Maximal persistence in random clique complexes

We study the persistent homology of an Erdős–Rényi random clique complex filtration on $n$ vertices. Here, each edge $e$ appears at a time $p_e\in [0,1]$ chosen uniform randomly in the interval, and the \emph{persistence} of a cycle $\sigma$ is defined as $p_2\mathrm{/}{p}_1$, where $p_1$ and $p_2$ are the birth and death times of the cycle respectively. We show that for fixed $k\ge 1$, with high probability the maximal persistence of a $k$-cycle is of order roughly $n^{1\mathrm{/}k(k+1)}$. These results are in sharp contrast with the random geometric setting where earlier work by Bobrowski, Kahle, and Skraba shows that for random Čech and Vietoris–Rips filtrations, the maximal persistence of a $k$-cycle is much smaller, of order ${(\log n\mathrm{/}\log \log n)}^{1\mathrm{/}k}$.