Percolation Thresholds in Hyperbolic Lattices

Abstract

We use invasion percolation to compute numerical values for bond and site percolation thresholds $p_c$ (existence of an infinite cluster) and $p_u$ (uniqueness of the infinite cluster) of tesselations $\{P,Q\}$ of the hyperbolic plane, where $Q$ faces meet at each vertex and each face is a $P$-gon. Our values are accurate to six or seven decimal places, allowing us to explore their functional dependency on $P$ and $Q$ and to numerically compute critical exponents. We also prove rigorous upper and lower bounds for $p_c$ and $p_u$ that can be used to find the scaling of both thresholds as a function of $P$ and $Q$.

extended journal version, 14 pages, 17 figures, 3 tables

References in corpus (18)

• Hyperbolic Geometry of Complex Networks
• Efficient Monte Carlo algorithm and high-precision results for percolation
• Continuum Percolation Thresholds in Two Dimensions
• Cayley Trees and Bethe Lattices, a concise analysis for mathematicians and physicists
• Percolation in the Hyperbolic Plane
• Random numbers for large scale distributed Monte Carlo simulations
• Geometric effects on critical behaviours of the Ising model
• Random walk on Bethe lattice and hyperbolic geometry
• Percolation on hyperbolic lattices
• Ising model on hyperbolic lattice studied by corner transfer matrix renormalization group method
• Hard Discs on the Hyperbolic Plane
• The dynamic exponent of the Ising model on negatively curved surfaces
• Weak correlation effects in the Ising model on triangular-tiled hyperbolic lattices
• Percolation in Finite Matching Lattices
• Crossing on hyperbolic lattices
• Diffusion on a heptagonal lattice
• Free Energy Analysis of Spin Models on Hyperbolic Lattice Geometries
• Bounds of percolation thresholds on hyperbolic lattices

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• Topological Percolation on Hyperbolic Simplicial Complexes
• Percolation Thresholds and Fisher Exponents in Hypercubic Lattices
• Percolation on branching simplicial and cell complexes and its relation to interdependent percolation
• Efficient space virtualisation for Hoshen--Kopelman algorithm
• Renormalization group for link percolation on planar hyperbolic manifolds
• Percolation thresholds on high dimensional $D_n$ and dense packing lattices
• What are the limits of universality?
• Uncovering Exceptional Contours in non-Hermitian Hyperbolic Matter
• Homology-changing percolation transitions on finite graphs
• Extremal $\{p,q\}$-Animals