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Extreme value statistics in strongly correlated systems

Final Activity Report Summary - EXVALSTAT (Extreme value statistics in strongly correlated systems)

Extreme value statistics in strongly correlated systems

Extreme value distributions for random variables are known. However, nature often presents us with processes which are strongly correlated. The main focus of our project was to extend extreme value theory for certain systems with strongly correlated random variables.

Much of our information about the physical world comes in the form of time series. Examples include temperature records or river levels. Such time series are well approximated by so-called Gaussian 1/f^{\alpha} signals, which also describe the steady-state fluctuations of various interfaces and are of inherent mathematical interest. The strength of the correlations along the signal can be tuned via the parameter alpha. In this project we studied the extreme value statistics (EVS) of 1/f^{\alpha} signals comprehensively.

We also considered two celebrated models from statistical mechanics, percolation and the Ising model. Percolation is in many respects an EVS problem, since the order parameter is the largest cluster.

Results

1. 1/f^{\alpha} signals: maximum relative to the average value (G. Gyorgyi, N. R. Moloney, K. Ozogany and Z. Racz, Phys. Rev. E 75, 021123 (2007))
We made a comprehensive numerical survey of the distribution of the maximum for a wide range of alpha values. In the stationary regime, we found pronounced finite-size corrections to the Fisher-Tippett-Gumbel (FTG) distribution. In the non-stationary regime, we developed theory for generalised random acceleration processes.

2. 1/f^{\alpha} signals: maximum relative to the initial value (including an external collaboration with T. W. Burkhardt of Temple University) (T. W. Burkhardt, G. Gyorgyi, N. R. Moloney and Z. Racz, Phys. Rev. E 76, 041119 (2007))
In this study we measured a signal's maximum relative to its initial value. In the stationary regime we refined Berman's result (as did Sabhapandit and Majumdar (2007)). In the non-stationary regime we calculated the EVS distribution analytically for the random walk, the random acceleration process, and the limit of the strongest possible correlations.

3. Finite-size scaling in extreme statistics
In the real world we only have access to finite samples. Therefore, a detailed finite-size scaling analysis is called for. For the case of iid variables, we developed an iterative procedure that can, in principle, be used to calculate the finite-size scaling correction functions to arbitrary order. In parallel, we formulated the question in the more intuitive renormalisation group framework. We applied our analysis to the distribution of the size of the largest cluster in subcritical percolation. For the case of Gaussian 1/f^{\alpha} non-stationary signals, we found power-law speed convergence to the limiting distribution.

Ongoing projects

1. Extreme value statistics in the critical Ising model
We applied the Wolff algorithm to study the EVS of the magnetisation of the critical two-dimensional Ising model.

2. Trends in trendless signals
Provoked by the climate change debate, we developed a project that tries to pose questions such as: how extreme is it that x record temperatures have been recorded in the last y years given an annual temperature increase of z degrees?

Conclusion
We now have a rather detailed understanding of EVS in 1/f^{\alpha} signals, and we know how correlations along the signal influence the shape of the EVS distribution. We are also able to calculate FSS corrections in the iid setting, and choose appropriate scaling variables for fast convergence for non-stationary 1/f^{\alpha} signals.

Finally, we would like to thank the European Commission for supporting Dr. Nicholas R. Moloney under a Marie Curie Fellowship. Although Eotvos University has a world-class Institute for Theoretical Physics, there are almost no resources for attracting foreign post-docs. Marie Curie Fellowships therefore provide a lifeline.