A friend of mine, obtained a lower bound for the trace norm of matrices described in this question (for the special case $a_{ij} = \pm 1$). That lower bound is $ \frac{f(n)}{2\pi}$ where $$ f(n) := \int_0^\infty \log\left( \frac{(1+t)^n +(1-t)^n}{2} +n(n-1) t(1+t)^{n-2}\right)t^{- 3/2} \ \mathrm{d}t $$ Numerical computaions suggest that $$ f(n) = 4 \pi n + o(n) $$ How to justify it? Moreover, is it possible to obtain a good rate of convergence?