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


Work by [Tamura](https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-33/issue-3/The-average-edge-order-of-triangulations-of-3-manifolds/ojm/1200787102.full) (extending results by [Luo and Stong](https://www.ams.org/journals/tran/1993-337-02/S0002-9947-1993-1134759-6/S0002-9947-1993-1134759-6.pdf)) 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 *any* closed 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.



What is known about the distribution of average edge-degrees for 3-manifold triangulations (with the number of 3-simplices less than a fixed constant)
Below is the data from Ben Burton's census of triangulations of $S^3$ with up to 9 tetrahedra. An Euler characteristic argument shows that the average edge degree is equal to $6\left(\frac{T}{V+T}\right)$, where $T$ is the number of tetrahedra and $V$ is the number of vertices, so the numbers below are all of that form. The data are given in (average edge degree, frequency) pairs. As Bruno guessed, it looks like the distribution weights heavily towards 6.


1 tetrahedron: 


* (2.0000000000000000, 1)
* (3.0000000000000000, 1)


2 tetrahedra:


* (2.0000000000000000, 1)
* (2.3999999999999999, 1)
* (3.0000000000000000, 1)
* (4.0000000000000000, 3)


3 tetrahedra:


* (2.5714285714285716, 1)
* (3.0000000000000000, 2)
* (3.6000000000000001, 9)
* (4.5000000000000000, 20)


4 tetrahedra:


* (2.6666666666666665, 2)
* (3.0000000000000000, 4)
* (3.4285714285714284, 16)
* (4.0000000000000000, 48)
* (4.7999999999999998, 128)


5 tetrahedra:


* (2.7272727272727271, 1)
* (3.0000000000000000, 4)
* (3.3333333333333335, 23)
* (3.7500000000000000, 110)
* (4.2857142857142856, 468)
* (5.0000000000000000, 1297)


6 tetrahedra:


* (2.7692307692307692, 1)
* (3.0000000000000000, 5)
* (3.2727272727272729, 36)
* (3.6000000000000001, 199)
* (4.0000000000000000, 1103)
* (4.5000000000000000, 4931)
* (5.1428571428571432, 13660)


7 tetrahedra:


* (2.7999999999999998, 1)
* (3.0000000000000000, 3)
* (3.2307692307692308, 39)
* (3.5000000000000000, 301)
* (3.8181818181818183, 2186)
* (4.2000000000000002, 13380)
* (4.6666666666666667, 62657)
* (5.2500000000000000, 169077)


8 tetrahedra:


* (2.8235294117647061, 1)
* (3.0000000000000000, 3)
* (3.2000000000000002, 51)
* (3.4285714285714284, 446)
* (3.6923076923076925, 3870)
* (4.0000000000000000, 28826)
* (4.3636363636363633, 180128)
* (4.7999999999999998, 829753)
* (5.3333333333333333, 2142197)


9 tetrahedra:


* (2.8421052631578947, 1)
* (3.0000000000000000, 3)
* (3.1764705882352939, 50)
* (3.3750000000000000, 567)
* (3.6000000000000001, 6046)
* (3.8571428571428572, 54876)
* (4.1538461538461542, 422860)
* (4.5000000000000000, 2612407)
* (4.9090909090909092, 11673471)
* (5.4000000000000004, 28691150)


