Critical percolation on the kagome hypergraph

TL;DR

This paper investigates the critical percolation surface of the kagome hypergraph, using critical polynomials to accurately estimate percolation thresholds for various related lattices, revealing insights into the method's precision.

Contribution

It introduces a high-precision method for analyzing the critical surface of the kagome hypergraph, including special cases and an exact point, enhancing understanding of percolation thresholds.

Findings

Critical points computed to 17-18 digit precision.

Method provides accurate estimations comparable to Monte Carlo.

Analysis explains why the method is highly accurate for certain lattices.

Abstract

We study the percolation critical surface of the kagome lattice in which each triangle is allowed an arbitrary connectivity. Using the method of critical polynomials, we find points along this critical surface to high precision. This kagome hypergraph contains many unsolved problems as special cases, including bond percolation on the kagome and $(3,12^2)$ lattices, and site percolation on the hexagonal, or honeycomb, lattice, as well as a single point for which there is an exact solution. We are able to compute enough points along the critical surface to find a very accurate fit, essentially a Taylor series about the exact point, that allows estimations of the critical point of any system that lies on the surface to precision rivaling Monte Carlo and traditional techniques of similar accuracy. We find also that this system sheds light on some of the surprising aspects of the method of critical polynomials, such as why it is so accurate for certain problems, like the kagome and $(3,12^2)$ lattices. The bond percolation critical points of these lattices can be found to 17 and 18 digits, respectively, because they are in close proximity, in a sense that can be made quantitative, to the exact point on the critical surface. We also discuss in detail a parallel implementation of the method which we use here for a few calculations.