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The highest and lowest values of a function, relative to a reference axis, are known as extreme values. These include absolute maximum and absolute minimum values, which represent the highest and lowest points the function reaches across its entire domain. Within a restricted portion of the function, the highest and lowest values are referred to as local maximum and local minimum values, respectively.Periodic functions, such as sine and cosine, show extreme values at infinitely many points due...
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Exact extreme-value statistics at mixed-order transitions.

Amir Bar1, Satya N Majumdar2, Grégory Schehr2

  • 1Department of Complex Systems, Weizmann Institute, Rehovot, Israel.

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Summary

This study analyzes extreme-value statistics in models with mixed-order phase transitions (MOT). We found novel distributions governing largest domain lengths at critical and ferromagnetic points, verified by simulations.

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Area of Science:

  • Statistical Mechanics
  • Complex Systems

Background:

  • Mixed-order phase transitions (MOT) blend characteristics of first-order and second-order transitions.
  • Understanding extreme-value statistics is crucial for characterizing complex systems.

Purpose of the Study:

  • To investigate extreme-value statistics of domain lengths in a prototypical MOT model.
  • To identify and compute the distributions governing the largest domain length.

Main Methods:

  • Analytical study of domain length distributions in the truncated inverse distance squared Ising model.
  • Examination of the paramagnetic, critical, and ferromagnetic phases.

Main Results:

  • In the paramagnetic phase, largest domain length distribution converges to Gumbel.
  • Novel, exactly computed distributions govern largest domain length fluctuations at critical and ferromagnetic points.

Conclusions:

  • The study provides exact analytical results for extreme-value statistics in MOT models.
  • Numerical simulations confirm the theoretical findings, validating the discovered distributions.