Scaling Limits for Pinned Gaussian Random Interfaces in the Presence of Two Possible Candidates

Abstract

The macroscopic shape of crystals is usually described by variational problems. We first explain those characterizing the Wulff shape, the Winterbottom shape and also those with two media, with a pinning effect. We give examples of minimizers in a pinning case. Then, we explain underlying microscopic models such as the Ising model and the ∇φ-interface model. Macroscopic variational problems and microscopic models are linked by a large deviation principle, or a law of large numbers. We will focus on the ∇φ-interface model with a pinning. For such model, results for d = 1, n ≥ 1 (obtained with Bolthausen et al. (Probab Theory Relat Fields 143:441–480, 2009) and Funaki and Otobe (J Math Soc Jpn 62:1005–1041, 2010)) and those for d ≥ 3, n = 1 (obtained with Bolthausen et al. (J Math Soc Jpn 67:1359–1412, 2015. Special issue for Kiyosi Ito)) will be presented, where d is the dimension of the base space, while n is the dimension of the value (target) space. See Funaki and Spohn (Commun Math Phys 185:1–36, 1997) and Funaki (Stochastic interface models. In: Picard J (ed) Lectures on probability theory and statistics. Ecole d’Ete de Probabilites de Saint-Flour XXXIII – 2003. Lecture notes in mathematics, vol 1869. Springer (2005), pp 103–274) for the ∇φ-interface model.