Stochastic Ising networks with asymmetric and time-delayed interactions
Hantao Zhang, Sidra Gibeault, Matthew W. Daniels, Philippe Talatchian, Ursula Ebels, Advait Madhavan, Mark D. Stiles
Stochastic Ising networks are widely used to describe collective dynamics in complex systems and to compute, yet physical implementations of these networks often contain both asymmetric interactions and nonzero signal propagation times that fall outside conventional theories of their behavior. Here, we study stochastic Ising networks with asymmetric and time-delayed interactions motivated by experimental observations in coupled superparamagnetic tunnel junctions. Time delay fundamentally reshapes the dynamics induced by antisymmetric couplings, producing strong oscillatory temporal correlations consistent with experiment. Sufficiently long delays strongly suppress the non-uniformity of steady-state probability distributions, a behavior explained by an exponential suppression of the first-order equal-time cross-correlation between pairs of spins. Residual non-uniformity is due to the equal-time cross-correlation mediated by a third spin that contributes only at higher order in the coupling strength. These relatively featureless probability distributions coexist with pronounced delayed temporal correlations, distinguishing them from equilibrium high-temperature behavior. We further show analytically that these delay-induced near-uniform distributions emerge in a broad class of stochastic networks that lack symmetry-breaking bias fields. Introducing symmetry-breaking bias fields restores a strong dependence of the steady state on interactions with qualitatively modified behavior. Simulations of networks with five coupled spins demonstrate that these effects persist beyond minimal systems with only two spins.
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