Glass transition of hard spheres in high dimensions

We have investigated analytically and numerically the liquid-glass transition of hard spheres for dimensions $d\to \mathrm{\infty }$ in the framework of mode-coupling theory. The numerical results for the critical collective and self nonergodicity parameters $f_c(k;d)$ and $f_c^{(s)}(k;d)$ exhibit non-Gaussian $k$ -dependence even up to $d=800$. $f_c^{(s)}(k;d)$ and $f_c(k;d)$ differ for $k\sim d^{1\mathrm{/}2}$, but become identical on a scale $k\sim d$, which is proven analytically. The critical packing fraction ${\varphi }_c(d)\sim d^2{2}^{-d}$ is above the corresponding Kauzmann packing fraction ${\varphi }_K(d)$ derived by a small cage expansion. Its quadratic pre-exponential factor is different from the linear one found earlier. The numerical values for the exponent parameter and therefore the critical exponents $a$ and $b$ depend on $d$, even for the largest values of $d$.