Hard-core configurations on a triangular lattice and Eisenstein primes

A. Mazel, I. Stuhl, Y. Suhov

Published 2018-03-11Version 1

We study the Gibbs statistics of high-density hard-core random configurations on a triangular lattice. Depending on certain arithmetic properties of the repulsion diameter $D$ (related to Eisenstein integers), we identify, for a large fugacity, the extreme periodic Gibbs measures and analyze their properties. (a) For the values of $D$ belonging to certain arithmetic classes a complete phase diagram is established. (b) For the remaining values of $D$ we prove non-uniqueness of a pure phase and provide some additional information. We argue that in general the list of extreme periodic Gibbs measures can vary according to the arithmetic structure of $D$. This argument is supported by the analysis of several specific values of $D$ (outside the classes from (a)) for which the complete phase diagram is established, using in part a computer assistance. The proofs are achieved by applying Zahradnik's extension of the Pirogov--Sinai theory.