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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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