Loop-erased walks and total positivity

By Sergey Fomin. 2000

88 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/00c51e83c8cd1079d4ac868a6c52a23503cd3c37

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

Total positivity, Grassmannians, and networks (2006, methodology): “k,2k such that [k],[k + 1,2k] ∈ M. Such cells are exactly the double Bruhat cells of Fomin -Zelevinsky, see Remark 3.11.”
The configurational measure on mutually avoiding SLE paths (2006, background): “Fomin [4] showed that the ν(A; x1, y1) × · · · × ν(A; xn, yn) measure of the set of n-tuples of excursions (ω(1), .”
Euler Integrals for Commuting SLEs (2005, background): “Then it is easy to see that solutions (functions annihilated by those operators) are polynomials in x1, . . . x2n homogeneous of degree n, of partial degree at most 1 in each variable.”
Commutation relations for Schramm‐Loewner evolutions (2004, methodology): “As pointed out in [16], duality follows from these convergence and Wilson’s algorithm, that gives an exact relation between UST and LERW at the discrete level ([25]).”
Random planar curves and Schramm-Loewner evolutions (2003, background): “See also, Fomin’s paper [50] for another approach to some of these exponents.”
A Formula for Plücker Coordinates Associated with a Planar Network (2008, background): “Fomin [1] whose setup differed from Postnikov’s in that the analogues of boundary measurements did not involve any signs.”