This is my first question on mathoverflow! It relates to a project I'm undertaking with a student.

Work by Tamura (extending results by Luo and Stong) shows that for any closed 3-manifold $M$ and any rational number $4.5 < r < 6$ there is a triangulation $T$ of $M$ for which that the average edge-degree $\mu(T)$ is $r$. Here the *degree* of an edge $e$ is the number of 3-simplices having $e$ as a face.

Now, fix a closed 3-manifold $M$ and consider the (necessarily finite) set of all triangulation of $M$ containing at most $K$ 3-simplices. Call this set $\mathcal{T}_K(M)$.

QUESTION: Does anyone know of results concerning the "distribution" of the average edge-degree $\mu(T)$ for $T\in \mathcal{T}_K(M)$? That is, what is known about the fraction of triangulations from $\mathcal{T}_K(M)$ for which the edge-degree lies in a given small interval $[r - \epsilon, r+\epsilon] \subset (4.5,6)$? Results concerning

anyclosed 3-manifold $M$ would be fine.

I suspect this is extremely difficult to answer in a precise way. However, I'd be very interested in knowing any asymptotic ($K\rightarrow \infty$) or approximate results as well. Thanks for your help!

NOTE: For the project we plan to use the Metropolis algorithm to sample from $\mathcal{T}_K(M)$ where $M$ is the 3-torus. Using these samples we hope to empirically estimate the distribution in question.