Percolation on the square lattice in aL�M geometry

Roberto Monetti, Ezequiel Vicente Albano

M. Goldberg

Yang Li, WU Da-yan, Xianshan Huang +1 more

Andrew Stewart Caldin

FINDING: Topological color code accuracy thresholds on hexagonal and square-octagonal lattices are determined via mapping to spin models, revealing lattice-dependent percolation thresholds. | MATH: The mapping reduces the quantum error correction threshold to a classical percolation problem on the dual lattice. For the square-octagonal lattice, the site percolation threshold \( p_c \) is approximately 0.5 (exact for square lattice) but modified by the octagonal tiling geometry; the hexagonal lattice yields \( p_c \approx 0.6527 \) (site percolation on triangular lattice). The threshold for the color code is given by \( p_{\text{th}} = 1 - p_c \) for the hexagonal case, and analogous for square-octagonal. Key constants: hexagonal lattice percolation threshold \( p_c = 0.5 \) (bond) or 0.5927 (site); square-octagonal lattice percolation threshold is not exactly known but lies near 0.5–0.6. | CONNECTION: The square-octagonal lattice is a semiregular tiling (Archimedean) with vertex config Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Xiaomei Feng, Youjin Deng, Henk W. J. Blöte

We investigate bond- and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome, and diced lattices with nearest-neighbor bonds, and the square lattice with nearest- and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb, and diced lattices. We also include the bond- and site-percolation thresholds for the square lattice with nearest- and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension X_{t2}=4 predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent. The amplitude of the power-law correction associated with X_{t2}=4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.

C. M. Chandrashekar, Thomas Busch

Quantum percolation describes the problem of a quantum particle moving through a disordered system. While certain similarities to classical percolation exist, the quantum case has additional complexity due to the possibility of Anderson localisation. Here, we consider a directed discrete-time quantum walk as a model to study quantum percolation of a two-state particle on a two-dimensional lattice. Using numerical analysis we determine the fraction of connected edges required (transition point) in the lattice for the two-state particle to percolate with finite (non-zero) probability for three fundamental lattice geometries, finite square lattice, honeycomb lattice, and nanotube structure and show that it tends towards unity for increasing lattice sizes. To support the numerical results we also use a continuum approximation to analytically derive the expression for the percolation probability for the case of the square lattice and show that it agrees with the numerically obtained results for the discrete case. Beyond the fundamental interest to understand the dynamics of a two-state particle on a lattice (network) with disconnected vertices, our study has the potential to shed light on the transport dynamics in various quantum condensed matter systems and the construction of quantum information processing and communication protocols.