CONFORMAL MEASURE ENSEMBLES AND PLANAR ISING MAGNETIZATION: A REVIEW FEDERICO CAMIA, JIANPING JIANG, AND CHARLES M. NEWMAN Abstract. We provide a review of results on the critical and near-critical scaling limit of the planar Ising magnetization field obtained in the past dozen years. The results are presented in the framework of coupled loop and measure ensembles, and some new proofs are provided. 1. Synopsis In [18] the first and third authors introduced the concept of Conformal Measure En- semble (CME) as the scaling limit of the collection of appropriately rescaled counting measures of critical FK-Ising clusters. They proposed to use a representation of the Ising magnetization field in terms of such a CME to study its existence, uniqueness and con- formal properties in the critical scaling limit. Initial results and work in progress with Christophe Garban were presented by the first author at the Inhomogeneous Random Systems 2010 conference (Institut Henri Poincar´ e, Paris) and described in [6]. CMEs for Bernoulli and FK-Ising percolation were first constructed in [7]. The results on the two-dimensional Ising model discussed or conjectured in [6] have now been fully proved and have appeared in various papers by a combination of different authors [7, 11–14]. Those results, and more, were recently presented in a talk at the Inhomogeneous Ran- dom Systems 2020 conference (Institut Curie, Paris), which was the inspiration for the present paper, whose main goal is to review the results of [11–14] and present them in the unifying CME framework. While the results presented in this paper are not new, in some cases their formulation is somewhat different than what has previously appeared in the literature, and whenever we provide a detailed proof of a result, the proof is new. 2. Introduction and historical remarks The Ising model was introduced by Lenz in 1920 [37] to describe ferromagnetism, and is nowadays one of the most studied models of statistical mechanics. The one-dimensional version of the model was studied by Ising in his Ph.D. thesis [28] and in his subsequent paper [29], but it was not until Peierls’ and Onsager’s famous investigations of the two- dimensional version that the model gained popularity. In 1936 Peierls [43] proved that the two-dimensional model undergoes a phase transition; then in 1941 Kramers and Wan- nier [32] located the critical temperature of the model defined on the square lattice, and in 1944 Onsager [42] derived its free energy. Since then, the two-dimensional Ising model has played a special role in the theory of critical phenomena. Its phase transition has been extensively studied by both physicists and mathematicians, becoming a prototypical example and a test case for developing ideas and techniques and for checking hypotheses. Ferromagnetism is one of the most interesting phenomena in solid state physics; it refers to the tendency, observed in some metals such as iron and nickel, of the atomic spins to become spontaneuosly polarized in the same direction, generating a macroscopic 2010 Mathematics Subject Classification. Primary: 60K35, 82B20; Secondary: 82B27, 81T27, 81T40. 1 arXiv:2009.08129v1 [math.PR] 17 Sep 2020