the first question on p. 6 of our note in the affirmative
arxiv created 2022/04/19 · arxiv updated 2022/04/21
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 ≥ 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.