Wigner Theorem for Random Matrices

We give the main steps for a proof of the theorem.

1. The Catalan number

For any natural integer n, we define the nth Catalan number given by

Let q be a natural integer.

• A sequence of natural numbers (S_p)_{p in {0,1,...,q}} is a Bernoulli walk if S_0=0 and |S_{p+1}-S_p|=1 for all p in {0,1,...,q-1}. That is there are increments of +/-1 at each step.
• A Dyck path of length 2q is a Bernoulli walk with non-negative values ending at 0, i.e. S_{2q}=0.

Result 1: The number of Dyck paths of length 2q is C_q.

Result 2: We have the series expansion:

2. Words

Let n be a natural integer. The set {1,2,...,n} can be seen as an alphabet. Any sequence s_1, ..., s_q of letters in {1,2,...,n} is a word. Let M be the set of words from the alphabet {1,2,...,n}. Two words m and m' are equivalent if there is a bijection between their letters (thus 12321 and 42524 are equivalent), and we write m~m'.

the binary relation ~ is an equivalence relation, that is it is reflexive (m~m), symmetrical (m~m' if and only if m'~m) and transitive (if m~m' and m'~m'', then m~m''). There exists a partition of M made of equivalence classes, each of these having a representant (arbitrary).

Let M_q,p be the set of such representants of the words of length q+1 having p distinct letters and such that each pair of letters is consecutively repeated at least twice, no matter the order.

Result 3: the cardinal (number of elements) of M_2q,q+1 is C_q.

3. Wigner Theorem

Let M_n be a n x n symmetric random matrix such that its elements are Z_i,jI/sqrt(n), and the Z_i,j are all independent and identically distributed random variables with finite moments (see below Wigner Theorem for more precisions). We let λ_n,i be the ith eigenvalue (increasing order) of M_n.

Let

Consider \epsilon>0. With the Bienaymé-Tchebychev inequality, we have:

It is tedious to prove that the right-hand-side tends to 0, so that the sequence (γ_n,2q)_n in fact tends to the number C_q (as (E(γ_n,2q))_n does).

Using the Markov and Bienaymé-Tchebychev inequalities, we can prove that

for any positive integer k and B>0.

Now, let f be a continuous and bounded function on R. According to the Wierstrass approximation theorem, there exists a polynomial P such that (uniform norm):

We finally have:

Three points:

• The inequality (4) allows us saying that the limit of the first term of the right-hand-side of the inequality (7) tends to 0.
• We have the following facts

and bearing with this in mind, the second term tends to 0.

• The inequality (5) allows us to say that the third term tends to 0.

Hence, we deduce the theorem:

For all the theoretical justifications, find them in the formidable book: