The incipient infinite cluster of the FK-Ising model in dimensions d ≥ 3 and the susceptibility of the high-dimensional Ising model

• 成果类型: Article
• 署名作者: Panis, Romain
• 署名单位: Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Jean Monnet; Universite Lyon 1; Ecole Centrale de Lyon; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-025-01452-y
• 发表日期: 2026-04
• 页码: 2055-2097
• 关键词: self-avoiding walk; INVARIANT GIBBS-STATES; critical-behavior; phase-transitions; REFLECTION POSITIVITY; oriented percolation; SCALING LIMITS; lattice trees; inequalities; triviality

摘要

We consider the critical FK-Ising measure phi beta c on Z(d) with d >= 3. We construct the measure phi infinity:=lim|x|->infinity phi beta c[|0 <--> x] and prove it satisfies phi infinity[0 <-->infinity]=1. This corresponds to the natural candidate for the incipient infinite cluster measure of the FK-Ising model. Our proof uses a result of Lupu and Werner (Electron. Commun. Probab., 2016) that relates the FK-Ising model to the random current representation of the Ising model, together with a mixing property of random currents recently established by Aizenman and Duminil-Copin (Ann. Math., 2021). We then study the susceptibility chi(beta)of the nearest-neighbour Ising model on Zd . When d>4, we improve a previous result of Aizenman (Comm. Math. Phys., 1982) to obtain the existence of A>0 such that, for beta