Quantum dynamics on the sparse Erdős-Rényi graph

We analyse the large-time asymptotics of the Schrödinger time evolution of a free quantum particle on the Erdős-Rényi graph $\mathbb{G}(N,d/N)$, where the quantum Hamiltonian is given by the adjacency matrix of $\mathbb{G}$. Starting from a state localized at a vertex, the wave function is approximately radial, and we characterize precisely its asymptotic form. Our results in particular cover vertices whose degree is close to $2d$, hence probing the emergence of bound states from a dynamical perspective. This provides a dynamical counterpart of the coexistence of localized and delocalized states established in [arXiv:1905.03243,arXiv:2005.14180,arXiv:2106.12519,arXiv:2109.03227,arXiv:2305.16294], with the former giving rise to a localized mode and the latter to a dispersive wave. Our results hold in the regime $\sqrt{\log N} \ll d \ll N^{o(1)}$, which is the full region in which such dynamical phenomena may exist, and for times $1 \ll t \ll T_N$ for some explicit $T_N \gg 1$.