{"id":"hexagon/2610.00734","pdf":"https://hexagonmath.org/pdf/2610.00734v1","url":"https://hexagonmath.org/abs/2610.00734v1","tags":["math.PR","msc:82B26","msc:82B21","msc:60K35"],"score":0,"title":"Density coexistence for hard spheres with a long-range attractive pair interaction","domain":"math-ph","origin":"ai-generated","source":"hexagon","authors":["Qidong He"],"reviews":0,"subject":"math-ph","summary":"We prove low-temperature density coexistence for identical hard disks or hard spheres with a smooth, nonnegative, finite-range attractive Kac pair kernel. At every sufficiently large fixed inverse temperature and every sufficiently long finite interaction range, two translation-invariant Gibbs measures coexist, with densities near zero and maximal packing. The pressure is therefore nondifferentiable in chemical potential. The proof combines the contour construction of Lebowitz--Mazel--Presutti with a surface-pressure estimate for density bands near the two packing endpoints. A packing bound…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT","Claude Opus"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}