Applied benchmark tests for the famous ‘subgraph isomorphism problem’ empirically discovered interesting phase transitions in random graphs. This motivates our rigorous study of two variants of the induced subgraph isomorphism problem for two independent binomial random graphs with constant edge-probabilities $p_1,p_2$. In particular, (i) we prove a sharp threshold result for the appearance of $G_{n,p_1}$ as an induced subgraph of $G_{N,p_2}$, (ii) we show two-point concentration of the size of the maximum common induced subgraph of $G_{N,p_1}$ and $G_{N,p_2}$, and (iii) we show that the number of induced copies of $G_{n,p_1}$ in $G_{N,p_2}$ has an unusual ‘squashed lognormal’ limiting distribution.

These results resolve several open problems of Chatterjee and Diaconis, and confirm simulation-based predictions of McCreesh, Prosser, Solnon and Trimble. The proofs are based on careful refinements of the first and second moment method, using extra twists to (a) take some non-standard behaviors into account, and (b) work around the large variance issues that prevent standard applications of the second moment method, using in particular pseudorandom properties and multi-round exposure arguments to tame the variance.

Based on joint work with my PhD students Erlang Surya and Emily Zhu; see arXiv:2305.04850.