A Cavity Master Equation for the continuous time dynamics of discrete spins models

Abstract

We present a new method to close the Master Equation representing the continuous time dynamics of Ising interacting spins. The method makes use of the the theory of Random Point Processes to derive a master equation for local conditional probabilities. We analytically test our solution studying two known cases, the dynamics of the mean field ferromagnet and the dynamics of the one dimensional Ising system. We then present numerical results comparing our predictions with Monte Carlo simulations in three different models on random graphs with finite connectivity: the Ising ferromagnet, the Random Field Ising model, and the Viana-Bray spin-glass model.

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• The matrix product approximation for the dynamic cavity method
• Improved mean-field dynamical equations are able to detect the two-steps relaxation in glassy dynamics at low temperatures
• Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise
• The Cavity Master Equation: average and fixed point of the ferromagnetic model in random graphs
• Small-Coupling Dynamic Cavity: a Bayesian mean-field framework for epidemic inference
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• Biased thermodynamics can explain the behaviour of smart optimization algorithms that work above the dynamical threshold
• Connected components in networks with higher-order interactions
• Dynamics of epidemic models from cavity master equations
• From random point processes to hierarchical Cavity Master Equations for the stochastic dynamics of disordered systems in Random Graphs: Ising models and epidemics
• Global thermodynamic manifold for conservative control of stochastic systems