Loop-erased walks and total positivity (Q2716150)
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scientific article; zbMATH DE number 1602206
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Loop-erased walks and total positivity |
scientific article; zbMATH DE number 1602206 |
Statements
Loop-erased walks and total positivity (English)
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6 June 2001
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total positivity
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loop-erased walk
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hitting probability
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resistor network
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nonnegative matrices
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planar directed weighted graphs
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Brownian motion
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acyclic directed networks
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Markov chains
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0.8992254
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0.8734168
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0.8644186
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0.86064476
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The author considers matrices whose elements enumerate weights of walks in planar directed weighted graphs. These matrices are totally nonnegative. A combinatorial explanation of this phenomenon involves loop-erased walks. Applications include total positivity of hitting matrices of Brownian motion in planar domains.NEWLINENEWLINENEWLINEThe paper is organized as follows. Sections 2-5 are devoted to preliminaries of various kinds. Section 2 introduces walk matrices and hitting matrices of directed networks. Section 3 reviews classical results by Karlin-McGregor and Lindström on total positivity of walk matrices of acyclic directed networks or associated Markov chains. Section 4 gives an account of some of the results obtained on resistor networks and their Dirichlet-to-Neumann maps. Section 5 introduces loop-erased walks. The main results are presented in Sections 6-7.
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