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Eric I Corwin1, Robin Stinchcombe, M F Thorpe

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This study explores bond percolation on diverse lattices, revealing a smooth transition towards Erdős-Rényi random graphs in high dimensions. Results demonstrate consistent evolution of key statistical measures with increasing dimensionality.

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Area of Science:

  • Statistical Mechanics
  • Network Science
  • Materials Science

Background:

  • Percolation theory studies the connectivity of random networks.
  • Erdős-Rényi graphs serve as a fundamental model for random networks.
  • Understanding lattice behavior in high dimensions is crucial for various scientific domains.

Purpose of the Study:

  • To investigate bond percolation on lattices across a wide range of dimensions (2D to 14D).
  • To analyze the convergence of lattice percolation to Erdős-Rényi graph behavior.
  • To examine the dimensional evolution of statistical properties in bond-diluted systems.

Main Methods:

  • Simulations of bond percolation on various lattice types.
  • Analysis of results in the limit of large dimension (d) and high coordination number (z).
  • Inclusion of bond-diluted hypersphere pack results up to nine dimensions.

Main Results:

  • Bond percolation on lattices smoothly approaches Erdős-Rényi graph properties at high dimensions.
  • Mean coordination, excess kurtosis, and skewness evolve consistently with dimension.
  • Observed smooth transitions in statistical measures towards the Erdős-Rényi limit.

Conclusions:

  • Lattice percolation behavior is unified under the Erdős-Rényi model in high-dimensional limits.
  • The study provides a comprehensive view of percolation across dimensions.
  • Dimensionality plays a key role in determining network properties.