We have investigated analytically and numerically the liquid-glass transition of hard spheres for dimensions $d\to \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 $k\sim d^{ 1/2}$, but become identical on a scale $k\sim d$, which is proven analytically. The critical packing fraction $\phi_{ c}(d) \sim d^{ 2}2^{ -d}$ is above the corresponding Kauzmann packing fraction $\phi_{ K}(d)$ 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 $d$.
Categories: cond-mat.soft cond-mat.stat-mech