Percolation on hyperbolic lattices

Abstract

The percolation transitions on hyperbolic lattices are investigated numerically using finite-size scaling methods. The existence of two distinct percolation thresholds is verified. At the lower threshold, an unbounded cluster appears and reaches from the middle to the boundary. This transition is of the same type and has the same finite-size scaling properties as the corresponding transition for the Cayley tree. At the upper threshold, on the other hand, a single unbounded cluster forms which overwhelms all the others and occupies a finite fraction of the volume as well as of the boundary connections. The finite-size scaling properties for this upper threshold are different from those of the Cayley tree and two of the critical exponents are obtained. The results suggest that the percolation transition for the hyperbolic lattices forms a universality class of its own.

Phase transition ofq-state clock models on heptagonal lattices

“…Around that point, the hyperbolic lattice begins to manifest itself more as a surface. In contrary to the tree case above, for example, the energy cost at a domain wall increases roughly logarithmically with the cluster size [10], opening the possibility for T c to be finite. As a consequence, we observe these three phases in general: an ordered phase, a disordered phase but having a diverging susceptibility, and a normal disordered phase with a finite susceptibility.…”

“…Note that the summation is limited by the number of generations, n. In other words, the susceptibility diverges with χ n ∝ n at R = 1/ √ 2. Recalling differences between with and without loops in percolation phenomena [10], we may expect only a qualitative understanding for the heptagonal lattice from studying the Cayley tree rather than a quantitative agreement. Although the presence of closed loops will presumably alter the results described above, the essential parts of these arguments could be conveyed to our heptagonal lattice.…”

“…This argument is based on the fact that a negatively curved surface contains a huge amount of boundary points: that is, for a negatively curved surface, the ratio of surface area to perimeter (which is the * Corresponding author, E-mail: beomjun@skku.edu two-dimensional example of the so-called surface-volume ratio in general dimension) remain nonvanishing even in the large-system limit. Since it was pointed out that a system may have a novel behavior due to the presence of a nonvanishing boundary [7], there have been ongoing studies to clarify this issue [4,6,[8][9][10][11]. While the boundary effects can be sometimes excluded, for example, by using a periodic boundary condition [12] or by mathematical abstractions [5,[13][14][15], it is often crucial to understand how a boundary affects the physical properties since it may give the most important contribution to an observed behavior as will be explained in this work.…”

Quantum erasure-correcting codes and percolation on regular tilings of the hyperbolic plane

“…Interest in percolation on hyperbolic tilings was raised in a number of papers e.g. [1,3] and determining their critical probability is highly non-trivial. Note that all graphs G(m) are self-dual like the square lattice G(m).…”

Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network

“…A critical phase, if it exists, lies between a disordered phase withξ < 1/y d and an ordered phase withξ = ∞. Such a phase with a fractal exponent 0 < ψ < 1 is actually observed in the percolation transitions in enhanced trees [5], hyperbolic lattices [10], hierarchical graphs [11,12], and growing random networks [13].…”