In compressed sensing, we are interested in the restricted isometry property. Suppose the design matrix is $n$ by $p$, consisting of $np$ iid $\mathcal{N}(0, 1/n)$ entries. Assume both $n$ and $p$ are growing to infinity but one may be much larger than the other. Then, my question is, what is the best known upper bound on $k$ such that the restricted isometric constant vanishes asymptotically? 



Vanishing Restricted Isometric Constant
We know that your matrix is $(k,\delta)$-RIP provided $n\geq (C/\delta^2)k\log(p/k)$, where $C$ is an absolute constant. We also know that this bound is tight up to constants (see [this book](http://www.springer.com/us/book/9780817649470) for both results). If you want $\delta$ to vanish asymptotically, the implied bound on $k$ would then depend on how quickly you want $\delta$ to vanish.


