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Waiting for a Bat to Fly By (in Polynomial Time)

Published online by Cambridge University Press:  31 July 2006

ITAI BENJAMINI
Affiliation:
Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, POB 26, Rehovot 76100, Israel (e-mail: itai.benjamini@weizmann.ac.il)
GADY KOZMA
Affiliation:
Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540, USA (e-mail: gady@ias.edu)
LÁSZLÓ LOVÁSZ
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA (e-mail: lovasz@microsoft.com)
DAN ROMIK
Affiliation:
Department of Statistics, 367 Evans Hall, University of California, Berkeley, CA 94720-3860, USA (e-mail: romik@stat.berkeley.edu)
GÁBOR TARDOS
Affiliation:
Rényi Institute, Hungarian Academy of Sciences, Pf. 127, H-1354 Budapest, Hungary (e-mail: tardos@renyi.hu)
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Abstract

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We observe returns of a simple random walk on a finite graph to a fixed node, and would like to infer properties of the graph, in particular properties of the spectrum of the transition matrix. This is not possible in general, but at least the set of eigenvalues can be recovered under fairly general conditions, e.g., when the graph has a node-transitive automorphism group. The main result is that by observing polynomially many returns, it is possible to estimate the spectral gap of such a graph up to a constant factor.

Type
Paper
Copyright
2006 Cambridge University Press