The Distribution of Sandpile Groups of Random Graphs with their Pairings

We determine the distribution of the sandpile group (also known as the Jacobian) of the Erd\H{o}s-R\'{e}nyi random graph $G(n,q)$ along with its canonical duality pairing as $n$ tends to infinity, fully resolving a conjecture from 2015 due to Clancy, Leake, and Payne and generalizing the result by Wood on the groups. In particular, we show that a finite abelian $p$-group $G$ equipped with a perfect symmetric pairing $\delta$ appears as the Sylow $p$-part of the sandpile group and its pairing with frequency inversely proportional to $|G||\mathrm{Aut}(G,\delta)|$, where $\mathrm{Aut}(G,\delta)$ is the set of automorphisms of $G$ preserving the pairing $\delta$. While this distribution is related to the Cohen-Lenstra distribution, the two distributions are not the same on account of the additional algebraic data of the pairing. The proof utilizes the moment method: we first compute a complete set of moments for our random variable (the average number of epimorphisms from our random object to a fixed object in the category of interest) and then show the moments determine the distribution. To obtain the moments, we prove a universality result for the moments of cokernels of random symmetric integral matrices whose dual groups are equipped with symmetric pairings that is strong enough to handle both the dependence in the diagonal entries and the additional data of the pairing. We then apply results due to Sawin and Wood to show that these moments determine a unique distribution.