Loop-erased walks and total positivity

2000/04/12 by Sergey Fomin, Fomin, Sergey · 1 citation
Mathematics · #15A48 (Primary) 05C50 #31C20 #60J65 (Secondary} #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Rings and Algebras (math.RA) #math.CO #math.PR #math.RA #msc:05C50 #msc:15A48 #msc:31C20 #msc:60J65

paper · pdf · doi:10.48550/arxiv.math/0004083

20 pages; 7 figures

arxiv created 2000/04/12 · openalex publication_date 2000/04/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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• Uniform spanning trees and random matrix statistics

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