Rooted trees and moments of large sparse random matrices

In late 50th E. Wigner studied the moments of N N random symmetric matrices whose entries are independent identically distributed real random variables [Wig55]. He observed that if the law of these variables is symmetric, then after certain normalization, the leading contribution to the 2k-th moment M N 2k as N ∞ is described by the set of simple random walks of 2k steps in the upper half-plane starting and ending at zero. Moreover, the limit limN ∞ M N 2k is proportional to the number of these walks. Later this description was combined with the graph theory tools to study the spectral norm of large random matrices of this class known as the Wigner ensemble [BY88, FK81]. The use of the graph theory is possible here due to the one-to-one correspondence between the simple half-plane random walks and the set of rooted trees Tk with k edges drawn in the upper half-plane. Another version of the random walks representation is used to prove the universal character of extreme eigenvalue statistics of large random matrices of the Wigner ensemble [Sos99]. The common feature of these works is that one considers the moments M N 2k in the limit when k increases proportionally to some power of N [Gem]. Then one has to take into account not only the leading contribution to M N 2k , but also next terms of 1 N-expansion of it. To do this, modifications of the method of [Wig55] were proposed [BY88, FK81, Sos99] that involve additional combinatorial constructions. In paper [Kho01] it was shown that the trees still represent a simple and convenient description of the corrections to M N 2k . Namely, it was proved that the rooted trees added by the procedure of vertex gluing and shift of cycles describe all terms of 1 N-expansion of M N 2k . On this way one can separate two different classes of graphs obtained from trees: (A) those that have gluings of children with different parents and (B) those that glue the children of the same parent.