There exists extensive theory for the concentration of Gaussian measure. Through that, it can be easily proved that the square of the $\ell_2$ norm of a length $n$ zero mean Gaussian vector ${\bf x}$ with covariance matrix ${\bf I}_n$ is concentrated around $n$ with overwhelming probability. Such a result also follows immediately from the Restricted Isometry Property that holds for Gaussian matrices. I was wondering if any concentration results could be inferred for inner products of Gaussian i.i.d random variables. Namely, if $\Pr({\bf x}{\bf y}<\alpha)$ is overwhelming (or exponentially vanishing in $n$) for some meaningful $\alpha$.

If you have 2 standard Gaussians in $\mathbb{R}^n$, their inner product is the sum of $n$ i.i.d. variables, with their common distribution fixed (and having finite moments), so you will get convergence to the appropriate Gaussian distribution in line with the central limit theorem, with exponential bounds coming from Hoeffding's inequality, say. Do you need tight bounds or asymptotics is enough? 


I am not sure if I am understanding you quite clearly: you have i.i.d Gaussian random variables $x_i, y_i$ so that for any $\alpha > 0$ the quantity $P[\sum_i x_i y_i < \alpha]$ goes to zero. What you are looking for is the rate of convergence to zero ? Or you are looking for a function $\phi$ that gives nontrivial bounds like $P[\sum_{i=1}^n x_i y_i < \alpha] \leq \phi(\alpha, n)$ ? 

