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CRAMER'S RULE AND LOOP ENSEMBLES A. ABDESSELAM AND D. C. BRYDGES
 

Summary: CRAMER'S RULE AND LOOP ENSEMBLES
A. ABDESSELAM AND D. C. BRYDGES
Abstract. We review a 1986 result of G.X Viennot that expresses
a ratio of generating functions for disjoint oriented loops in a finite
graph in terms of the generating function of a single path in the
graph weighted according to loops in the path, defined by loop
erasure. The result is a generalisation of Cramer's formula for
the inverse of a matrix. We show that it arises from the Mayer
expansion.
1. Introduction
The result reviewed here was noted by G.X. Viennot as an immediate
corollary of his theory of heaps and pieces in [Vie86, Proposition 6.3],
but perhaps not many people in statistical mechanics have realised
that his result is an interesting statement about correlations for loop
ensembles such as one encounters in the contour expansion of the Ising
model.
It is also a generalisation of Cramer's formula for the inverse of a
matrix; indeed we rediscovered it in this context by the methods of
statistical mechanics, notably the Mayer expansion. In conversations
with combinatorialists we have since learned of another proof based on

  

Source: Abdesselam, Abdelmalek - Department of Mathematics, University of Virginia
Brydges, David - Department of Mathematics, University of British Columbia

 

Collections: Mathematics