Clustering Bounds on n-Point Correlations for Unbounded Spin Systems

2009/01/29 by Abdelmalek Abdesselam, Aldo Procacci, Benedetto Scoppola · 1 voice
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.PR

paper · pdf · doi:10.1007/s10955-009-9789-y

arxiv published 2009/01/29 · arxiv updated 2009/01/29 · openalex publication_date 2009/07/27 · crossref created 2009/07/27 · crossref issued 2009/07/28 · crossref published 2009/07/28 · crossref published-online 2009/07/28 · crossref published-print 2009/08/01 · openalex created_date 2016/06/24 · crossref deposited 2020/05/21 · crossref indexed 2025/10/01 · openalex updated_date 2026/07/28

Abstract

We prove clustering estimates for the truncated correlations, i.e., cumulants of an unbounded spin system on the lattice. We provide a unified treatment, based on cluster expansion techniques, of four different regimes: large mass, small interaction between sites, large self-interaction, as well as the more delicate small self-interaction or `low temperature' regime. A clustering estimate in the latter regime is needed for the Bosonic case of the recent result obtained by Lukkarinen and Spohn on the rigorous control on kinetic scales of quantum fluids.

Discussions

• The QFT cluster expansion is a bit different from the one used in statistical mechanics. The main use is to rigorously control the infinite volume limit $\Lambda\nearrow\mathbb{Z}^d$ for the pressure [stackexchange, 4 points]

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