Cavity master equation for the continuous time dynamics of discrete-spin models

We present an alternate method to close the master equation representing the continuous time dynamics of interacting Ising spins. The method makes use of 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 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.

Figure 1

Evolutionary dynamics of the global magnetization parameter for three spin models defined on a single instance of a random graph. (a) the Ising ferromagnet ($T_c\approx 0.96$). (b) the RFIM ($T_c\approx 0.78$). (c) the Viana-Bray model ($T_{SG}=0.506$). Dots represent kinetic MC simulation on a single instance realization of an ER graph, averaged over ${10}^5$ runs starting with the same initial conditions. Solid lines represent the cavity master equation approach presented in the main text. The insets show the mean error in local magnetization, as defined in the main text.

Figure 2

Dynamics of the global EA parameter for three spin models defined on a single instance of a random graph. (a) The Ising ferromagnet ($T_c\approx 0.96$). (b) The RFIM ($T_c\approx 0.78$). (c) The Viana-Bray model ($T_{SG}=0.506$). Dots represent kinetic MC simulations on a single instance realization of an ER graph, averaged over ${10}^5$ runs starting with the same initial conditions. Solid lines represent the cavity master equation approach presented in the main text. The behavior for low temperatures in the Viana-Bray model shows that even though global magnetization is close to zero for long times [see Fig. 1], spins are locally magnetized for long times, as it is expressed by the non zero value of $q_{EA}\left(t\right)$ for $T=0.25$. The insets show the mean error of local quantities, as defined in the main text.

Figure 3

Maximum mean error dependence with temperature between MC simulations and the CME approach presented in the main text for three spin models defined on a single instance of a random graph. (a) The Ising ferromagnet ($T_c\approx 0.96$). (b) The RFIM ($T_c\approx 0.78$). (c) The Viana-Bray model ($T_{SG}=0.506$). Simulation details are the same as for Figs. 1 and 2. For the ferromagnet and the RFIM, error increases before and decreases after the ferromagnetic transition as the temperature changes. For the Viana-Bray spin glass, error grows monotonically lowering the temperature.