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Random interlacements and the Gaussian free field. (English) Zbl 1261.60095
From the Introduction: The author considers continuous time random interlacements on \(\mathbb Z^d,\;d\geq 3\), where each doubly infinite trajectory modulo time-shift in the interlacement is decorated by i.i.d. exponential variables with parameter one which specify the time spent by the trajectory at each step. The author is interested in the random field of occupation times, i.e., the total time spent at each site of \(\mathbb Z^d\) by the collection of trajectories with label at most \(u\) in the interlacement point process. When \(d = 3\), the author relates this stationary random field to the two-dimensional Gaussian free field pinned at the origin, by looking at the properly scaled field of differences of occupation times of long rods of size \(N\), when the level \(u\) is either proportional to \(\log N/N\) or much larger than \(\log N/N\). The choice of \(u\) proportional to \(\log N/N\) corresponds to a non-degenerate probability that the interlacement at level \(u\) meets a given rod. In the asymptotic regime, ths brings into play an independent proportionality factor of the Gaussian free field, which is distributed as a certain time-marginal of a zero-dimensional Bessel process. This random factor disappears from the description of the limiting random field, when instead \(uN/\log N\) tends to infinity. For arbitrary \(d\geq 3\), the author also relates the field of occupation times at a level tending to infinity, to the \(d\)-dimensional Gaussian free field.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J27 Continuous-time Markov processes on discrete state spaces
60F05 Central limit and other weak theorems
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