Level Sets of the Gaussian Free Field and the Random Walk Loop Soup

The Gaussian free field is an ubiquitous object in probability theory and mathematicalphysics. It plays a central role in many different situations, and we aim to study furtherits geometric properties. This process is a random two-dimensional surface, arisingspontaneously from the stochastic behavior of various systems, to describe their randomfluctuations. It turns out to be extremely rough, which makes its geometric featuresquite challenging to study.Bernoulli percolation is a classical model for random media, which amounts to open orclose the vertices of a lattice in an independent fashion. For some parameter p, eachvertex is open with probability p, and it is closed with probability 1-p. As one lets theparameter vary, a phase transition can be observed, i.e. a drastic change of behavior, atsome critical threshold pc: for pcomponents, while for p>pc, a giant (infinite) connected component of open verticesappears. Percolation theory is widely used in applications, to understand spatialcorrelations in random systems.We are interested in the percolative properties of two-sided level sets of the Gaussianfree field, where the field remains not too large nor too small, that is, between twovalues -L and L. For this purpose, we want to bring techniques and ideas frompercolation theory, and combine them with other tools developed recently for the randomwalk loop soup. This latter planar process is intimately related to the Gaussian free field,and its Poisson structure, together with elementary properties of random walks, allowsfor powerful recombination arguments on loops.In particular, in the subcritical random walk loop soup, we plan to study the sharpness ofthe transition as one lets the threshold L increase, in analogy to Bernoulli percolation.Next we want to study the behavior of the loop soup for supercritical intensities. Finally,we aim to improve our understanding of the critical case, which is the one directly linkedto the Gaussian free field, and where geometric considerations seem to be the mostsubtle.