# Large holes in particle systems : forbidden regions, large deviations and potential theory

# Large holes in particle systems : forbidden regions, large deviations and potential theory

In particles systems, a "hole'' of size R is defined to be a ball of radius R that is devoid of particles. The study of how the probability of having such a hole decays to 0 (as R -> infty) is an important and well-studied question in particle systems. In this talk, we ask what causes a large hole to appear? In other words, conditioned on having a large hole, how does the configuration of particles outside the hole look like? Surprisingly, very little is understood about this question, except in the very special case of Gaussian random matrix ensembles, where there is an accumulation of particles at the edge of the hole, and equilibrium intensity beyond. We study this question in the context of zeros of Gaussian random polynomials, and provide a complete description of the intensity profile of the outside particles. A remarkable feature that we find is the appearance of a curious "forbidden region'' between the accumulation at the edge of the hole and the equilibrium intensity far beyond. This is in stark contrast to the case of Gaussian random matrices, and seems to be novel even in the wider setting of statistical physics models. Our methods connect to effective versions of large deviation principles for random polynomials, potential theory and constrained optimization on measure spaces. These ideas can also be applied to other problems, including Jancovici-Lebowitz-Manificat laws for Coulomb systems at general temperatures, and understanding over and under-crowding phenomena for Gaussian zeros. Based on joint works with Alon Nishry.