On the norms of the random walks on planar graphs

Andrzej Żuk

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 5, page 1463-1490
  • ISSN: 0373-0956

Abstract

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We consider the nearest neighbor random walk on planar graphs. For certain families of these graphs, we give explicit upper bounds on the norm of the random walk operator in terms of the minimal number of edges at each vertex. We show that for a wide range of planar graphs the spectral radius of the random walk is less than one.

How to cite

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Żuk, Andrzej. "On the norms of the random walks on planar graphs." Annales de l'institut Fourier 47.5 (1997): 1463-1490. <http://eudml.org/doc/75270>.

@article{Żuk1997,
abstract = {We consider the nearest neighbor random walk on planar graphs. For certain families of these graphs, we give explicit upper bounds on the norm of the random walk operator in terms of the minimal number of edges at each vertex. We show that for a wide range of planar graphs the spectral radius of the random walk is less than one.},
author = {Żuk, Andrzej},
journal = {Annales de l'institut Fourier},
keywords = {random walks; norm of the operator; planar graphs; upper bounds},
language = {eng},
number = {5},
pages = {1463-1490},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the norms of the random walks on planar graphs},
url = {http://eudml.org/doc/75270},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Żuk, Andrzej
TI - On the norms of the random walks on planar graphs
JO - Annales de l'institut Fourier
PY - 1997
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 5
SP - 1463
EP - 1490
AB - We consider the nearest neighbor random walk on planar graphs. For certain families of these graphs, we give explicit upper bounds on the norm of the random walk operator in terms of the minimal number of edges at each vertex. We show that for a wide range of planar graphs the spectral radius of the random walk is less than one.
LA - eng
KW - random walks; norm of the operator; planar graphs; upper bounds
UR - http://eudml.org/doc/75270
ER -

References

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  11. [11] R.S. STRICHARTZ, Analysis of the Laplacian on the Complete Riemannian Manifold, Journal of Functional Analysis, 52 (1983), 48-79. Zbl0515.58037MR84m:58138
  12. [12] W. WOESS, Random walks on infinite graphs and groups — a survey of selected topics, Bull. London Math. Soc., 26 (1994), 1-60. Zbl0830.60061MR94i:60081
  13. [13] W. WOESS, A note on tilings and strong isoperimetric inequality, preprint, 1996. 
  14. [14] A. ŻUK, A remark on the norms of a random walk on surface groups, Coll. Math., 72, n° 1 (1997), 195-206. Zbl0872.60052MR98d:60133b
  15. [15] A. ŻUK, A generalized Følner condition and the norms of random walks operators on groups, preprint, 1996. Zbl0990.43001

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