Nonlinear distributional fixed-point equations in statistical mechanics

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In numerous stochastic models, the probability distribution of quantities of interest can only be described implicitly, as solutions of so-called distributional fixed-point equations. While linear distributional fixed-point equations have been studied intensively there is no theory of nonlinear distributional fixed-point equations, although such equations appear prominently in active fields of research in statistical physics.The aim of my project is to study nonlinear distributional fixed-point equations, starting from concrete examples. Relevant questions concern the existence and uniqueness of solutions and their properties, such as smoothness or tail behaviour.These results will help to simplify calculating critical parameters in spin models and will provide new insights into equilibrium distributions of ideal gases.