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We determine the distribution of the sandpile group (also known as theJacobian) of the Erd\H{o}s-R\'{e}nyi random graph $G(n,q)$ along with itscanonical duality pairing as $n$ tends to infinity, fully resolving aconjecture from 2015 due to Clancy, Leake, and Payne and generalizing theresult 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 theSylow $p$-part of the sandpile group and its pairing with frequency inverselyproportional to $|G||\mathrm{Aut}(G,\delta)|$, where $\mathrm{Aut}(G,\delta)$is the set of automorphisms of $G$ preserving the pairing $\delta$. While thisdistribution is related to the Cohen-Lenstra distribution, the twodistributions are not the same on account of the additional algebraic data ofthe pairing. The proof utilizes the moment method: we first compute a completeset of moments for our random variable (the average number of epimorphisms fromour random object to a fixed object in the category of interest) and then showthe moments determine the distribution. To obtain the moments, we prove auniversality result for the moments of cokernels of random symmetric integralmatrices whose dual groups are equipped with symmetric pairings that is strongenough to handle both the dependence in the diagonal entries and the additionaldata of the pairing. We then apply results due to Sawin and Wood to show thatthese moments determine a unique distribution.

The Distribution of Sandpile Groups of Random Graphs with their Pairings