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Let $\mathcal H_n$ denote the space of Hermitian $n\times n$ matrices and let $\mu_{GUE}$ denote the following measure on $\mathcal H_n$: $$ \mu_{GUE}(dM) = \exp\left( -\frac{n}{2}\text{tr}\ M^2\right)\ dM, $$ where $dM$ denotes the Lebesgue measure on $\mathcal H_n$. For fixed $n$, $\mu_{GUE}$ has finite mass. The resulting normalized measure is the GUE ensemble studied in Random Matrix Theory.

Question: Can this measure be extended to the space of self-adjoint operators?

Fuzzy question: Do people study Gaussian ensembles of random self-adjoint operators?

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  • $\begingroup$ I'm not sure but wouldn't that measure pick up only the zero matrix? $\endgroup$
    – Phoenix87
    Commented Jun 27, 2015 at 23:21
  • $\begingroup$ The only measure I've defined is $\mu_{GUE}$ for finite $n$. This is clearly not a Dirac measure. Are you referring to a potential extension of the measure to the space of self-adjoint operators? $\endgroup$
    – pre-kidney
    Commented Jun 27, 2015 at 23:36
  • $\begingroup$ pre-kidney indeed $\endgroup$
    – Phoenix87
    Commented Jun 28, 2015 at 8:41
  • $\begingroup$ What's the difference between a matrix and an operator? $\endgroup$
    – André Henriques
    Commented Jun 28, 2015 at 13:38
  • $\begingroup$ The point is to extend the measure from a finite-dimensional manifold to an infinite-dimensional manifold. $\endgroup$
    – pre-kidney
    Commented Jun 28, 2015 at 14:06

3 Answers 3

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One way of looking at this is that, up to normalization, the GUE is standard Gaussian measure on the space of Hermitian matrices, equipped with the Hilbert-Schmidt or Frobenius inner product. So what you're asking for is a special case of extending the notion of standard Gaussian measure to a particular infinite-dimensional space. But you can't do that.

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A classic problem in this connection is to derive GOE or GUE statistics for the spectrum of a random self-adjoint operator of the form $H=-\nabla^2 + V(\vec{r})$, in some bounded domain of $\mathbb{R}^3$. The measure is the Gaussian measure for $V(\vec{r})$, of zero mean and given two-point correlation function. Since this is a real operator, one would expect GOE statistics, to obtain GUE statistics one would replace $\nabla\mapsto \nabla+i\vec{B}\times\vec{r}$ for some given vector $\vec{B}$.

This problem was solved by Konstantin Efetov in 1982, as described in much detail in his book on Supersymmetry in Disorder and Chaos. The GOE or GUE statistics is found to hold only over a limited range $E_{0}$ (the so-called Thouless energy): eigenvalues that differ by more than $E_0$ become uncorrelated, reverting to Poisson statistics.

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Do you mean Free Probability Theory (See e.g. Voiculescu, D. V.; Dykema, K. J.; Nica, A. Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups) ?

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  • $\begingroup$ Please be more specific. At the moment I am learning about free probability, which is what inspired this question. $\endgroup$
    – pre-kidney
    Commented Jun 28, 2015 at 4:21

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