Bounds of percolation thresholds on hyperbolic lattices

Area of Science:

• Statistical physics
• Geometric combinatorics
• Hyperbolic geometry

Background:

• Percolation theory investigates connectivity in random networks.
• Hyperbolic lattices offer unique geometric properties distinct from Euclidean space.
• Understanding phase transitions in these lattices is crucial for various scientific domains.

Purpose of the Study:

• To analytically determine the critical occupation probability (p(c2)) for bond percolation on specific hyperbolic lattices.
• To establish lower bounds for p(c2) in order-5 square, its dual, and order-5-4 rhombille tilings.
• To contribute to the understanding of percolation phenomena in non-Euclidean geometries.

Main Methods:

• Analytical study of bond percolation.
• Application of the substitution method.
• Utilizing known bounds for the order-5 pentagonal tiling.

Main Results:

• Established a lower bound for p(c2) of 0.382508 for the order-5 square tiling.
• Determined a lower bound for p(c2) of 0.472043 for the dual of the order-5 square tiling.
• Calculated a lower bound for p(c2) of 0.275768 for the order-5-4 rhombille tiling.

Conclusions:

• The study provides precise lower bounds for percolation thresholds on hyperbolic lattices.
• These findings enhance the theoretical framework for percolation on complex geometric structures.
• The results are applicable to fields involving network science and statistical mechanics on curved spaces.