In nonequilibrium statistical mechanics, the asymmetric simple exclusion process (ASEP) serves as a paradigmatic example. We investigate the spectral characteristics of the ASEP, focusing on the spectral boundary of its generator matrix. We examine finite ASEP chains of length L, under periodic boundary conditions (PBCs) and open boundary conditions (OBCs). Notably, the spectral boundary exhibits L spikes for PBCs and L+1 spikes for OBCs. Treating the ASEP generator as an interacting non-Hermitian fermionic model, we extend the model to have tunable interaction. In the noninteracting case, the analytically computed many-body spectrum shows a spectral boundary with prominent spikes. For PBCs, we use the coordinate Bethe ansatz to interpolate between the noninteracting case to the ASEP limit and show that these spikes stem from clustering of Bethe roots. The robustness of the spikes in the spectral boundary is demonstrated by linking the ASEP generator to random matrices with trace correlations or, equivalently, random graphs with distinct cycle structures, both displaying similar spiked spectral boundaries.