On the Gaussian fluctuations of the critical Curie-Weiss model in statistical mechanics. (English) Zbl 0684.60080

It is known, due to some recent results, that the fluctuations of the critical Curie-Weiss model are not Gaussian. In this paper the author shows that for a large class of probability measures \(\rho\) there is considerable Gaussian structure in the internal fluctuations of the critical model. In the Curie-Weiss model all large subsystems fluctuate and the implications of this can be described in terms of weak convergence in the space of paths.
The main result of this paper is focussed on fluctuations of the error term, which may be called the second order fluctuations. It is proved that the second order fluctuations are asymptotically those of a Brownian bridge collapsing to the t-axis, and on the other hand, that the polygonal process converges in distribution to a Brownian motion with randomised shift Y.
Reviewer: V.Tigoiu


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
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[1] Billingsley, P.: Convergence of probability measures. New York: Wiley 1968 · Zbl 0172.21201
[2] Chaganty, N.R., Sethuraman, J.: Limit theorems in the area of large deviations for some dependent random variables. Ann. Probab. 15, 628-645 (1987) · Zbl 0637.60044
[3] Comets, F.: Nucleation for a long range magnetic model. Ann. Inst. Henri PoincarĂ© 23, 135-178 (1987) · Zbl 0633.60110
[4] Dawson, D.A.: Critical dynamics and fluctuations for a mean-field model of cooperative behaviour. J. Stat. Phys. 31, 29-85 (1983)
[5] Eisele, T., Ellis, R.S.: Symmetry breaking and random waves for magnetic systems on a circle. Z. Wahrscheinlichkeitstheor. Verw. Geb. 63, 297-348 (1983) · Zbl 0494.60097
[6] Ellis, R.S.: Entropy, large deviations and statistical mechanics. Berlin Heidelberg New York: Springer 1985 · Zbl 0566.60097
[7] Ellis, R.S., Newman, C.M.: Fluctuationes in Curie-Weiss examplis. In: Dell’Antonio, G., Doplicher, S., Jona-Lasinio, G. (eds.) Mathematical problems in theoretical physics. Conference, Rome 1977. (Lect. Notes Phys., vol. 80, pp. 312-324) Berlin Heidelberg New York: Springer 1978
[8] Ellis, R.S., Newman, C.M.: The statistics of Curie-Weiss models. J. Statist. Phys. 19, 149-161 (1978)
[9] Ellis, R.S., Newman, C.M.: Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrscheinlichkeitstheor. Verw. Geb. 44, 117-139 (1978) · Zbl 0364.60120
[10] Ellis, R.S., Newman, C.M., Rosen, J.S.: Limit theorems for sums of dependent random variables occurring in statistical mechanics, II: conditioning, multiple phases and metastability. Z. Wahrscheinlichkeitstheor. Verw. Geb. 51, 153-169 (1980) · Zbl 0404.60096
[11] Jeon, J.W.: Weak convergence of processes occurring in statistical mechanics. J. Korean Statist. Soc. 12, 10-17 (1983)
[12] Kac, M.: Mathematical mechanisms of phase transition. In: Statistical Physics: phase transitions and superfluidity, Vol. 1, pp. 241-305. Brandeis University Summer Institute in Theoretical Physics 1966. Chretien, M., Gross, E.P., Deser, S. (eds.). New York: Gordon and Breach 1968
[13] Parthasarathy, K.R.: Probability measures on metric spaces. New York: Academic Press 1967 · Zbl 0153.19101
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