Shotgun threshold for sparse Erdős-Rényi graphs

In the shotgun assembly problem for a graph, we are given the empirical profile for rooted neighborhoods of depth $r$ (up to isomorphism) for some $r\ge 1$ and we wish to recover the underlying graph up to isomorphism. When the underlying graph is an Erdős-Rényi $\mathcal{G}(n,\frac{\lambda }{n})$, we show that the shotgun assembly threshold $r_{\ast }\approx \frac{\log n}{\log ({\lambda }^2{\gamma }_{\lambda }{)}^{-1}}$ where ${\gamma }_{\lambda }$ is the probability for two independent Poisson-Galton-Watson trees with parameter $\lambda$ to be rooted isomorphic with each other. Our result sharpens a constant factor in a previous work by Mossel and Ross (2019) and thus solves a question therein.