2023/04/17 by A. Mazel, Mazel, A., I. Stuhl +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2304.08642
openalex publication_date 2023/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the hard-core model of statistical mechanics on a unit cubic lattice ℤ3, which is intrinsically related to the sphere-packing problem for spheres with centers in ℤ3. The model is defined by the sphere diameter D>0 which is interpreted as a Euclidean exclusion distance between point particles located at spheres centers. The second parameter of the underlying model is the particle fugacity u. For u>1 the ground states of the model are given by the dense-packings of the spheres. The identification of such dense-packings is a considerable challenge, and we solve it for D2=2, 3, 4, 5, 6, 8, 9, 10, 11, 12 as well as for D2=2ℓ2, where ℓ∈ℕ. For the former family of values of D2 our proofs are self-contained. For D2=2ℓ2 our results are based on the proof of Kepler's conjecture. Depending on the value of D2, we encounter three physically distinct situations: (i) finitely many periodic ground states, (ii) countably many layered periodic ground states and (iii) countably many not necessarily layered periodic ground states. For the first two cases we use the Pirogov-Sinai theory and identify the corresponding periodic Gibbs distributions for D2=2,3,5,8,9,10,12 and D2=2ℓ2, ℓ∈ℕ, in a high-density regime u>u_*(D2), where the system is ordered and tends to fluctuate around some ground states. In particular, for D2=5 only a finite number out of countably many layered periodic ground states generate pure phases.