Abstract
During the last decade many attempts have been made to characterize absence of spontaneous breaking of continuous symmetry for the Heisenberg model on graphs by using suitable classifications of random walks (refs. 4 and 10). We propose and study a new type problem for random walks on graphs, which is particularly interesting for disordered graphs. We compare this classification with the classical one and with an analogous one introduced in ref. 4. Various examples, that are not space-homogeneous, are provided.
Similar content being viewed by others
References
M. Barlow, T. Coulhon, and A. Grigor'yan, Manifolds and graphs with slow heat kernel decay, Invent. Math. 144:609–649 (2001).
R. Burioni and D. Cassi, Universal properties of spectral dimension, Phys. Rev. Lett. 76:1091–1093 (1996).
R. Burioni, D. Cassi, and C. Destri, Spectral partitions on infinite graphs, J. Phys. A 33:3627, 3636 (2000).
R. Burioni, D. Cassi, and A. Vezzani, The type-problem on the average for random walks on graphs, Eur. Phys. J. B 15:665(2000).
D. Cassi, Local vs. average behavior on inhomogeneous structures: Recurrence on the average and a further extension of MerminüWagner theorem on graphs, Phys. Rev. Lett. 76:2941(1996).
D. Cassi and L. Fabbian, The spherical model on graphs, J. Phys. A 32:L93(1999).
T. Coulhon and A. Grigor'yan, Random walks on graphs with regular volume growth, Geom. Funct. Anal. 8:656–701 (1998).
A. Grigor'yan and A. Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, Math. Ann. 324:521–556 (2002).
A. Grigor'yan and A. Telcs, Sub-Gaussian estimates of heat kernels on infinite graphs, Duke Math. J. 109:451–510 (2002).
F. Merkl and H. Wagner, Recurrent random walks and the absence of continuous symmetry breaking on graphs, J. Stat. Phys. 75:153–165 (1994).
E. Hille, Analytic Function Theory, Vol. I (Chelsea Publ. Co., New York, 1959).
J. F. C. Kingman, The exponential decay of Markov transition probabilities, Proc. London Math. Soc. 13:337–358 (1963).
E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978).
T. Lyons, A simple criterion for transience of a reversible Markov chain, Ann. Probab. 11:393–402 (1983).
G. Pólya, Uber eine Aufgabe der Wahrscheinlichkeitstheorie betreffend die Irrfahrt im Straßennetz, Math. Ann. 84:149–160 (1921).
W. Rudin, Principles of Mathematical Analysis (McGrawüHill, 1953).
A. Telcs, Isoperimetric inequalities for random walks, Potential Anal. 19:237–249 (2003).
W. Woess, Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Mathematics, Vol. 138 (Cambridge University Press, 2000).
M. Yamasaki, Discrete potentials on an infinite network, Mem. Fac. Sci., Shimane Univ. 13:31–44 (1979).
F. Zucca, Mean value properties for harmonic functions on graphs and trees, Ann. Mat. Pura Appl. (4) 181:105–130 (2002).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bertacchi, D., Zucca, F. Classification on the Average of Random Walks. Journal of Statistical Physics 114, 947–975 (2004). https://doi.org/10.1023/B:JOSS.0000012513.55697.65
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000012513.55697.65