**Contemporary Mathematics**

Volume: 206;
1997;
181 pp;
Softcover

MSC: Primary 31;
Secondary 43; 60

Print ISBN: 978-0-8218-0605-0

Product Code: CONM/206

List Price: $46.00

AMS Member Price: $36.80

MAA Member Price: $41.40

**Electronic ISBN: 978-0-8218-7797-5
Product Code: CONM/206.E**

List Price: $46.00

AMS Member Price: $36.80

MAA Member Price: $41.40

# Harmonic Functions on Trees and Buildings

Share this page *Edited by *
*Adam Korányi*

This volume presents the proceedings of the workshop
“Harmonic Functions on Graphs” held at the Graduate Center of CUNY in
the fall of 1995. The main papers present material from four
minicourses given by leading experts: D. Cartwright,
A. Figà-Talamanca, S. Sawyer and T. Steger. These minicourses are
introductions which gradually progress to deeper and less known
branches of the subject. One of the topics treated is buildings,
which are discrete analogues of symmetric spaces of arbitrary rank;
buildings of rank are trees. Harmonic analysis on buildings is a
fairly new and important field of research. One of the minicourses
discusses buildings from the combinatorial perspective and another
examines them from the \(p\)-adic perspective. The third minicourse
deals with the connections of trees with \(p\)-adic analysis. And the
fourth deals with random walks, i.e., with the probabilistic side of
harmonic functions on trees.

The book also contains the extended abstracts of 19 of the 20
lectures given by the participants on their recent results. These
abstracts, well detailed and clearly understandable, give a good
cross-section of the present state of research in the
field.

#### Readership

Graduate students and research mathematicians interested in potential theory.

# Table of Contents

## Harmonic Functions on Trees and Buildings

- Contents vii8 free
- Preface ix10 free
- Part I. Minicourses. 112 free
- Part II. Abstracts of Lectures. 109120
- Heat kernel for regular trees 111122
- Tensor product of spherical representations of the group of automorphisms of a homogeneous tree 113124
- The horocyclic Radon transform on trees 117128
- Eigenfunctions of the Laplacian on a homogeneous tree 121132
- Exotic convergence in theorems of Fatou type 125136
- A spectral gap property for transfer operators 129140
- Harmonic functions on discrete subgroups of semi-simple Lie groups 133144
- Biased random walks and harmonic functions on the Lamplighter group 137148
- Characterization of the eigenfunctions of the Laplace operators for an affine building of rank 2 141152
- Free product of representations 145156
- The Jacobian of a finite graph 149160
- Flows and harmonic functions on graphs 153164
- Applications of diffusion processes on trees to mathematical finance 157168
- Characterizing harmonic functions by mean value properties on trees and symmetric spaces 161172
- Factors from buildings 165176
- Harnack and Liouville properties on graphs 169180
- The spectrum of a directed Cayley graph of a free group 173184
- Factorization of the Green's kernel for non-nearest neighbor random walks 175186
- Harmonic functions for group-invariant random walks 179190