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

Vibrational density of states of jammed packing: mean-field theory