Stochastic dynamics of N bistable elements with global time-delayed interactions: Towards an exact solution of the master equations for finite N

We consider a network of N noisy bistable elements with global time-delayed couplings. In a two-state description, where elements are represented by Ising spins, the collective dynamics is described by an infinite hierarchy of coupled master equations which was solved at the mean-field level in the thermodynamic limit. When the number of elements is finite, as is the case in actual laser networks, an analytical description was deemed so far intractable and numerical studies seemed to be necessary. In this paper we consider the case of two interacting elements and show that a partial analytical description of the stationary state is possible if the stochastic process is time symmetric. This requires some relationship between the transition rates to be satisfied.