According to [this paper](https://www.math.uwaterloo.ca/~nwormald/papers/girth.pdf), the probability that a random $d$-regular graph of order $n$ has no cycles of length $c\_1,c\_2,\ldots,c\_t$ is $$P=\exp\left(-\sum\_{i=1}^t\mu\_i+o(1)\right)$$ as $n\rightarrow\infty$, where $\mu\_i=\frac{(d-1)^{c\_i}}{2c\_i}$. So it seems that as $d$ gets larger, $\mu\_i$ also gets larger and thus $P$ gets smaller. But this looks very counterintuitive, since as $d$ gets larger there are more edges in the graph, so the probability of having a cycle of any fixed length should increase. What am I missing?



Probability of having no cycles of fixed length in $d$-regular graphs
I can't comment, but $P$ is the probability that there are NO cycles of those lengths. So this probability goes down. Your intuition holds.


