The umpteen operator and its Lifshitz tails

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.