The Expected Order of Saturated RNA Secondary Structures

TL;DR

This paper analyzes the expected order of saturated RNA secondary structures, showing it is logarithmic in size and sharply concentrated, indicating traditional models accurately reflect their overall shape.

Contribution

It establishes the expected order of saturated RNA secondary structures and demonstrates their behavior aligns with traditional models regarding overall shape.

Findings

Expected order is approximately log_4(n)

Order is sharply concentrated around the mean

Traditional models accurately predict the structure's overall shape

Abstract

We show the expected order of RNA saturated secondary structures of size $n$ is $\log_4n(1+O(\frac{\log_2n}{n}))$, if we select the saturated secondary structure uniformly at random. Furthermore, the order of saturated secondary structures is sharply concentrated around its mean. As a consequence saturated structures and structures in the traditional model behave the same with respect to the expected order. Thus we may conclude that the traditional model has already drawn the right picture and conclusions inferred from it with respect to the order (the overall shape) of a structure remain valid even if enforcing saturation (at least in expectation).