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Title:
Correlations in many-body time series: The 2-D Ising model as paradigm
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Abstract:
We have shown previously, that the eigenvalues for correlations matrices constructed from time series established for each spin in a 2-D Ising Model with periodic boundary conditions under a standard Metropolis time evolution have at critical temperature apower law distribution for large eigenvalues and a Marchenko-Pastur distribution at hightemperatures. We analytically showed that for any periodic structure, which has power law correlations in space, the spectrum of the correlation matrix again displays a power law and the exponent of the latter is a function of the exponent of the former and the dimension ofthe respective state [1]. Very long time series where needed for the corresponding numerics, even though we also used the fact that the time evolution is stochastic After briefly showing these facts we shall proceed to the analysis of short time series, by means of the creation of a data ensemble. This ensemble is obtained by selecting randomly M different time series from the total set of N time series of length T<<N. The correlation matrix of the original time series will only have T-1 non-zero eigenvalues. We allow repletion of the same time series in different subsets and thus the total number of selections is binomial and for typical situations such as N=1000 and T= 100 we can choose say M=90 and we will have a huge number of subsets to form no-singular correlation matrices from. For numerical purposes this new ensemble is not practical but some small random subset will provide very good statistics. The question is, wether significant information can be puled. In [2] this ensembles defined and results for blockdiagonal random matrices where used. Finally we proceed to entirely new and promising calculations the 2.D Ising model. In this case the distribution for the largest eigenvalues in the data ensemble shows a power law at critical temperature, which we hope to relate to the power law discovered in the long time series. This argument in turn is based on the expected behaviour of power law banded matrices at the critical exponent 1 [3]. [1] Prosen, T., B. Buča, and T. H. Seligman. "Spectral analysis of finite-time correlation matrices near equilibrium phase transitions." EPL (Europhysics Letters) 108.2 (2014): 20006. [2] Vyas, Manan, T. Guhr, and T. H. Seligman. "Multivariate analysis of short time series in terms of ensembles of correlation matrices." Scientific reports 8.1 (2018): 14620. [3] Mirlin, Alexander D., et al. "Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices." Physical Review E 54.4 (1996): 3221.
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