I need help proving the following fact: Let $G$ be a random $d$-regular graph with adjacency matrix $A$. The smallest eigenvalue $\lambda_n$ of $A$ should satisfy $|\lambda_n| = o_d(d)$. (In particular, I think $|\lambda_n| = O(\sqrt{d})$.)
Exercise 7 in [1] is to prove this fact, so it seems like there should be a simple proof. However, I cannot think of a reasonably simple proof or find one anywhere, despite a lot of searching. In [2], they prove that $\max\{|\lambda_2|, |\lambda_n|\} = o(d)$, but their proof is very complicated and only focused on the $\lambda_2$ part -- further indicating that there is some simple proof I'm missing for the $\lambda_n$ part.
My attempts always seem to come to proving something harder than we started, like, $\mathbb{E}[x_ix_j] \ge -o(1)/n$ for adjacent $i, j$ and any assignment $x$ to the vertices. I have no ideas that seem to actually move me forward, so even a pointer would be greatly appreciated.
[1] https://www.sumofsquares.org/public/lec02-1_maxcut.pdf
[2] https://www.researchgate.net/publication/4355188_On_the_second_eigenvalue_of_random_regular_graphs
