RSB bounds on the maximum cut

Abstract

In the context of random regular graphs, the size of the maximum cut is probably the second most studied graph parameter after the independence ratio. Zdeborov\'a and Boettcher used the cavity method, a non-rigorous statistical physics technique, to predict one-step replica symmetry breaking (1-RSB) formulas. Coja-Ohglan et al. confirmed these predictions as rigorous upper bounds using the interpolation method. While these upper bounds were not expected to be exact, they may be very close to the true values. In this paper, we establish 2-RSB upper bounds and fine-tune their parameters to beat the aforementioned 1-RSB bounds.