If I know $Y$ is a random d-regular bipartite graph (tanner code in coding theory), can I show $Y^TY$ has large spectral gap with high probability?
More specifically: If I know $Y=AX \in \mathbb{R}^{d\times n}$, where $X\in\mathbb{R}^{k\times n}$, and each column of $X$ has exactly $d$ entries, generated uniformly at random, with value from $\{0,1\}$.
Is it possible for me to show that $Y^TY$ has some nice spectral property? e.g., It is nearly block-diagonal or has large eigen-gap, etc.