Odd cutsets and the hard-core model on \(\mathbb{Z}^{d}\)
The authors study the hardcore lattice gas on \(\mathbb Z^d\) and study in particular the high-dimensional, high-density situation, where there are multiple (Gibbs) measures. They improve on existing results by Galvin and Kahn, showing that there is a phase transition if the intensity parameter (``activity'') is larger than \(C d^{\frac{1}{3}} \log^2 d\), rather than the earlier known \(d^{\frac{1}{4}} (\log d)^{\frac{3}{4}}\). As the actual result is conjectured to behave like \(C d^{-1}\), possibly with logarithmic corrections, this result is not yet optimal, but the arguments are already quite ingenious. The authors use refined combinatorics on ``cutsets'' which play the roles of contours, separating the odd and even phases. Some of the arguments which they employ were earlier used by Peled.
scientific article; zbMATH DE number 6340415
scientific article