Mean field equation and relativistic Abelian Chern-Simons model on finite graphs

Abstract In this paper, we study the Mean field equation and the relativistic Abelian Chern-Simons equations (involving two Higgs particles and any two gauge fields) on the finite connected graphs. For the former equation, we establish the existence results and some uniqueness result. In particular, we find that there is no set of critical parameters for the Mean field equation on the finite graphs and the existence is ensured for any non-negative parameters, which is in contrast to the continuous case. In addition, we give the optimal constant which is the threshold for the uniqueness of the equation on the finite complete graphs with simple weight. A key observation is that the solution can take at most two values. While for the second problem, we study the existence of maximal condensates, and also establish the existence of multiple solutions, including a local minimizer for the transformed energy functional and a mountain-pass type solution.