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Loop-erased walks and total positivity
Author(s):
Sergey
Fomin
Journal:
Trans. Amer. Math. Soc.
353
(2001),
3563-3583.
MSC (2000):
Primary 15A48;
Secondary 05C50, 31C20, 60J65
Posted:
April 9, 2001
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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:
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
Posted:
April 9, 2001
Additional Notes:
Supported in part by NSF grant \#DMS-9700927
Copyright of article:
Copyright
2001,
by Sergey Fomin
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