Tails of Condition Number Distributions

Let $\kappa$ be the condition number of an $m$-by-$n$ matrix with independent standard Gaussian entries, either real ($\beta = 1$) or complex ($\beta = 2$). The major result is the existence of a constant $C$ (depending on $m$, $n$, and $\beta$) such that $P[\kappa > x] < C \, x^{-\beta}$ for all $x$. As $x \rightarrow \infty$, the bound is asymptotically tight. An analytic expression is given for the constant $C$, and simple estimates are given, one involving a Tracy--Widom largest eigenvalue distribution. All of the results extend beyond real and complex entries to general $\beta$.