The dimensional evolution of structure and dynamics in hard sphere liquids

The formulation of the mean-field, infinite-dimensional solution of hard sphere glasses is a significant milestone for theoretical physics. How relevant this description might be for understanding low-dimensional glass-forming liquids, however, remains unclear. These liquids indeed exhibit a complex interplay between structure and dynamics, and the importance of this interplay might only slowly diminish as dimension $d$ increases. A careful numerical assessment of the matter has long been hindered by the exponential increase of computational costs with $d$. By revisiting a once common simulation technique involving the use of periodic boundary conditions modeled on $D_d$ lattices, we here partly sidestep this difficulty, thus allowing the study of hard sphere liquids up to $d=13$. Parallel efforts by Mangeat and Zamponi [Phys. Rev. E 93, 012609 (2016)] have expanded the mean-field description of glasses to finite $d$ by leveraging standard liquid-state theory, and thus help bridge the gap from the other direction. The relatively smooth evolution of both structure and dynamics across the $d$ gap allows us to relate the two approaches, and to identify some of the missing features that a finite-$d$ theory of glasses might hope to include to achieve near quantitative agreement.