Sunday, October 25, 2015

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''.

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