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 $r\geq 1$ and we wish to recover the underlying graph up to isomorphism. When the underlying graph is an Erdős-Rényi $\mathcal G(n, \fracλ{n})$, we show that the shotgun assembly threshold $r_* \approx \frac{ \log n}{\log (λ^2 γ_λ)^{-1}}$ where $γ_λ$ is the probability for two independent Poisson-Galton-Watson trees with parameter $λ$ 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.

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