On the norms of the random walks on planar graphs
Annales de l'Institut Fourier, Volume 47 (1997) no. 5, pp. 1463-1490.

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.

On considère la marche aléatoire simple sur les graphes planaires. Pour certaines familles de ces graphes, on donne des bornes supérieures explicites de la norme de l’opérateur de marche aléatoire en terme du nombre minimal des arêtes à chaque sommet. On démontre que pour un grand nombre de graphes planaires le rayon spectral de cette marche aléatoire est plus petit que un.

@article{AIF_1997__47_5_1463_0,
     author = {\.Zuk, Andrzej},
     title = {On the norms of the random walks on planar graphs},
     journal = {Annales de l'Institut Fourier},
     pages = {1463--1490},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {5},
     year = {1997},
     doi = {10.5802/aif.1606},
     mrnumber = {99g:60127},
     zbl = {0897.60079},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1606/}
}
TY  - JOUR
AU  - Żuk, Andrzej
TI  - On the norms of the random walks on planar graphs
JO  - Annales de l'Institut Fourier
PY  - 1997
SP  - 1463
EP  - 1490
VL  - 47
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1606/
DO  - 10.5802/aif.1606
LA  - en
ID  - AIF_1997__47_5_1463_0
ER  - 
%0 Journal Article
%A Żuk, Andrzej
%T On the norms of the random walks on planar graphs
%J Annales de l'Institut Fourier
%D 1997
%P 1463-1490
%V 47
%N 5
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1606/
%R 10.5802/aif.1606
%G en
%F AIF_1997__47_5_1463_0
Żuk, Andrzej. On the norms of the random walks on planar graphs. Annales de l'Institut Fourier, Volume 47 (1997) no. 5, pp. 1463-1490. doi : 10.5802/aif.1606. http://www.numdam.org/articles/10.5802/aif.1606/

[1] A. Ancona, Positive harmonic functions and hyperbolicity. Potential theory, surveys and problems, Lecture Notes in Math., 1344, ed. J. Král et al., Springer, Berlin, 1988, 1-23. | MR | Zbl

[2] L. Bartholdi, S. Cantat, T. Ceccherini Silberstein, P. De La Harpe, Estimates for simple random walks on fundamental groups of surfaces, Coll. Math., 72, n° 1 (1997), 173-193. | MR | Zbl

[3] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, Ganning (ed.) Princeton Univ. Press., 1970, 195-199. | MR | Zbl

[4] P.A. Cherix, A. Valette, On spectra of simple random walks on one-relator groups, Pacific J. Math. (to appear). | Zbl

[5] Y. Colin De Verdière, Spectres de graphes, cours de DEA, Grenoble, 1995.

[6] J. Dodziuk, Difference Equations, Isoperimetric Inequality and Transience of Certain Random Walks, Trans. Amer. Math. Soc., 284, n° 2 (1984), 787-794. | Zbl

[7] V. Kaimanovich, Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators, Analysis, 1 (1992), 61-82. | Zbl

[8] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc., 92 (1959), 336-354. | MR | Zbl

[9] P. Papasoglu, Strongly geodesically automatic groups are hyperbolic, Invent. Math., 121 (1995), 323-334. | MR | Zbl

[10] P.M. Soardi, Recurrence and transience of the edge graph of a tiling of the Euclidean plane, Math. Ann., 287 (1990), 613-626. | MR | Zbl

[11] R.S. Strichartz, Analysis of the Laplacian on the Complete Riemannian Manifold, Journal of Functional Analysis, 52 (1983), 48-79. | MR | Zbl

[12] W. Woess, Random walks on infinite graphs and groups — a survey of selected topics, Bull. London Math. Soc., 26 (1994), 1-60. | MR | Zbl

[13] W. Woess, A note on tilings and strong isoperimetric inequality, preprint, 1996.

[14] A. Żuk, A remark on the norms of a random walk on surface groups, Coll. Math., 72, n° 1 (1997), 195-206. | MR | Zbl

[15] A. Żuk, A generalized Følner condition and the norms of random walks operators on groups, preprint, 1996. | Zbl

Cited by Sources: