Approximate capacities of two-dimensional codes by spatial mixing

Abstract

We apply several techniques developed in recent years for counting algorithms and statistical physics to study the spatial mixing property of two-dimensional codes arising from local hard (independent set) constraints, including: hard-square, hard-hexagon, read/write isolated memory (RWIM), and nonattacking kings (NAK). For these constraints, the existence of strong spatial mixing implies the existence of polynomial-time approximation scheme (PTAS) for computing the capacity. The existence of strong spatial mixing and PTAS was previously known for the hard-square constraint. We show the existence of strong spatial mixing for hard-hexagon and RWIM constraints by establishing the strong spatial mixing along self-avoiding walks, which implies PTAS for computing the capacities of these codes. We also show that for the NAK constraint, the strong spatial mixing does not hold along self-avoiding walks.

Counting independent sets in amenable groups

“…There have been many recent works related to the study of correlation decay properties and their relation to approximation algorithms for the free energy (and related quantities such as pressure, capacity, and entropy) in the infinite setting [8, 16, 34–36, 40, 53]. In this work, we put all these results in a single framework, which also encompasses the results from Weitz, Sly and Sun, and Sinclair et al , and at the same time generalizes them.…”

“…Related results were also proven by Pavlov in [40], who developed an approximation algorithm for the hard square entropy, that is, the free energy of the hardcore model in the Cayley graph of Z 2 with canonical generators and activity λ = 1. Later, there were also some explorations due to Wang et al [53] in Cayley graphs of Z 2 with respect to other generators (e.g., the non-attacking kings system) in the context of information theory and algorithms for approximating capacities.…”