  Let $M^m$ be a minimal properly immersed submanifold in an ambient space
close, in a suitable sense, to the space form $\mathbb{N}^n_k$ of curvature
$-k\le 0$. In this paper, we are interested in the relation between the density
function $\Theta(r)$ of $M^m$ and the spectrum of the Laplace-Beltrami
operator. In particular, we prove that if $\Theta(r)$ has subexponential growth
(when $k<0$) or sub-polynomial growth ($k=0$) along a sequence, then the
spectrum of $M^m$ is the same as that of the space form $\mathbb{N}^m_k$.
Notably, the result applies to Anderson's (smooth) solutions of Plateau's
problem at infinity on the hyperbolic space $\mathbb{H}^n$, independently of
their boundary regularity. We also give a simple condition on the second
fundamental form that ensures $M$ to have finite density. In particular, we
show that minimal submanifolds of $\mathbb{H}^n$ with finite total curvature
have finite density.
