a harmonic function on a graph which is bounded below and nonconstant
There exists no harmonic function on all of the -dimensional grid which is bounded below and nonconstant. This categorises a particular property of the grid; below we see that other graphs can admit such harmonic functions.
Let be a 3-regular tree. Assign “levels” to the vertices of as follows: Fix a vertex , and let be a branch of (an infinite simple path) from . For every vertex of there exists a unique shortest path from to a vertex of ; let be the length of this path.
Now define . Without loss of generality, note that the three neighbours of satisfy (“ is the parent of ”), (“ are the siblings of ”). And indeed, .
So is a positive nonconstant harmonic function on .
|Title||a harmonic function on a graph which is bounded below and nonconstant|
|Date of creation||2013-03-22 12:44:26|
|Last modified on||2013-03-22 12:44:26|
|Last modified by||drini (3)|