On the norms of the random walks on planar graphs
type: Item
instance of: scholarly article
title: On the norms of the random walks on planar graphs
zbMATH Open document ID: 0897.60079
author: Andrzej Zuk
publication date: 1997-12-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75270
review text: Let us consider a connected planar graph \(X\) such that the degree of each vertex, i.e. the number of edges adjacent to the vertex, is finite and such that there are a finite number of vertices in any compact subset of the plane. Let us also suppose that there are no loops or multiple edges. On this graph we consider a random walk that goes from a vertex to one of its neighbors picked uniformly at random. We associate with this random walk a random walk operator \(M\), \(Mf(q)= {1\over N(q)} \sum_{p\sim q} f(p)\) for \(f\in \ell^2 (X,N)\), where \(N(q)\) is the degree of vertex \(q\) and where \(p\sim q\) means that \(\{q,p\}\) is an edge, i.e. \(p\) and \(q\) are neighbors. The operator \(M\) is self-adjoint on the space \(\ell^2 (X,N)\). We establish some upper bounds for \(\| M\|\), the norm of this operator, acting on \(\ell^2 (X,N)\). If \(P^n (q,p)\) is the probability of going from \(q\) to \(p\) in \(n\) steps, then we know [see \textit{W. Woess}, Bull. Lond. Math. Soc. 26, No. 1, 1-60 (1994; Zbl 0830.60061)] that \(\| M\|= \lim_{n\to \infty} (P^{2n} (q,q))^{1/2n}\).
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 05B45
zbMATH DE Number: 1092701
zbMATH Keywords: random walks
zbMATH Keywords: norm of the operator
zbMATH Keywords: planar graphs
zbMATH Keywords: upper bounds
MaRDI profile type: Publication
OpenAlex ID: W2320383199
cites work: Q3832832
cites work: Estimates for simple random walks on fundamental groups of surfaces
cites work: Q5614192
cites work: On spectra of simple random walks on one-relator groups. With an appendix by Paul Jolissaint
cites work: Difference Equations, Isoperimetric Inequality and Transience of Certain Random Walks
cites work: Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators.
cites work: Symmetric Random Walks on Groups
cites work: Strongly geodesically automatic groups are hyperbolic
cites work: Recurrence and transience of the edge graph of a tiling of the Euclidean plane
cites work: Analysis of the Laplacian on a complete Riemannian manifold
cites work: Random Walks on Infinite Graphs and Groups - a Survey on Selected topics
cites work: A remark on the norm of a random walk on surface groups
cites work: A generalized Følner condition and the norms of random walk operators on groups
DOI: 10.5802/AIF.1606
Recommended article: ON BORWEIN’S CONJECTURES FOR PLANAR UNIFORM RANDOM WALKS
Recommended article: Random walks on graphs and approximation of \(L^2\)-invariants
Recommended article: On the trace of random walks on random graphs
Recommended article: RANDOM WALKS ON REGULAR POLYHEDRA AND OTHER DISTANCE–REGULAR GRAPHS
Recommended article: Random walks on graphs with a strong isoperimetric property
Recommended article: Remarks on random walks on graphs and the Floyd boundary
Recommended article: Q5365110
Recommended article: Q3479330
Recommended article: On the diameter of random planar graphs
Recommended article: On the Diameter of Random Planar Graphs
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1092701