Multi-body quenched disordered $XY$ and $p$-clock models on random graphs

The $XY$ model with four-body quenched disordered interactions and its discrete $p$-clock proxy is studied on bipartite random graphs by means of the cavity method. The phase diagrams are determined from the ordered case to the spin-glass case. Dynamic, spinodal and thermodynamic transition lines are identified by analyzing free energy, complexity and tree reconstruction functions as temperature and disorder are changed. The study of the convergence of the $p$-clock model to the $XY$ model is performed down to temperature low enough to determine all relevant transition points for different connectivity.