an:06307347
Zbl 1303.60022
Sznitman, Alain-Sol
On scaling limits and Brownian interlacements
EN
Bull. Braz. Math. Soc. (N.S.) 44, No. 4, 555-592 (2013).
00332492
2013
j
60F05 60J65 60G60 60J27
Brownian interlacements; scaling limits; massless Gaussian free field; isomorphism theorems
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 \(d=3\).
Ivan Podvigin (Novosibirsk)
Zbl 1247.60135