Fractional moment asymptotics for characteristic polynomials of the real elliptic ensemble

Let $A_{N,\tau}$ be a matrix from the real elliptic ensemble with symmetry parameter $\tau\in(-1,1)$, and let $\mu$ lie in the real bulk interval $(-(1+\tau),1+\tau)$. Let $G$ denote the Barnes $G$-function. Fractional moment asymptotics. For any $\ell\in(-1,\infty)$,

$$\mathbb{E}|\det(A_{N,\tau}-\mu I)|^{\ell}=e^{\frac{N \ell}{2} \left(\frac{\mu^2}{1+\tau}-1\right)}N^{\ell(\ell-1)/4}C_{\tau}(\ell)(1+o(1)),$$

where

$$C_{\tau}(\ell)=\left(\frac{(1+\tau)^{\ell/2}}{(1-\tau^2)^{\ell(\ell+1)/4}}\right)\left(\frac{(2\pi)^{\ell/2}G\left(\frac{1}{2}\right)}{2^{\ell(\ell-1)/4}G\left(\frac{\ell}{2}+1\right)G\left(\frac{\ell}{2}+\frac{1}{2}\right)}\right).$$

This extends the corresponding conjecture for the real Ginibre ensemble to the elliptic ensemble and generalizes the preceding integer-moment asymptotics; the fractional-moment regime remains conjectural.