The following does not decide on whether the infimum is $0$, but it does give a 
rough bound. 

Let $v$ be a normalized eigenvector. Suppose  $\|v\|_1<n^{\alpha}$ 
for some $\alpha<1/2$. Let
$I=\{i: |v_i|<n^{-\beta}\}$ with $\beta>\alpha$ t.b.d. By the no-gap delocalization property of Rudelson-Vershynin (see https://arxiv.org/pdf/1506.04012.pdf), with high probability, for al such vectors,
$$C(|I|/n)^{12}\leq \sum_{i\in I} |v_i|^2\leq n^{-\beta} \sum_{i\in I} |v_i|\leq n^{\alpha-\beta}.$$
(I am simplifying a bit, out of laziness; in reality you have to make sure $I$ is not too small, but the argument still goes through. See the difference between Theorem 1.3 and 1.5 there.) 
Therefore,
$$|I|\leq  n\cdot (n^{(\alpha-\beta)})^{1/12}.$$
Thus, if $\alpha<\beta$ then $|I|/n\to 0$. That is, the cardinality of the set of coordinates larger than $n^{-\beta}$ is larger than (say) $n/2$. But then,
$$n^{\alpha}\geq \sum_i |v_i|\geq \sum_{i\notin I} |v_i|\geq n^{1-\beta}/2.$$
Thus, we get that $\alpha\geq 1-\beta$. Since $\beta$ is arbitrary subject to the constraint $\beta>\alpha$, it follows that $\alpha>1/2$, which contradicts $\alpha<1/2$.

**Added in edit:** I realize that I was writing about a matrix with independent centered entries; the Laplacian matrix does not fall in that framework, although its no gap delocalization  is treated (with weaker bounds) in Eldan's work, reference 9 in the cited paper of Rudelson-Vershunin.