## Monday, December 14, 2015

### Duality in Random Matrix Theory

I provide examples of a duality between the Gaussian symplectic ensemble and the Gaussian orthogonal ensemble.
For example, in the Gaussian symplectic ensemble (GSE), one can calculate the following expectation values: \begin{align*} \mathbb{E}[ \mathrm{tr} 1 ] &= 2 n\\ \mathbb{E}[ \mathrm{tr} (H^{2}) ] &= n (2 n-1)\\ \mathbb{E}[ \mathrm{tr} (H^{4}) ] &= \frac{1}{2} n \left(8 n^2-10 n+5\right)\\ \mathbb{E}[ \mathrm{tr} (H^{6}) ] &= \frac{1}{4} n \left(40 n^3-88 n^2+104 n-41\right)\\ \mathbb{E}[ \mathrm{tr} (H^{8}) ] &= 28 n^5-93 n^4+187 n^3-\frac{345 n^2}{2}+\frac{509 n}{8}\\ \mathbb{E}[ \mathrm{tr} (H^{10}) ] &=\frac{1}{16} n \left(1344 n^5-6176 n^4+18320 n^3-28600 n^2+24286 n-8229\right)\\ \mathbb{E}[ \mathrm{tr} (H^{12}) ] &=\frac{1}{32} n \left(8448 n^6-50752 n^5+204768 n^4-470080 n^3+668592 n^2-516958 n+166377\right) \end{align*} In a previous blog post I calculated these expectation values in the Gaussian orthogonal ensemble (GOE): \begin{align*} \mathbb{E}[ \mathrm{tr} 1 ] &= 2 n\\ \mathbb{E}[ \mathrm{tr} (H^{2}) ] &= n (2 n+1)\\ \mathbb{E}[ \mathrm{tr} (H^{4}) ] &= \frac{1}{2} n \left(8 n^2+10 n+5\right)\\ \mathbb{E}[ \mathrm{tr} (H^{6}) ] &= \frac{1}{4} n \left(40 n^3+88 n^2+104 n+41\right)\\ \mathbb{E}[ \mathrm{tr} (H^{8}) ] &= 28 n^5+93 n^4+187 n^3+\frac{345 n^2}{2}+\frac{509 n}{8}\\ \mathbb{E}[ \mathrm{tr} (H^{10}) ] &=\frac{1}{16} n \left(1344 n^5+6176 n^4+18320 n^3+28600 n^2+24286 n+8229\right)\\ \mathbb{E}[ \mathrm{tr} (H^{12}) ] &=\frac{1}{32} n \left(8448 n^6+50752 n^5+204768 n^4+470080 n^3+668592 n^2+516958 n+166377\right) \end{align*} If one formally replaces $n$ by $-n$ in the expectation values in the GSE, one gets the expectation values in the GOE (up to sign). With a formula one can write $$\mathbb{E}[ \mathrm{tr} (H^{2 p}) ]_{\mathrm{GSE}(2n)} =(-1)^{p+1} \mathbb{E}[ \mathrm{tr} (H^{2 p}) ] _{\mathrm{GOE}(-2n)}$$ This is a duality between the Gaussian orthogonal ensemble and the Gaussian symplectic ensemble. I find this a bizarre formula because the natural interpretation of the right hand side seems to involve matrices of "negative dimension". Nevertheless, this connection between the orthogonal group and the symplectic group shows up in various places in mathematics and physics. For example, one has proven that $Sp(2N)$ and $SO(−2N)$ gauge theories are equivalent (see reference below).

Similary, I can calculate expectation values in the Gaussian unitary ensemble: \begin{align*} \mathbb{E}[ \mathrm{tr} (H^{2}) ] &= n^2\\ \mathbb{E}[ \mathrm{tr} (H^{4}) ] &= 2 n^3+n\\ \mathbb{E}[ \mathrm{tr} (H^{6}) ] &= 5 n^2 \left(n^2+2\right)\\ \mathbb{E}[ \mathrm{tr} (H^{8}) ] &= 7 n \left(2 n^4+10 n^2+3\right)\\ \mathbb{E}[ \mathrm{tr} (H^{10}) ] &=84 n^2 \left(32 n^4+80 n^2+23\right)\\ \mathbb{E}[ \mathrm{tr} (H^{12}) ] &=66 n \left(256 n^6+1120 n^4+784 n^2+45\right)\\ \end{align*} Now one sees that the expectation values are invariant (up to sign) under the transformation $n \mapsto -n$. With a formula $$\mathbb{E}[ \mathrm{tr} (H^{2 p}) ]_{\mathrm{GUE}(n)} =(-1)^{p+1} \mathbb{E}[ \mathrm{tr} (H^{2 p}) ] _{\mathrm{GUE}(-n)}$$ One could say that the Gaussian unitary ensemble is self-dual.

• The above expectation values are for matrices of size $2 n \times 2 n$ in the GSE and GOE, and matrices of size $n \times n$ in the GUE.