We present the first study of disordered jammed hard-sphere packings infour-, five- and six-dimensional Euclidean spaces. Using a collision-drivenpacking generation algorithm, we obtain the first estimates for the packingfractions of the maximally random jammed (MRJ) states for space dimensions$d=4$, 5 and 6 to be $\phi_{MRJ} \simeq 0.46$, 0.31 and 0.20, respectively. Toa good approximation, the MRJ density obeys the scaling form $\phi_{MRJ}=c_1/2^d+(c_2 d)/2^d$, where $c_1=-2.72$ and $c_2=2.56$, which appears to beconsistent with high-dimensional asymptotic limit, albeit with differentcoefficients. Calculations of the pair correlation function $g_{2}(r)$ andstructure factor $S(k)$ for these states show that short-range orderingappreciably decreases with increasing dimension, consistent with a recentlyproposed ``decorrelation principle,'' which, among othe things, states thatunconstrained correlations diminish as the dimension increases and vanishentirely in the limit $d \to \infty$. As in three dimensions (where $\phi_{MRJ}\simeq 0.64$), the packings show no signs of crystallization, are isostatic,and have a power-law divergence in $g_{2}(r)$ at contact with power-lawexponent $\simeq 0.4$. Across dimensions, the cumulative number of neighborsequals the kissing number of the conjectured densest packing close to where$g_{2}(r)$ has its first minimum. We obtain estimates for the freezing andmelting desnities for the equilibrium hard-sphere fluid-solid transition,$\phi_F \simeq 0.32$ and $\phi_M \simeq 0.39$, respectively, for $d=4$, and$\phi_F \simeq 0.19$ and $\phi_M \simeq 0.24$, respectively, for $d=5$.
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