A new lower bound on the hard sphere packing density in large space dimensions was recently proven [1]. I will try to explain why, in my opinion, this is a huge breakthrough in our understanding of the sphere packing problem in high dimensions. To this aim, I will informally review some rigorous mathematical results [2], and some physics results coming from both replica theory [3] and numerical computations [4]. I conclude with a review of possible scenarios for the hard sphere phase diagram in large dimensions [5].
[1] B.Klartag, arxiv.org:2504.05042
[2] See e.g. H.Cohn, arXiv:1603.05202 and 1611.01685
[3] G.Parisi, F.Zamponi - Rev. Mod. Phys. 82, 789 (2010)
[4] P.Charbonneau, C.M.Gish, R.S.Hoy, P.K.Morse - Eur.Phys.J. E 44, 101 (2021)
[5] P.Charbonneau, P.K.Morse, W.Perkins, F.Zamponi - Phys.Rev.E 104, 064612 (2021)