Abstract
The rate at which the spin-spin autocorrelation function goes to zero as the time goes to infinity in stochastic ISING models with ferro-magnetic interactions and attractive flip rates is investigated. It is first shown that if this rate is exponential then the exponent is the same as the gap between 0 and the rest of the spectrum of the infinitesimal generator as an operator on the L 2 space of the equilibrium measure. It is easy to get upper bounds on the gap in the spectrum. The best known such bounds are related to the susceptibility of the magnetization. In two dimensions the spin-spin autocorrelation function is investigated when the temperature is exactly at the critical temperature. The rate of convergence to 0 is no longer exponential in this case, and again we obtain only upper bounds. No lower bounds are known at any temperatures near the critical temperature.
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References
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Holley, R. (1991). On the Asymptotics of the Spin-Spin Autocorrelation Function In Stochastic Ising Models Near the Critical Temperature. In: Alexander, K.S., Watkins, J.C. (eds) Spatial Stochastic Processes. Progress in Probability, vol 19. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0451-0_5
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DOI: https://doi.org/10.1007/978-1-4612-0451-0_5
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