Scaling limits and extreme values of Gibbs measures

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In the area of probability, an increasingly important role has been played in recent years by random systems in which the randomness is observed in the spatial structure. Random systems defined on lattices have been introduced as discrete models that describe phase transitions for various phenomena, ranging from liquid in porous media to the spread of disease. Our understanding of some of these models, such as percolation and Ising model, has been improved greatly in the last decades, and works around it have led to Fields medals in 2006 and 2010.The aim of the proposed research is to open new directions for several long standing open questions in random systems on lattices. One circle of the questions concern the gradient Gibbs measures, which is a model of random surface introduced in the 1970s by Brascamp, Lebowitz and Lieb as a model for crystal interfaces. A long standing universality conjecture states that the large scale statistical properties of these random surfaces behave like a Gaussian free field. This has been partially confirmed by the work of Naddaf and Spencer (and others). The PI intends to improve the understanding of the gradient Gibbs measures, by quantifying the existing fluctuation theorems, settling the 20-year-old conjectures in surface tension (that describes the energy of a surface profile with a global tilt), and to establish some universality conjectures of the extremes of log-correlated fields.The second circle of questions concern the XY and the Villain models, which are mathematical models of liquid crystals, liquid helium and superconductors. Works around it have led to the Nobel Prize in Physics (Kosterlitz and Thouless) in 2016. Physicists predict that at low temperature the large scale property of these models are closely related to the Gaussian free field. This is known as the Gaussian spin wave conjecture. Some mathematical progress was made towards the conjecture in the 1970s and the early 1980s, building around the works of Frohlich, Simon and Spencer. However, methods developed in these papers (infrared bounds and Coulomb gas renormalization) were not sufficient to complete the proof of this conjecture. The PI intends to resolve this long-standing Gaussian spin wave conjecture for the XY and the Villain models in dimension three and higher.In doing so, the PI will develop a robust framework to study the scaling limits, fluctuations and large deviations of a large class of Gibbs measures. New bridges will be built between probability, statistical mechanics and mathematical analysis.