# a harmonic function on a graph which is bounded below and nonconstant

There exists no harmonic function^{} on all of the $d$-dimensional grid ${\mathbb{Z}}^{d}$ 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 ${\mathcal{T}}_{3}=({V}_{3},{E}_{3})$ be a 3-regular tree. Assign “levels” to the vertices of ${\mathcal{T}}_{3}$ as follows: Fix a vertex $o\in {V}_{3}$, and let $\pi $ be a branch of ${\mathcal{T}}_{3}$ (an infinite simple path) from $o$. For every vertex $v\in {V}_{3}$ of ${\mathcal{T}}_{3}$ there exists a *unique* shortest path from $v$ to a vertex of $\pi $; let $\mathrm{\ell}(v)=\left|\pi \right|$ be the length of this path.

Now define $f(v)={2}^{-\mathrm{\ell}(v)}>0$. Without loss of generality, note that the three neighbours ${u}_{1},{u}_{2},{u}_{3}$ of $v$ satisfy $\mathrm{\ell}({u}_{1})=\mathrm{\ell}(v)-1$ (“${u}_{1}$ is the parent of $v$”), $\mathrm{\ell}({u}_{2})=\mathrm{\ell}({u}_{3})=\mathrm{\ell}(v)+1$ (“${u}_{2},{u}_{3}$ are the siblings of $v$”). And indeed, $\frac{1}{3}\left({2}^{\mathrm{\ell}(v)-1}+{2}^{\mathrm{\ell}(v)+1}+{2}^{\mathrm{\ell}(v)+1}\right)={2}^{\mathrm{\ell}(v)}$.

So $f$ is a positive nonconstant harmonic function on ${\mathcal{T}}_{3}$.

Title | a harmonic function on a graph which is bounded below and nonconstant |
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Canonical name | AHarmonicFunctionOnAGraphWhichIsBoundedBelowAndNonconstant |

Date of creation | 2013-03-22 12:44:26 |

Last modified on | 2013-03-22 12:44:26 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 6 |

Author | drini (3) |

Entry type | Example |

Classification | msc 30F15 |

Classification | msc 31C05 |

Classification | msc 31B05 |

Classification | msc 31A05 |