Ordering through learning in two-dimensional Ising spins

Abstract: We study two-dimensional Ising spins, evolving through reinforcement learning using their state, action, and reward. The state of a spin is defined as whether it is in the majority or minority with its nearest neighbours. The spin updates its state using an {\epsilon}-greedy algorithm. The parameter {\epsilon} plays the role equivalent to the temperature in the Ising model. We find a phase transition from long-ranged ordered to a disordered state as we tune {\epsilon} from small to large values. In analogy with the phase transition in the Ising model, we calculate the critical {\epsilon} and the three critical exponents {\beta}, {\gamma}, {\nu} of magnetization, susceptibility, and correlation length, respectively. A hyper-scaling relation d{\nu} = 2{\beta} + {\gamma} is obtained between the three exponents. The system is studied for different learning rates. The exponents approach the exact values for two-dimensional Ising model for lower learning rates.