Sharpness of the disordered sphere-packing bound

Determine whether the asymptotic lower bound of order d log d divided by 2^{d+1} is sharp up to constant factors for highly disordered sphere packings.

Background

The paper discusses sphere packings whose density is asymptotically at least (1-o(1))d log d/2{d+1}. These packings are described as highly disordered, in contrast with structured lattice packings. The authors indicate that it is plausible that this lower bound captures the correct scale for such packings, but do not establish the corresponding upper bound.

References

It moreover seems reasonable to conjecture that the bound~eq:our-spheres is actually sharp, up to constant factors, for such ``highly disordered'' sphere packings.

eq:our-spheres:

θ(d)(1−o(1))dlog⁡d2d+1 .\theta(d) (1-o(1)) \frac{d\log d}{2^{d+1}}\, .

— Probabilistic combinatorics at exponentially small scales  (2512.15077 - Sahasrabudhe, 17 Dec 2025) in Section 1, Introduction; discussion of sphere packing in high dimensions