On the moments of the Gaussian orthogonal ensemble

I am currently reading Mehta's book on random matrices and decided to implement a Mathematica program to calculate expectation values in the Gaussian orthogonal ensemble (GOE). This is the set of symmetric \( n \times n \) matrices \( H \) with probability measure \begin{equation*} P(H) dH = \mathcal{N} \prod_{ i \le j} dH_{ij} \ \exp \left( -\frac{1}{2}\ \mathrm{tr}(H^2)\right) \end{equation*} My program uses recursion based on Wick's theorem (also called Isserlis' theorem according to Wikipedia), and also some rules for summing over indices in \( n \) dimensions. I used ideas from Derevianko's program Wick.m

Some expectation values in the Gaussian orthogonal ensemble

I calculate some expectation values if the probability measure is given by \begin{equation*} P(H) dH = \mathcal{N} \prod_{ i \le j} dH_{ij} \ \exp \left( -\frac{1}{2}\ \mathrm{tr}(H^2)\right) \end{equation*} Hereby are \( H \) symmetric \( n \times n \) matrices and \(\mathcal{N}\) is the normalization factor. This is a special case of the Gaussian orthogonal ensemble (GOE).

Invariance of the Gaussian orthogonal ensemble

On page 17 in his book , Mehta proves the following result about the ensemble of symmetric \( n \times n \) matrices \( H \)

1. If the ensemble is invariant under every transformation \( H \mapsto R H R^T \) with \( R \) an orthogonal matrix
2. and if all components \( H_{ij}, i \le j \) are independent

then the probability measure has the form \begin{equation}\label{eq:20151025e} \prod_{ i \le j} dH_{ij} \ \exp \left( -a\ \mathrm{tr}(H^2) + b\ \mathrm{tr} H + c \right) \end{equation} with \( a, b \) and \(c \) constants.

I prove here the converse, namely, the probability measure \eqref{eq:20151025e} is invariant under transformations \( H \mapsto R H R^T \).

Gaussian orthogonal ensemble

I am currently reading Mehta's book on random matrices (the first edition because it is thinner than the third). I plan to write some blog posts while studying this book. In chapter 2, Metha defines the Gaussian orthogonal ensemble. This is the set of symmetric \( n \times n \) matrices \( H \) with probability density \begin{equation}\label{eq:20151025a} \prod_{ i \le j} dH_{ij} \ \exp \left( -a\ \mathrm{tr}(H^2) + b\ \mathrm{tr} H + c \right) \end{equation} with \( a, b \) and \(c \) constants. It can be calculated that this density function is invariant under transformations \begin{equation}\label{eq:20151025b} H \mapsto R H R^T \end{equation} with \( R \) an orthogonal matrix.

 This is completely equivalent with the vector case. In the vector case the probability density is \begin{equation}\label{eq:20151025c} \prod_{ i} dx_i \ \exp \left( -a\ \sum_i x^2_i + c \right) \end{equation} This density is invariant under rotations \begin{equation}\label{eq:20151025d} x \mapsto R x \end{equation}

One can see that \eqref{eq:20151025d} is the vector representation of the orthogonal group and \eqref{eq:20151025b} is the representation on symmetric matrices. Because symmetric matrices do not form an irreducible representation of the orthogonal group - I can namely subtract the trace - I wonder at this point if one also studies something like ''Gaussian orthogonal ensemble on traceless symmetric matrices''.

Sylvester, On Arithmetical Series, 1892

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In Chebyshev's Mémoire, which is a famous paper in the history of number theory, Chebyshev obtains bounds on the function \begin{equation*} \psi(x) = \sum_{p^{\alpha} \le x} \log p \end{equation*} where the sum is over prime powers. Sylvester improved Chebyshev's bounds in his paper On Arithmetical Series by applying two strategies:

1. Whereas Chebyshev analyzed the expression \begin{equation*} T(x) - T\left( \frac{x}{2} \right)- T\left( \frac{x}{3} \right)- T\left( \frac{x}{5} \right)+ T\left( \frac{x}{30} \right) \end{equation*} where \( T(x) = \sum_{ 1 \le n \le x} \log n\ \), Sylvester analyzed more complex expressions, for example \begin{equation*} T(x) - T\left( \frac{x}{2} \right)- T\left( \frac{x}{3} \right)- T\left( \frac{x}{5} \right)+ T\left( \frac{x}{6} \right)-T\left( \frac{x}{7} \right)+ T\left( \frac{x}{70} \right)- T\left( \frac{x}{210} \right) \end{equation*}
2. Sylvester also applied an iterative method so that he could strengthen bounds he obtained in a recursive manner.

Method (1) is explained in part II, §1 of the paper; method (2) in part II, §2. Sylvester obtained the best bounds by combining (1) and (2). The whole exercise is now more of historical interest because the prime number theorem, which is the statement \( \psi(x) \sim x \), has been proved four years after the publication of Sylvester's paper. Nevertheless, in this post I will see how method (2) works in a simple situation. I will of course obtain bounds that are weaker than Sylvester's. More elaborations on method (1) can be found in this post.