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 Abstract:  The high-density hard-core configuration model has attracted attention for quite a long time; first rigorous results about the phase transition on a lattice where published by Dobrushin in late 1960s. Since then, various aspects of the model became important in a number of applications. We propose a solution for the model on a triangular lattice. The phase diagram (i.e., the collection of shift-periodic pure phases) depends on arithmetic properties of the exclusion distance \(D\); a convenient classification of possible cases can be given in terms of Eisenstein primes.

For two classes of values of \(D\) the phase diagrams is completely specified:

(I) when either \(D\) or \(D/{\sqrt 3}\) is a positive integer whose prime decomposition does not contain factors of the form \(6k+1\),

(II) when \(D^2\) is an integer whose prime decomposition contains (i) a single prime of the form \(6k+1\), and (ii) other primes, if any, in even powers, except for the prime \(3\). For the remaining values of \(D\) we offer some partial results supported by computer simulations. The main method of proof is the Pirogov-Sinai theory complemented by Zahradnik's argument and the Dominated ground state theory.

This is a joint work with A. Mazel and Y. Suhov.