[Paper Review] Reply to Comment on " Universal Fluctuations in Correlated Systems"

Bramwell, S. T., Kim Christensen+8 more|arXiv (Cornell University)|Sep 17, 2002

Advanced Thermodynamics and Statistical Mechanics3 citations

TL;DR

This paper responds to criticisms of their earlier work on universal fluctuations in correlated systems, arguing that deviations from the Freidlin–Tougaard–Gumbel (FTG) distribution are not merely finite-size effects but stem from strong correlations in many-body systems. It demonstrates that extremal statistics of correlated variables—unlike independent ones—can yield non-FTG distributions such as the BHP distribution, suggesting a deeper link between extreme value theory and global fluctuations in correlated systems.

ABSTRACT

Reply to the comment, cond-mat/0209398 by by N.W. Watkins, S.C. Chapman, and G. Rowlands

Motivation & Objective

• To clarify that observed deviations from the FTG distribution in correlated systems are not due to slow convergence in finite-size systems, as suggested by critics.
• To argue that simple extremal statistics of independent variables cannot explain the observed fluctuations in systems like the 2D-XY model.
• To establish that only when extremal statistics are applied to complex, correlated many-body objects—rather than independent variables—do they yield non-FTG distributions.
• To support the hypothesis that strong correlations lead to persistent deviations from FTG even in the thermodynamic limit.
• To highlight the open problem of connecting global sum fluctuations with extremal statistics, despite apparent agreement with FTG in some cases.

Proposed method

• Analyzes the extreme value statistics of a 2D-XY model in the low-temperature phase, where the order parameter fluctuation PDF is known to be the BHP distribution.
• Contrasts the behavior of extremal statistics on independent variables (yielding FTG) with those on correlated variables (yielding BHP or similar forms).
• Uses renormalization group analysis to show that long-range correlated signals renormalize extreme value tails from exp(−y) (FTG) to y exp(−y), matching the BHP asymptotic form.
• Examines the Sneppen depinning model to show that avalanche size extremes follow the BHP distribution across scales, indicating a strong correlation regime.
• Evaluates finite-size corrections in Gaussian-distributed variables to distinguish weak correlation effects from true strong correlation phenomena.
• Compares the thermodynamic limit behavior of correlated systems with the asymptotic convergence of independent variables to isolate the role of correlations.

Experimental results

Research questions

• RQ1 Are deviations from the FTG distribution in correlated systems primarily due to finite-size effects or intrinsic strong correlations?
• RQ2 Can extremal statistics of correlated many-body systems explain the observed non-FTG distributions like the BHP distribution?
• RQ3 Does the thermodynamic limit of correlated systems still exhibit deviations from FTG, indicating a fundamental departure from standard extreme value theory?
• RQ4 What is the role of renormalization group flow in shaping the tail behavior of extreme value distributions in long-range correlated systems?
• RQ5 Is there a viable theoretical connection between global sum fluctuations and extremal statistics, even when both appear to follow the FTG distribution?

Key findings

• The BHP distribution for order parameter fluctuations in the 2D-XY model is not explained by extremal statistics of independent variables, which instead yield the FTG distribution.
• Finite-size corrections alone cannot account for the observed deviations from FTG; strong correlations are necessary to produce persistent deviations in the thermodynamic limit.
• In the Sneppen depinning model, extreme avalanche sizes follow the BHP distribution across scales, indicating a strong correlation regime where FTG fails.
• Renormalization group analysis shows that long-range correlated signals renormalize extreme value tails from exp(−y) (FTG) to y exp(−y), matching the exact BHP asymptotic form.
• The connection between global sum fluctuations and extremal statistics remains unproven, even when both appear to follow the FTG distribution.
• The results support the original hypothesis from [1] that correlations in many-body systems can generate extreme value statistics that deviate from FTG, providing a potential link to universal fluctuations.