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On the spatial Markov property of soups of unoriented and oriented loops. (English) Zbl 1370.60192
Donati-Martin, Catherine (ed.) et al., Séminaire de probabilités XLVIII. Cham: Springer (ISBN 978-3-319-44464-2/pbk; 978-3-319-44465-9/ebook). Lecture Notes in Mathematics 2168. Séminaire de Probabilités, 481-503 (2016).
Let \(\Gamma\) be a directed graph with \(d(x)<\infty\) edges going out of each vertex \(x\in \Gamma\). A rooted \(n\)- steps loop \(l\) from a vertex \(l_0\) is \(l=(l_0,e_1,l_1,\dots,l_{n-1}, e_{n-1},l_n),\) where \(l_n=l_0\) and \(e_i\) is an edge from \(l_{i-1}\) to \(l_i\). Then, \[ p_l= \frac{1}{\prod_{i=0}^{n-1} d(l_i)} \] is the probability that the \(n\)-step trajectory of the random walk on \(\Gamma\) starting from \(l_0\) is exactly the loop \(l\). Next, on the set of all loops \(l\) of length \(n\) the measure \(\rho\) is defined by \(\rho_l=\frac{p_l}{n}.\) In a similar manner, a measure on the set of loops of an undirected graph is defined. The paper focuses on the description and the discussion of the spatial Markov property of the loops, accompanied by an analytical review of the relevant literature.
For the entire collection see [Zbl 1359.60007].

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
60G60 Random fields
60J27 Continuous-time Markov processes on discrete state spaces
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
05C81 Random walks on graphs
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