We consider sequences of large sparse random graphs whose degree distribution approaches a limit with finite mean. This model includes both the random regular graphs and the Erdös-Renyi graphs of constant average degree. We prove that the maximum bisection ratio of such a graph sequence converges almost surely to a deterministic limit. We extend this result to so-called 2-spin spin glasses in the paramagnetic to ferromagnetic regime. Our work generalizes the graph interpolation method to some non-additive graph parameters.
"Convergence of maximum bisection ratio of sparse random graphs." Electron. Commun. Probab. 23 1 - 10, 2018. https://doi.org/10.1214/18-ECP164