Some probabilistic trees with algebraic roots

TL;DR

This paper studies various probabilistic processes generating random graphs, revealing that the probability of these graphs being trees has a surprisingly simple, parameter-independent formula, and discusses open questions arising from these findings.

Contribution

It introduces new probabilistic processes for generating random graphs and derives simple, parameter-independent formulas for the probability of these graphs being trees.

Findings

Probability of the graph being a tree is given by a simple expression.

The probability formula is independent of most parameters.

Raises open questions about the underlying processes and their properties.

Abstract

In this article we consider several probabilistic processes defining random grapha. One of these processes appeared recently in connection with a factorization problem in the symmetric group. For each of the probabilistic processes, we prove that the probability for the random graph to be a tree has an extremely simple expression, which is independent of most parameters of the problem. This raises many open questions.