Torquato–Stillinger conjecture on high-dimensional disordered packing terminal density

For sufficiently large $d$, let $g_2({\bf r})$ be a hard-core nonnegative tempered distribution satisfying

$$g_2({\bf r})=1+{\cal O}(|{\bf r}|^{-d-\varepsilon})$$

for some $\varepsilon>0$. Let $\rho$ be the number density, let $h({\bf r})=g_2({\bf r})-1$, and define the structure factor by

$$S({\bf k})\equiv 1+\rho\widetilde h({\bf k}).$$

Torquato–Stillinger conjecture. The function $g_2({\bf r})$ is a pair correlation function of a translationally invariant disordered sphere packing in $\mathbb{R}^d$ at number density $\rho$ if and only if $S({\bf k})\geq 0$. The maximum achievable density is the terminal density $\phi_*$. This conjecture asserts that, in sufficiently high dimensions, the stated necessary conditions are also sufficient and that the terminal density characterizes the maximum density achievable by such disordered packings.