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Percolation thresholds in hyperbolic lattices.

Stephan Mertens1,2, Cristopher Moore1

  • 1Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe, New Mexico 87501, USA.

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Summary
This summary is machine-generated.

We computed percolation thresholds for hyperbolic plane tessellations using invasion percolation. Our accurate numerical values reveal how these thresholds depend on P and Q, and we derived bounds for scaling analysis.

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Area of Science:

  • Hyperbolic geometry
  • Statistical physics
  • Computational mathematics

Background:

  • Percolation theory studies the formation of clusters in random systems.
  • Hyperbolic tessellations provide a unique geometric framework for studying percolation.
  • Understanding percolation thresholds is crucial for various scientific fields.

Purpose of the Study:

  • To compute numerical values for bond and site percolation thresholds (p_c and p_u) of {P,Q} tessellations of the hyperbolic plane.
  • To explore the functional dependency of these thresholds on P and Q.
  • To numerically compute critical exponents and establish rigorous bounds for scaling analysis.

Main Methods:

  • Utilizing the invasion percolation algorithm for numerical computation.
  • Analyzing {P,Q} tessellations of the hyperbolic plane.
  • Deriving rigorous upper and lower bounds for percolation thresholds.

Main Results:

  • Accurate numerical values for bond and site percolation thresholds (p_c, p_u) were obtained, precise to six or seven decimal places.
  • The functional dependency of p_c and p_u on P and Q was explored.
  • Critical exponents were numerically computed, and rigorous bounds for threshold scaling were established.

Conclusions:

  • The study provides precise numerical data and theoretical bounds for percolation on hyperbolic tessellations.
  • The findings facilitate a deeper understanding of the scaling behavior of percolation thresholds in these complex geometric structures.
  • This research contributes to the fields of statistical physics and computational geometry.