2018/03/11 by A. Mazel, Mazel, A., I. Stuhl +5 · 1 citation
Mathematics · Physics and Astronomy · #60G60 #82B20 #82B26 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60G60 #msc:82B20 #msc:82B26
paper · pdf · doi:10.48550/arxiv.1803.04041
Analysis of the model on the hexagonal lattice has been added. Assisting programs are in Ancillary files
openalex publication_date 2018/03/11 · arxiv created 2020/10/21 · arxiv updated 2020/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We perform a rigorous study of the Gibbs statistics of high-density hard-core random configurations on a unit triangular lattice \mathbbA2 and a unit honeycomb graph ℍ2, for any value of the (Euclidean) repulsion diameter D>0. Only attainable values of D are relevant, for which D2=a2+b2+ab, a, b ∈ℤ (Löschian numbers). Depending on arithmetic properties of D2, we identify, for large fugacities, the pure phases (extreme Gibbs measures) and specify their symmetries. The answers depend on the way(s) an equilateral triangle of side-length D can be inscribed in \mathbbA2 or ℍ2. On \mathbbA2, our approach works for all attainable D2; on ℍ2 we have to exclude D2 = 4, 7, 31, 133, where a sliding phenomenon occurs, similar to that on a unit square lattice ℤ2. For all values D2 apart from the excluded ones we prove the existence of a first-order phase transition where the number of co-existing pure phases grows at least as O(D2). The proof is based on the Pirogov--Sinai theory which requires non-trivial verifications of key assumptions: finiteness of the set of periodic ground states and the Peierls bound. To establish the Peierls bound, we develop a general method based on the concept of a re-distributed area for Delaunay triangles. Some of the presented proofs are computer-assisted. As a by-product of the ground state identification, we solve the disk-packing problem on \mathbbA2 and ℍ2 for any value of the disk diameter D.