What does the "hard edge" and the "soft edge" of a spectrum refer to?

Question

This comes up in the context of the spectrum of random matrices, and probably other places. Searching around I can't find a reference for what "hard" or "soft" means--can anyone help?

Answer 1

I literally put "hard edge soft edge random matrix spectrum" into Google and this paper was one of the top hits:

Forrester, "Hard and soft edge spacing distributions for random matrix ensembles with orthogonal and symplectic symmetry", Nonlinearity 19(12), 2006.

The largest eigenvalue is thus to leading order at $\sqrt{2N}$, but to higher order the eigenvalue density is nonzero to the right of this point so it is referred to as the soft edge.

...

The Laguerre ensembles have their origin in positive definite Hermitian matrices and as such all eigenvalues are positive. Because of this constraint the neighbourhood of the origin in the Laguerre ensembles is referred to as the hard edge.

It seems that an edge is "hard" or "soft" depending on whether the eigenvalue density drops completely to zero or not.

Answer 2

Quote from Non-asymptotic theory of random matrices: extreme singular values

In Section 2 we review estimates on the largest singular value (the soft edge). The more difficult problem of estimating the smallest singular value (the hard edge) is discussed in Section 3...