We consider the spectral gap of a uniformly chosen random (d1,d2)\documentclass[12pt]{minimal}
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\begin{document}$$(d_1,d_2)$$\end{document}-biregular bipartite graph G with |V1|=n,|V2|=m\documentclass[12pt]{minimal}
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\begin{document}$$|V_1|=n, |V_2|=m$$\end{document}, where d1,d2\documentclass[12pt]{minimal}
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\begin{document}$$d_1,d_2$$\end{document} could possibly grow with n and m. Let A be the adjacency matrix of G. Under the assumption that d1≥d2\documentclass[12pt]{minimal}
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\begin{document}$$d_1\ge d_2$$\end{document} and d2=O(n2/3),\documentclass[12pt]{minimal}
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\begin{document}$$d_2=O(n^{2/3}),$$\end{document} we show that λ2(A)=O(d1)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _2(A)=O(\sqrt{d_1})$$\end{document} with high probability. As a corollary, combining the results from (Tikhomirov and Yousse in Ann Probab 47(1):362–419, 2019), we show that the second singular value of a uniform random d-regular digraph is O(d)\documentclass[12pt]{minimal}
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\begin{document}$$O(\sqrt{d})$$\end{document} for 1≤d≤n/2\documentclass[12pt]{minimal}
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\begin{document}$$1\le d\le n/2$$\end{document} with high probability. Assuming d2\documentclass[12pt]{minimal}
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\begin{document}$$d_2$$\end{document} is fixed and d1=O(n2)\documentclass[12pt]{minimal}
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\begin{document}$$d_1=O(n^2)$$\end{document}, we further prove that for a random (d1,d2)\documentclass[12pt]{minimal}
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\begin{document}$$(d_1,d_2)$$\end{document}-biregular bipartite graph, |λi2(A)-d1|=O(d1)\documentclass[12pt]{minimal}
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\begin{document}$$|\lambda _i^2(A)-d_1|=O(\sqrt{d_1})$$\end{document} for all 2≤i≤n+m-1\documentclass[12pt]{minimal}
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\begin{document}$$2\le i\le n+m-1$$\end{document} with high probability. The proofs of the two results are based on the size biased coupling method introduced in Cook et al. (Ann Probab 46(1):72–125, 2018) for random d-regular graphs and several new switching operations we define for random bipartite biregular graphs.
