Cut-off method for endogeny of recursive tree processes

TL;DR

This paper introduces a new method to determine endogeny in recursive tree processes, demonstrating its effectiveness on various models including hierarchical graphs and mean-field optimization problems.

Contribution

The paper presents a novel approach for proving endogeny in recursive tree processes, applicable to diverse models such as multiplicative cascades and mean-field optimization problems.

Findings

Established endogeny for random metrics on hierarchical graphs

Proved endogeny for mean-field matching and TSP problems

Applicable to various recursive distributional processes

Abstract

Given a solution to a recursive distributional equation, a natural (and non-trivial) question is whether the corresponding recursive tree process is endogenous. That is, whether the random environment almost surely defines the tree process. We propose a new method of proving endogeny, which applies to various processes. As explicit examples, we establish endogeny of the random metrics on non-pivotal hierarchical graphs defined by multiplicative cascades and of mean-field optimization problems as the mean-field matching and travelling salesman problems in pseudo-dimension q>1.