Large-scale universal behaviour of the Ising model

This research lies in the area of probability theory, specifically discrete probability, and the main focus is on the development of rigorous mathematical tools to analyse spin models, which arise in mathematical physics. Spin models are primarily used to explain the observed phenomenon whereby the strength of magnets gradually decreases as the temperature increases, eventually dissipating when a critical temperature is reached. Mathematically, spin models are defined as complex systems comprising multiple microscopic magnetic components known as spins, which interact with one another. Each spin has a random orientation and magnitude whose values are constrained by a potential. Despite the wide range in the choice of the potential, these models often exhibit similar large-scale behaviour, and are, therefore, said to belong to the same Universality Class. The Ising model, the most prominent among spin models where spins have only orientation, is considered universal for a wide class of models. Its study has flourished over the last few decades due to its fundamental link with the seemingly unrelated area of percolation theory, which models how fluids flow through a porous medium. This link allows relating important objects of interest in the study of the Ising model, such as its correlation functions, to geometric percolation objects, and has led to several recent breakthroughs in the understanding of its critical behaviour. The proposed research programme will make a major contribution to solving the long-standing problem of establishing that the large-scale behaviour of the Ising model is universal. Despite the impressive progress mentioned above, existing results rely to a large extent on delicate properties of the Ising model specifically, and therefore showing their universality remains elusive. Our strategy will be to focus on some of the most fundamental examples of spin models, which share essential similarities with the Ising model, but lack some of the particular properties that facilitated many of the results on its critical phenomena: the f4 and Blume-Capel models. Their study poses new challenges due to the presence of spins that have both orientation and magnitude. This leads to exciting new features which are observed by a wide variety of spin models. More specifically, the concrete objectives, all tied together by the overarching goal of determining the large-scale behaviour of spin models, are outlined as follows: - I will prove the exponential decay of truncated correlations at large scales in the whole supercritical phase. - I will show the large-scale decay of correlations at the critical temperature in dimension 2 and deduce that the magnets lose their magnetisation. This extends the result obtained in my recent work in dimensions 3 and above. - I will establish that the Pfaffian structure of the correlations of the planar Ising model is universal at large scales. A key tool for accomplishing these objectives will be the novel percolation representations introduced in my recent works, which lay the foundations for analysing these models. These representations enable the use of well-established percolation machinery in new contexts. Furthermore, the proposed approach will utilise the close connection with the Ising model to import cutting-edge techniques pivotal for the recent developments in its study.