Rejection-free Glauber Monte Carlo for the 2D Random Field Ising Model via Hierarchical Probabilistic Counters

Abstract: We present an efficient Monte Carlo algorithm for the simulation of the two-dimensional Random Field Ising Model (RFIM). The method combines the event-driven, rejection-free character of the Bortz Kalos-Lebowitz (BKL) algorithm with Glauber transition probabilities, introducing hierarchical probabilistic counters to perform spin selection in O(log N) operations. This enables efficient sampling of the system's dynamics, especially in the low-temperature and low-disorder regime, where traditional Metropolis updates suffer from critical slowing down. Furthermore, this approach allows a proper dynamical simulation of the Ising system's behavior even in the presence of a Random Field (RF), unlike the BKL method. RFIM simulations with Gaussian field distributions reproduce the expected reduction of the pseudo-critical temperature with increasing disorder. Benchmarking shows speedups exceeding two orders of magnitude compared to the Metropolis algorithm in the low-temperature regime. The proposed method provides an efficient and dynamically faithful tool for studying both equilibrium and non-equilibrium phenomena in disordered spin systems.