| In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to mean-field statistical physics models), the central requirement is to solve a fixed point equation of the form X g ((ξi, X i), i ≥ 1), where (ξi) i≥1 and g (·) are given and (Xi)i≥1 are independent copies of X with unknown distribution. We call such an equation a recursive distributional equation. Exploiting the natural recursive structure one can associate a tree-indexed process with every solution, and such a process is called a recursive tree process. This process in some sense is a solution of an infinite system of recursive distributional equations.; The dissertation is devoted to the study of such fixed point equations and the associated recursive tree process when the given function g (·) is essentially a “maximum” or a “minimum” function. Such equations arise in the context of optimization problems and branching random walks. The present work mainly concentrates on the theoretical question of endogeny: the tree-indexed process being measurable with respect to the given i.i.d innovations (ξ i). We outline some basic general theory which is natural from a statistical physics point of view and then specialize to study some concrete examples arising from various different contexts. |
