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On the spatial Markov property of soups of unoriented and oriented loops
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We describe simple properties of some soups of unoriented Markov loops and of some soups of oriented Markov loops that can be interpreted as a spatial Markov property of these loop-soups. This property of the latter soup is related to well-known features of the uniform spanning trees (such as Wilson's algorithm) while the Markov property of the former soup is related to the Gaussian Free Field and to identities used in the foundational papers of Symanzik, Nelson, and of Brydges, Fr\"ohlich and Spencer or Dynkin, or more recently by Le Jan.
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Loop clusters on complete graphs
In the loop soup on the complete graph K_n, the number of clusters of size d converges to a mixed Poisson law with mixing variable exp(-dZ/κ), and clusters larger than n^{1-ε} appear almost surely.
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