Smoothed-Cubic Spin-Glass Model of Random Lasers

We study the equilibrium glassy behavior of a multimode random laser model with nonlinear four-body quenched disordered interactions and a global smoothed-cubic constraint on mode intensities. This constraint, which provides a more realistic representation of gain saturation than the commonly used spherical constraint, prevents intensity condensation while preserving the dense, long-range interaction structure characteristic of many multistate random lasers. The model effective Hamiltonian is a function of mode amplitudes with random frequencies and is defined on a complete mode-locked graph. Using large-scale GPU-accelerated Monte Carlo simulations with the Parallel Tempering algorithm, we analyze systems of varying sizes to probe their thermodynamic-limit behavior. Finite-size scaling of the specific heat, of the Parisi overlap distributions, and of the inverse participation ratio's reveals a spin-glass transition, with critical exponents matching the mean-field Random Energy Model universality class. The smoothed-cubic constraint produces broad, non-condensed intensity distributions, avoiding the pseudo-condensation seen in spherical models on the same interaction graph. Our results show that more realistic gain-saturation constraints preserve spin-glass characteristics while enabling simulations of larger, more dilute systems, providing a robust framework for studying glassy random lasers with self-starting mode-locking.