# A result on power of random symmetric matrices and vector multiplication: u^T S^n v

I have a rough intuition about some results regarding random matrices theory, but I'm not sure if this is correct - if it exists already somewhere or if there is a way to prove it clearly.

Given:

• $$n,N \in \mathbb N$$
• $$S_N$$ a random symmetric matrix with normal distribution and dimension $$N$$
• $$u,v \in \mathbb R^N$$ two unitary vectors
• we define: $$X_{N,n} = u^T \frac{S^{2n}}{N^n} v$$

then $$X_{N,n}$$ is a random variable such that:

• For $$N$$ large enough, $$\mathbb E[X_{N,n}] \simeq C_n ~u^T v$$ with C_n Catalan number $$n$$
• $$\text {Var}[X_{N,n}]$$ is decreasing with $$N$$

A few ideas that make me feel it is correct are the followings: first, with $$S_N$$ can be written as $$S_N = O^T D O$$ where $$O$$ is a orthogonal matrix and $$D$$ is a diagonal matrix - furthermore, the distribution of the eigenvalues is well known (in particular when $$N\to \infty$$).

Then, writing $$U=Ou$$ and $$V=Ov$$, we have $$X_{N,n} = U^T \frac{D^{2n}}{N^n} V$$. Also, because $$O$$ is some kind of "random" orthogonal matrix, if we look at the elements of $$U$$ for $$N$$ sufficiently large, I guess it would be pretty much as if:

• $$\sqrt N U_i \sim \mathcal N (0,1)$$
• $$\sqrt N V_i \sim \mathcal N (0,1)$$
• $$\text{Cov}(\sqrt N U_i, \sqrt N V_i) = u^T v$$.

Of course, this is wrong and just an intuition. But now, we would also have: $$X_{N,n} = \frac{1}{N} \sum_{i=1}^N (\sqrt N U_i) \frac{D_{i,i}^{2n}}{N^n} (\sqrt N V_i)$$

With the rough assumption that $$D_{i,i}$$ is independant of $$U_i,V_i$$ (maybe from some rotational invariance arguments?) and that the $$D_{i,i}U_iV_i$$ are i.i.d (which is obviously wrong), then it looks like a setup to apply law of large numbers and argue that for $$N$$ sufficiently large, with Wigner semicircle law, this should get close to: $$\mathbb E\left[ (\sqrt N U_i) \frac{D_{i,i}^{2n}}{N^n} (\sqrt N V_i)\right] = \text{Cov}(\sqrt N U_i, \sqrt N V_i) \mathbb E\left[\frac{D_{i,i}^{2n}}{N^n}\right] = C_n u^T v$$

Anyway, I'm not yet completely familiar with random matrices theory but I guess that if this is true, then there should be some results like this somewhere in some textbooks. Thank you in advance for your help!

Let $$S_N$$ be random symmetric matrix of size $$N\times N$$ whose "free" elements are independent and normally distributed, with expectation zero and variance such that the joint probability distribution of the elements of the matrix is $$C e^{- \mathrm{tr}\ X^2}$$ where $$C$$ is the normalizing constant. It can be shown that the eigenvectors and the eigenvalues are independent. The upper distribution is invariant under the conjugation of any orthogonal matrix. It also can be shown that $$\lim_N \mathbb E [\mathrm{tr}(S_N^{2n}) / N^n] = \int x^n d\rho_{sc}(x)dx= C_n$$ where $$C_n$$ is the $$n^{th}$$ Catalan number and $$\rho_n$$ is the semicircle density. (Note that depending on the definition it might be off by a factor of $$2^{2n}$$.) This is a classical result know as the semicircle law, it was first proved by E. Wigner, later generalized by many (Arnold, Bai, Tao etc.). The results you are interested in are modification of this classical theorem.