While experimenting with positive-definite functions, I was led to the following:

Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider the (positive-definite) matrix $$M_{ij}=\frac{1}{1+|x_i-x_j|}.$$ Now I wish to know:

How do I obtain an estimate for the smallest eigenvalue $\lambda_n$ of $M$?

Preliminary experiments (see plot; x-axis: $n$, y-axis: $\lambda_n$) suggest that $\lambda_n \approx 1/n^2$, but how do I prove that or perhaps a more accurate result?