Large Deviations for Random Trees

TL;DR

This paper establishes a Large Deviation Principle for the degree distribution in large random trees under Gibbs measures, providing explicit rate functions and implications for RNA secondary structure analysis.

Contribution

It introduces a rigorous LDP framework for vertex degrees in Gibbs-distributed random trees, with explicit rate functions and a Law of Large Numbers, advancing understanding of complex tree structures.

Findings

Explicit LDP rate function for vertex degrees

Law of Large Numbers for degree distribution

Application to RNA secondary structure analysis

Abstract

We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures.