Nonlinear distributional fixed-point equations in statistical mechanics

在许多随机模型中，感兴趣的量的概率分布只能隐式地描述为所谓的分布不动点方程的解。虽然线性分布不动点方程已经被深入研究，但非线性分布不动点方程的理论还没有出现，尽管非线性分布不动点方程在统计物理学的活跃研究领域中占有突出地位。我的项目的目的是研究非线性分布不动点方程，从具体的例子开始。相关问题涉及解的存在性和唯一性及其性质，如光滑性或尾部行为。这些结果将有助于简化自旋模型中关键参数的计算，并将为理想气体的平衡分布提供新的见解。

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.