Exponential Decay of Truncated Correlations for the Ising Model in any Dimension for all but the Critical Temperature (original) (raw)
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Long-range components of the interaction in statistical mechanical systems may affect the critical behavior, raising the system's 'effective dimension'. Presented here are explicit implications to this effect of a collection of rigorous results on the critical exponents in ferromagnetic models with one-component lsing (and more generally Griffiths-Simon class) spin variables. In particular, it is established that even in dimensions d < 4 if a ferromagnetic Ising spin model has a reflection-positive pair interaction with a sufficiently slow decay, e.g. as Jx = 1/Ixl a+~ with 0 < a~< d/2, then the exponents ~, 6, ? and A 4 exist and take their mean-field values. This proves rigorously an early renormalization-group prediction of Fisher, Ma and Nickel. In the converse direction: when the decay is by a similar power law with try> 2, then the long-range part of the interaction has no effect on the existent critical exponent bounds, which coincide then with those obtained for short-range models.
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We present results of the study of the factorial moments in the two-dimensional ]sing model. We have related their intermittent behaviour to the short distance singularity of the correlation functions. Analogies and differences with the similar phenomenon in the multiparticle production are discussed. Possible application to study the finite temperature gauge theory is also outlined.
Journal of Statistical Physics, 1996
We find the asymptotic decrease of correlations (aA +y, aB), y 9 Z" + t l Yl ~ oo, in the Ising model at high temperatures. For the case when monomials an and a n both are odd, using the saddle-point method, we find the asymptotics of the correlations for any dimension v. For even monomials aA, an we formulate a general hypothesis about the form of the asymptotics and confirm it in two cases: (1) v= 1 and the vector y has an arbitrary direction, (2) y is directed along a fixed axis and arbitrary v. Here we use besides the saddle-point method, some arguments from scattering theory.
Correlation matrices at the phase transition of the Ising model
We study spectral densities for systems on lattices, which, at a phase transition display, power-law spatial correlations. Constructing the spatial correlation matrix we prove that its eigenvalue density shows a power law that can be derived from the spatial correlations. In practice time series are short in the sense that they are either not stationary over long time intervals or that they are not available over long time intervals. Also we usually do not have time series for all variables available. We shall make numerical simulations on a 2-D Ising model with the usual Metropolis algorithm as time-evolution. Using all spins on a grid with periodic boundary conditions we find a power law, that is, for large grids, compatible with the analytic result. We still find a power law even if we choose a fairly small subset of grid points at random. The exponents of the power laws will be smaller under such circumstances. For very short time series leading to singular correlation matrices we use a recently developed technique to lift the degeneracy at zero in the spectrum and find a significant signature of critical behavior even in this case as compared to high temperature results which tend to those of random matrix models.