Volume 108, Number 2, October 2014
|Number of page(s)||5|
|Published online||17 October 2014|
Spectral analysis of finite-time correlation matrices near equilibrium phase transitions
1 Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México - C.P. 62210 Cuernavaca, México
2 Department of Physics, FMF, University of Ljubljana - Jadranska 19, 1000 Ljubljana, Slovenia
3 Centro Internacional de Ciencias - C.P. 62210 Cuernavaca, México
Received: 23 June 2014
Accepted: 1 October 2014
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 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 two-dimensional 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.
PACS: 05.45.Tp – Time series analysis / 64.60.Ht – Dynamic critical phenomena / 02.50.Sk – Multivariate analysis
© EPLA, 2014
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