  Several mean-field theories predict that Hessian matrices of amorphous solids
can be written by using the random matrix in the limit of the large spatial
dimensions $d\to\infty$. Motivated by these results, we here propose a way to
map a Hessian of the amorphous solid to a random matrix. This is possible by
determining the coefficients of a random matrix so that the trace of the random
matrix coincides with the Hessian of the original system. We compare our result
with that of previous numerical simulations of harmonic spheres in several
spatial dimensions $d=3$, $5$, and $9$. For small pressure $p\ll 1$ (near
jamming), we find a good agreement even in $d=3$, and obtain better agreements
in larger $d$, suggesting that the approximation indeed becomes exact in the
limit of large spatial dimensions.

Disordered Systems and Neural Networks;Soft Condensed Matter;Statistical Mechanics