#
On measures of average degree for lattices

#### by

## Sven Erick Alm

#### Uppsala University

##### U.U.D.M. Report 2003:6 ISSN 1101-3591

pdf

##### In *Combinatorics, Probability and Computing* (2006) **6**, 477-488.

### Abstract

The usual definition of average degree for a non-regular lattice has
the disadvantage that it takes the same value for many lattices with
clearly different connectivity.
We introduce an alternative definition of average degree, which better
separates different lattices.

These measures are compared on a class of lattices and are analyzed
using a Markov chain describing a random walk on the lattice.
Using the new measure, we conjecture the order of both the critical
probabilities for bond percolation and the connective constants for
self-avoiding walks on these lattices.

2006-07-31, Sven Erick Alm, sea@math.uu.se