I am looking for literature on the question whether a randomly chosen sequence of $k$-regular graphs converges in the Benjamini-Schramm sense to the universal covering with probability one.

There are several ways to make this a precise question. I would be interested in any result. This is not my area, so it might well be common knowledge. I'd appreciate any help anyway.

Let me propose one way to make the question precise: Let $k,n$ be natural numbers and let $X(k,n)$ be the set of isomorphy classes of $k$-regular connected graphs with $n$ vertices. Then $v(k,n)=n\,|X(k,n)|$ is the total number of vertices in the union over $X(k,n)$.

Fix some $R>0$ and let $v(k,n)_{\ge R}$ denote the number of such vertices with $\mathrm{Inj}(v)\ge R$, where $\mathrm{Inj}(v)$ denotes the injectivity radius at $v$ (i.e., the radius of the largest disk around $v$ which is a tree). The question is, whether the quotient $$ \frac{v(k,n)_{\ge R}}{v(k,n)}$$ tends to 1, as $n\to\infty$, where $k$ remains fixed.

A related question of interest is the case when one fixes one finite connected $k$-regular graph and asks for all connected coverings of a degree $d$, when $d$ tends to infinity.