METASTABILITY FOR EXPANDING BUBBLES ON A STICKY SUBSTRATE

• 成果类型: Article
• 署名作者: Lacoin, Hubert; Yang, Shangjie
• 署名单位: Instituto Nacional de Matematica Pura e Aplicada (IMPA)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1763
• 发表日期: 2022
• 页码: 3408-3449
• 关键词: mixing times; cutoff; polymer; equilibrium; DYNAMICS

摘要

We study the dynamical behavior of a one dimensional interface interacting with a sticky impenetrable substrate or wall. The interface is subject to two effects going in opposite directions. Contacts between the interface and the substrate are given an energetic bonus while an external force with constant intensity pulls the interface away from the wall. Our interface is modeled by the graph of a one-dimensional nearest-neighbor path on Z(+), starting at 0 and ending at 0 after 2N steps, with the wall being the horizontal axis. At equilibrium each path xi = (xi(x))(x=0)(2N) is given a probability proportional to lambda(H(xi)) exp(sigma/N A(xi)), where H(xi) := #{x :xi(x) = 0} and A(xi) is the area enclosed between the path xi and the x-axis. We then consider the classical heat-bath dynamics which equilibrates the value of each xi(x) at a constant rate via corner-flip. Investigating the statics of the model, we derive the full phase diagram in lambda and sigma of this model, and identify the critical line which separates a localized phase where the pinning force sticks the interface to the wall and a delocalized one, for which the external force stabilizes xi around a deterministic shape at a macroscopic distance of the wall. On the dynamical side, we identify a second critical line, which separates a rapidly mixing phase (for which the system mixes in polynomial time) to a slow phase where the mixing time grows exponentially. In this slowly mixing regime, we obtain a sharp estimate of the mixing time on the log scale, and provide evidences of a metastable behavior.