Universality and conformal invariance of certain two dimensional statistical physics models

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My research is in the field of probability theory primarily focussing on problems arising from the field of statistical and mathematical physics. Statistical physics tries to justify various physical phenomena (for example phase transition like melting of ice, demagnetization of magnets at high temperature etc) using a model which is inherently probabilistic in nature. The basic idea is to provide a framework to describe a macroscopic phenomenon (like transition of state, which we can actually see in everyday life) as an outcome of the microscopic behaviour of the molecules and atoms. The premise is that the microscopic behaviour is random which can be described using a probabilistic model, and the large scale statistical behaviour of this model is precisely what we observe as a macroscopic phenomenon. My research is based on problems related to the universal large scale behaviours of such statistical physics models. The questions become particularly interesting at or near criticality, that is, at or near the precise point when the phase transition occurs.******During and around my PhD, I worked on problems built into a framework where the underlying geometry is also random, which leads to the topic of Liouville quantum gravity and random planar maps. Recently I am also interested in gradient models, particularly the dimer model, random homomorphisms and the six vertex model. Generally, one can think of these models as models of random functions and in the planar case, these functions can be viewed as a random surface. My focus is mainly in the planar case, where I am working on toy problems around several big conjectures regarding the universal fluctuations of these random surfaces.******My current research program consists primarily of two such models. One is concerned broadly with establishing universality of fluctuations in the dimer model (which is a model of random perfect matchings on graphs). With my collaborators in Cambridge and Paris, we have developed a novel technique which enables us to obtain a universality result. At the moment, we are writing a series of papers on problems along these lines, a couple of which are already submitted.******The second program involves random homomorphisms from the square lattice to integers. Along with my collaborators in Paris, we are trying to develop a renormalization technique to obtain information about the correlations of these models. Finding out precise information about these models is quite challenging and many questions have been open for a while. Our approach is to utilize the renormalization techniques recently developed to analyze a similar model.******These two programs give rise to several projects and sub-projects. In the upcoming years, my target would be to make progress on these questions of interest.**