تقرير
The umpteen operator and its Lifshitz tails
| العنوان: | The umpteen operator and its Lifshitz tails |
|---|---|
| المؤلفون: | Feldheim, Ohad N., Sodin, Sasha |
| سنة النشر: | 2020 |
| المجموعة: | Mathematics Mathematical Physics |
| مصطلحات موضوعية: | Mathematical Physics, Mathematics - Probability, Mathematics - Spectral Theory |
| الوصف: | 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 r\^ole 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 Schr\"odinger 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. Comment: 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 |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2010.01552 |
| رقم الأكسشن: | edsarx.2010.01552 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |