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Loop Erasure and Spanning Trees

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Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 2026)

Abstract

Recall that an oriented link g is a pair of points \((g^{-},g^{+})\,{\rm {such that}} \,C_{g}=C_{g^{-},g^{+}}\neq0\,. {\rm{Define}} \,-g=(g^{+},g^{-})\,.\,{\rm{Let} \,\mu_{\neq}^{x,y}\,{\rm{ be\, the\, measure\, induced\, by\, C\, on\, discrete\, self-avoiding\, paths\, between\, x\, and\, y\,}}: \,\mu_{\neq}^{x,y}(x,x_{2},...,x_{n-1}\,,y)=C_{x,x_{2}}C_{x_{1},x_{3}}... C_{x_{n-1},y}}.\)

Another way to define a measure on discrete self avoiding paths from x to y from a measure on paths from x to y is loop erasure defined in Sect. 3.1 (see also [16, 17, 39] and [31]). In this context, the loops, which can be reduced to points, include holding times, and loop erasure produces a discrete path without holding times.

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Correspondence to Yves Le Jan .

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© 2011 Springer-Verlag Berlin Heidelberg

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Jan, Y.L. (2011). Loop Erasure and Spanning Trees. In: Markov Paths, Loops and Fields. Lecture Notes in Mathematics(), vol 2026. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21216-1_8

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