  We consider the spectral gap of a uniformly chosen random
$(d_1,d_2)$-biregular bipartite graph $G$ with $|V_1|=n, |V_2|=m$, where
$d_1,d_2$ could possibly grow with $n$ and $m$. Let $A$ be the adjacency matrix
of $G$. Under the assumption that $d_1\geq d_2$ and $d_2=O(n^{2/3}),$ we show
that $\lambda_2(A)=O(\sqrt{d_1})$ with high probability. As a corollary,
combining the results from Tikhomirov and Youssef (2019), we showed that the
second singular value of a uniform random $d$-regular digraph is $O(\sqrt{d})$
for $1\leq d\leq n/2$ with high probability. Assuming $d_2$ is fixed and
$d_1=O(n^2)$, we further prove that for a random $(d_1,d_2)$-biregular
bipartite graph, $|\lambda_i^2(A)-d_1|=O(\sqrt{d_1})$ for all $2\leq i\leq
n+m-1$ with high probability. The proofs of the two results are based on the
size biased coupling method introduced in Cook, Goldstein, and Johnson (2018)
for random $d$-regular graphs and several new switching operations we defined
for random bipartite biregular graphs.
