Matrix Group Integrals, Surfaces, and Mapping Class Groups II: $\mathrm{O}\left(n\right)$ and $\mathrm{Sp}\left(n\right)$

Abstract

Let $w$ be a word in the free group on $r$ generators. The expected value of the trace of the word in $r$ independent Haar elements of $\mathrm{O}(n)$ gives a function ${\cal T}r_{w}^{\mathrm{O}}(n)$ of $n$. We show that ${\cal T}r_{w}^{\mathrm{O}}(n)$ has a convergent Laurent expansion at $n=\infty$ involving maps on surfaces and $L^{2}$-Euler characteristics of mapping class groups associated to these maps. This can be compared to known, by now classical, results for the GUE and GOE ensembles, and is similar to previous results concerning $\mathrm{U}\left(n\right)$, yet with some surprising twists. A priori to our result, ${\cal T}r_{w}^{\mathrm{O}}(n)$ does not change if $w$ is replaced with $\alpha(w)$ where $\alpha$ is an automorphism of the free group. One main feature of the Laurent expansion we obtain is that its coefficients respect this symmetry under $\mathrm{Aut}(\mathrm{\mathbf{F}}_{r})$. As corollaries of our main theorem, we obtain a quantitative estimate on the rate of decay of ${\cal T}r_{w}^{\mathrm{O}}(n)$ as $n\to\infty$, we generalize a formula of Frobenius and Schur, and we obtain a universality result on random orthogonal matrices sampled according to words in free groups, generalizing a theorem of Diaconis and Shahshahani. Our results are obtained more generally for a tuple of words $w_1,\ldots,w_\ell$, leading to functions ${\cal T}r_{w_{1},\ldots,w_{\ell}}^{\mathrm{O}}$. We also obtain all the analogous results for the compact symplectic groups $\mathrm{Sp}\left(n\right)$ through a rather mysterious duality formula.