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Universal scaling in real dimension.

Giacomo Bighin1, Tilman Enss1, Nicolò Defenu2

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We studied universality in physics on a complex, non-homogeneous graph, the long-range diluted graph (LRDG). Our findings reveal universal scaling behavior controlled by the spectral dimension, extending universality to new complex systems.

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Area of Science:

  • Complex systems physics
  • Statistical mechanics
  • Graph theory

Background:

  • Universality is a key concept in many-body physics, typically studied in homogeneous systems.
  • Understanding universality in non-homogeneous or complex systems remains a significant challenge.
  • The spectral dimension (ds) is a critical parameter for describing scaling in complex geometries.

Purpose of the Study:

  • To investigate universality on a non-homogeneous graph, the long-range diluted graph (LRDG).
  • To explore how the spectral dimension (ds) controls scaling theory in complex geometries.
  • To determine universal scaling exponents as continuous functions of the spectral dimension.

Main Methods:

  • Theoretical analysis of scaling theory on the LRDG.
  • Continuous tuning of the spectral dimension (ds) to non-integer values.
  • Extensive numerical simulations of symmetric models on the LRDG.

Main Results:

  • The spectral dimension (ds) was identified as the single controlling parameter for scaling theory on the LRDG.
  • Universal scaling exponents were found to be continuous functions of the spectral dimension.
  • Numerical simulations showed quantitative agreement with theoretical predictions for universal scaling.

Conclusions:

  • Universality can be extended to non-homogeneous systems like the LRDG.
  • The spectral dimension is a crucial parameter for understanding universal scaling on complex graphs.
  • This work provides a framework for studying universality in diverse complex systems.