Uniform erasure robustness conjecture for random Gabor frames

Let $(g,\Lambda)$ be a Gabor frame with $g$ uniformly distributed on the unit sphere $\mathbb{S}^{M-1}$, and let $\Lambda\subset\mathbb{Z}_M\times\mathbb{Z}_M$. For $p\in[0,1]$, define

$$\Delta(p)=\min_{\substack{\Lambda'\subset\Lambda,\\ |\Lambda'|\geq(1-p)|\Lambda|}}\sigma_{\min}^2(\Phi_{\Lambda'}^*).$$

Uniform erasure-robustness conjecture. Suppose that $|\Lambda|=O(M\log^\alpha M)$, where $\alpha\geq0$ is to be specified. Then, for every $p\in(0,1)$, $\Delta(p)\geq C$ with high probability, where $C>0$ depends only on $p$. Numerical experiments suggest that this lower bound is dimension-independent and would establish robustness of the Gabor frame under a fixed proportion of erasures; the permissible exponent $\alpha$ remains to be determined.