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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Loop-erased walks and total positivity


Author: Sergey Fomin
Journal: Trans. Amer. Math. Soc. 353 (2001), 3563-3583
MSC (2000): Primary 15A48; Secondary 05C50, 31C20, 60J65
Published electronically: April 9, 2001
MathSciNet review: 1837248
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Abstract: We consider matrices whose elements enumerate weights of walks in planar directed weighted graphs (not necessarily acyclic). These matrices are totally nonnegative; more precisely, all their minors are formal power series in edge weights with nonnegative coefficients. A combinatorial explanation of this phenomenon involves loop-erased walks. Applications include total positivity of hitting matrices of Brownian motion in planar domains.


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Additional Information

Sergey Fomin
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: fomin@math.lsa.umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02824-0
PII: S 0002-9947(01)02824-0
Keywords: Total positivity, loop-erased walk, hitting probability, resistor network
Received by editor(s): July 27, 2000
Received by editor(s) in revised form: January 19, 2001
Published electronically: April 9, 2001
Additional Notes: Supported in part by NSF grant #DMS-9700927
Article copyright: © Copyright 2001 by Sergey Fomin