Let $G$ be a finite, connected graph. Let $T$ be a uniform spanning tree, and let $e$ be a uniformly random edge not in $T$. When we add $e$ to $T$, we get a subgraph with a unique cycle, $C$. I am interested in the distribution of the length of the cycle $C$, i.e. the random variable $|C|$.

Section 5 of https://arxiv.org/pdf/1106.2226.pdf describes a lower bound on the probability that this cycle is at most $g$, where $g$ is a number such that every edge in $G$ lies on a cycle of length at most $g$.

~~ (I also don't understand their claim that $$P( \text{the cycle in the $UST^+$ has length} \leq g) \geq \frac{1}{d(d-1)^{g - 2}},$$ and I would appreciate it if someone could give a hint on how to derive that from the correctness of Wilson's algorithm.)~~ Resolved (I think) by Ben Barbers comment.

Are more precise statistics of this distribution known?

It seems that the distribution of the length of a LERW between two adjacent vertices of $G$ is very closely related to this problem. I would appreciate any information about that.

Some updates:

From this post Perimeters of random-walk polygons I learned about this paper https://journals.aps.org/pre/abstract/10.1103/PhysRevE.55.R2093 , which mentions that $|C|$ is expected to follow a power law. This is consistent with the experiments I conducted.

They relate the power law to certain scaling limit computations.

This paper argues (through what seems to be exact computation for small cases and then by computer simulation) about the distribution of the cycle in the square lattice: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.46.R4471

They also say that the burning time distribution can be used to easily compute the distribution of loops.

I don't follow most of what they are saying, but I'll study their methods.