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From loop clusters and random interlacements to the free field. (English) Zbl 1348.60141
Let \({\mathcal G}=(V,E)\) be a connected undirected graph, with each vertex of a finite degree. It is assumed that to each edge \(e\in E\) a positive conductance \(C(e)\) is prescribed, while the vertices are endowed with a nonnegative killing measure \(k(x), x\in V\). Accordingly, a continuous-time Markovian jump process \(X_t,\;0\leq t<\zeta \), on \({\mathcal G}\) is defined, such that the transition rate from \(x\) to \(y,\) \(x\sim y\), equals \(C(x,y)\) and the transition rate from \(x\in V\) to a cemetery point outside \(V\) is equal to \(k(x).\) The process may blow up at a finite time, so that \(\zeta\) is either \(+\infty\) or the first time \(X_t\) gets killed or blows up. Next, for \(\alpha>0\) a Poisson point process \({\mathcal L}_\alpha\) with intensity \(\alpha\mu\) is defined, where \(\mu\) is a loop measure on the set of loops on \({\mathcal G}\). As a consequence of the coupling scheme constructed for the above model, the author proves that \({\mathcal L}_{1/2}\) does not percolate.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G15 Gaussian processes
60G60 Random fields
60J75 Jump processes (MSC2010)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
05C90 Applications of graph theory
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