Abstract
We investigate the relation to random configurations and combinatorial maps of the Eulerian networks defined by Poissonian emsembles of Markov loops.
Highlights
We investigate the relation to random configurations and combinatorial maps of the Eulerian networks defined by Poissonian emsembles of Markov loops
The purpose of this note is to show that random networks, which can be defined as images of Poissonian Markov loop ensembles, are naturally associated to random configurations and combinatorial maps
The relation with configurations, defined as families of entering and exiting half edges attached to each vertex and coupled to form edges, follows from the distribution of the edge occupation field, given in [4]
Summary
The purpose of this note is to show that random networks, which can be defined as images of Poissonian Markov loop ensembles ( known as “loop soups”), are naturally associated to random configurations and combinatorial maps. Given any oriented edge (x, y) of the graph, we denote by Nx,y(l) the total number of jumps made from x to y by the loop l and set Nx(l) = y Nx,y(l). Given any oriented edge (x, y) of the graph, we denote by Nx,y(L) or Nx,y the total number of jumps made from x to y by the loops of L and set again Nx = y Nx,y. We define a network to be a N-valued function defined on oriented edges of the graph It is given by a matrix k with N-valued coefficients which vanishes on the diagonal and on entries (x, y) such that {x, y} is not an edge of the graph. It is obvious that the field N defines a random network which verifies the Eulerian property. See [10] for related Markov properties
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