Endogeny for the Logistic Recursive Distributional Equation

Abstract

In this article we prove the bivariate uniqueness property for a particular "max-type" recursive distributional equation (RDE). Using the general theory developed in [5] we then show that the corresponding recursive tree process (RTP) has no external randomness, more preciously, the RTP is endogenous. The RDE we consider is so called the Logistic RDE, which appears in the proof of the ζ(2)-limit of the random assignment problem [4] using the local weak convergence method. Thus this work provides a non-trivial application of the general theory developed in [5].

Random tree recursions: Which fixed points correspond to tangible sets of trees?

“…The coloring of the vertices of the random state tree is reminiscent of an interpretation, which also yields a coloring of the tree via (4). But note that for a given fixed point of Ψ, the random state tree coloring always exists, and it is a random coloring (on top of the randomness of the tree).…”

Replica symmetry of the minimum matching

“…However, the proofs in [20], [28], [41], [42] are very different from the approach in the physics literature and do not seem to generalize to d = 1. The original proof by Aldous [2] is the one that comes closest to justifying the replica symmetric ansatz (particularly in view of additional results in [6], [37]), but it seems to rely on finding a solution to (7).…”

“…We believe that in principle the method applies also when 0 < d < 1, but we have run into some difficulties that have so far prevented us from establishing (6) in that case.…”

A recursive distribution equation for the stable tree

“…and distributed as T , g is a measurable mapping, and ξ is independent of (T i , i ≥ 0). RDEs are pertinent in various contexts with recursive structures, including Galton-Watson branching processes [9], Poisson weighted infinite trees [10], and Quicksort algorithms [51].…”