On scaling limits and Brownian interlacements. (English) Zbl 1303.60022

The author investigates the scaling limit of the field of occupation times of continuous time interlacements on \(\mathbb{Z}^d\), \(d\geq3\). Let \(\mathcal{L}^N\), \(N\geq1\), be the random measures on \({\mathbb{R}^d}\) given by the equation \[ {\mathcal{L}^N=\frac{1}{dN^2}\sum_{x\in\mathbb{Z}^d}L_{x, u_N}\delta_{x/N}}, \] where \(L_{x, u_N}\) is the field of occupation times of random interlacements at level \(u_N\) and \(\{u_N\}_{N\geq1}\) is a suitably chosen sequence of positive numbers.
The first main result states that, in the constant intensity regime (\(u_N=d\alpha N^{d-2}\), \(\alpha>0\)), the random measures \({\mathcal{L}^N}\) converges in distribution to \({\mathcal{L}_\alpha,}\) as \({N\to\infty},\) where \({\mathcal{L}_\alpha}\) denotes the occupation-time measure of Brownian interlacements at level \(\alpha\).
The second main result states that, in the high intensity regime (\({u_NN^{d-2}\to\infty}\)), convergence in distribution to the massless Gaussian free field holds for \[ \hat{\mathcal{L}}^N=\sqrt{\frac{d}{2N^{2-d}u_N}}(\mathcal{L}^N-\mathbb{E}[\mathcal{L}^N]). \]
At the end of paper, there is the scaling limit theorem considered using the isomorphism theorem due to the author [Electron. Commun. Probab. 17, Paper No. 9, 9 p. (2012; Zbl 1247.60135)] and it is applied to the case \(d=3\).


60F05 Central limit and other weak theorems
60J65 Brownian motion
60G60 Random fields
60J27 Continuous-time Markov processes on discrete state spaces


Zbl 1247.60135
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