Shattering in the Ising p-spin glass model

• 成果类型: Article
• 署名作者: Gamarnik, David; Jagannath, Aukosh; Kizildag, Eren C.
• 署名单位: Massachusetts Institute of Technology (MIT); University of Waterloo; University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01414-4
• 发表日期: 2025
• 页码: 89-141
• 关键词: random-energy model; local algorithms; solvable model; bounds; phase

摘要

We study the Isingp-spin glass model for largep. We show that for any inversetemperature root ln 2infinity, there exists exponentially many well-separated clusters such that(a) each cluster has exponentially small Gibbs mass, and (b) the clusters collectivelycontain all but a vanishing fraction of Gibbs mass. Moreover, these clusters consist ofconfigurations with energy near beta. Range of temperatures for which shattering occursis within thereplica symmetricregion. To the best of our knowledge, this is the firstshattering result regarding the Isingp-spin glass models. Furthermore, we show thatfor any gamma>0 and any large enoughp, the model exhibits an intricate geometricalproperty known as the multi Overlap Gap Property above the energy value gamma root 2ln2.Our proofs are elementary, and in particular based on simple applications of the firstand the second moment methods.