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
R. Abe and A. Hatano, Dynamics of the Ising Model Near the Transition Point, Progress of Theoretical Physics 39 (1968), pp. 947–956.
M. Aizenman and R. Holley, Rapid Convergence to Equilibrium of Stochastic Ising Models in the Dobrushin Shlosman Regime, “Percolation Theory and Ergodic Theory of Infinite Particle Systems,” IMA Vols in Math. and its Appl. ed by Harry Kesten, Springer Verlag, 1987, pp. 1–11.
K. Binder, Monte Carlo Investigations of Phase Transitions and Critical phenomena, Phase Transitions and Critical Phenomena 5B (1976), pp. 1–105.
K. Binder and M.H. Kalos, Monte Carlo Studies of Relaxation Phenomena: Kinetics of Phase Changes and Critical Slowing Down, “Monte Carlo Methods in Statistical Physics,” 1979, pp. 225–260.
C.M. Fortuin, P.W. Kasteleyn, and J. Ginibre, Correlation Inequalities on Some Partially Ordered Sets, Commun. Math. Phys. 22 (1971), pp. 89–103.
R. Holley, Possible Rates of Convergence in Finite Range, Attractive Spin Systems, Contemporary Math. 41 (1985), pp. 215–234..
B.I. Helpern, and P.C. Hohenberg, Theory of Dynamic Critical Phenomena, Rev. Mod. Phys. 49 (1977), pp. 435–479..
R. Holley and D. W. Stroock, L 2 theory for the stochastic Ising model, Z. Wahr. verw. Geb. 35 (1976), pp. 87–101.
K. Kawasaki, Kinetics of Ising Models, Phase Transitions and Critical Phenomena 2 (1972), pp. 443–501.
T. M. Liggett, “Interacting Particle Systems,” Springer Verlag, 1985.
B.M. McCoy and T.T. Wu, “The Two-Dimensional Ising Model,” Harvard University Press, Cambridge, Massachusets, 1973.
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© 1991 Springer Science+Business Media New York
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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
Publisher Name: Birkhäuser, Boston, MA
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