Glass transition of hard spheres in high dimensions

We have investigated analytically and numerically the liquid-glass transition of hard spheres for dimensions d→∞ 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∼d^1/2, but become identical on a scale k∼d, which is proven analytically. The critical packing fraction φ_c(d)∼d²2^−d is above the corresponding Kauzmann packing fraction φ_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.