## Experimental Mathematics

- Experiment. Math.
- Volume 15, Issue 3 (2006), 367-381.

### Combinatorial Random Walks on 3-Manifolds

#### Abstract

We define a combinatorial, discrete-time random walk on a closed, triangulated 3-manifold. As one varies the triangulation, keeping the number of tetrahedra fixed, the maximal mean commute time of the random walk becomes a random variable on a finite, uniform probability space of triangulations. Using computer experiments, we obtain empirical density functions for these random variables. The densities are then applied in developing Bayes-type heuristics that allow a walking entity, moving randomly in an unknown 3-manifold, to obtain probabilistic information about which manifold it might be moving in. Mean commute times are calculated via the effective electrical resistance of certain quartic graphs associated with the random walk. As a by-product, we define a topological invariant, the electrical resistance, of a 3-manifold, which we interpret as a refined complexity measure with values in the rational numbers.

#### Article information

**Source**

Experiment. Math., Volume 15, Issue 3 (2006), 367-381.

**Dates**

First available in Project Euclid: 5 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1175789765

**Mathematical Reviews number (MathSciNet)**

MR2264473

**Zentralblatt MATH identifier**

1112.60004

**Subjects**

Primary: 60B99: None of the above, but in this section

Secondary: 60G50: Sums of independent random variables; random walks 57N10: Topology of general 3-manifolds [See also 57Mxx]

**Keywords**

Statistical topology 3-manifolds random walks electrical resistance

#### Citation

Banagl, Markus. Combinatorial Random Walks on 3-Manifolds. Experiment. Math. 15 (2006), no. 3, 367--381. https://projecteuclid.org/euclid.em/1175789765