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Non-backtracking loop soups and statistical mechanics on spin networks. (English) Zbl 1366.82020
The authors define and study a Markov field of which the free energy density can be computed exactly in any dimension, which turns to be a problem related to a random-loop model referred to as loop soup. The problem is tackled by introducing a new ingredient which is a non-backtracking condition on the open loops. The striking result of the paper is that the distribution of the random network with prescribed boundary conditions is defined by a Gibbs distribution with a local Hamiltonian. The ansatz of the paper is to express the partition function of the model in terms of determinants of two different matrices which, for some values of the edge weights, involve transition matrices of some Markov processes.

MSC:
 82B30 Statistical thermodynamics 82B31 Stochastic methods applied to problems in equilibrium statistical mechanics 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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