From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models (Q373571)

The authors consider a cluster growth model on \(\mathbb{Z}^d,\) called internal diffusion limited aggregation (internal DLA). The internal DLA cluster \(A(N)\) of volume \(N\) is obtained inductively as follows. Initially, it is assumed that the explored region is empty, i.e., \(A(0)=\emptyset\). Then, consider \(N\) independent discrete-time random walks \( S_1,\dots,S_N\) starting from \(0\). For \(k\leq N\), \(A(k-1)\) is obtained and define \(\tau_k=\inf\{t\geq0: S_k(t)\notin A(k-1)\}\) and \(A(k)=A(k-1)\cup \{S_k(\tau_k)\}\). The inner (outer) error \(\delta_I(n)\) (resp. \(\delta_O(n)\)) is such that NEWLINE\[NEWLINEn-\delta_I(n)=\sup\{r\geq 0:\mathbb{B}(0,r)\subset A(|\mathbb{B}(0,n)|)\}NEWLINE\]NEWLINE (resp. \( n+\delta_O(n)=\sup\{r\geq 0:A(|\mathbb{B}(0,n)|)\subset \mathbb{B}(0,r))\}\)).NEWLINENEWLINEThe main result is the following improvement of the main result [\textit{G. F. Lawler} et al., Ann. Probab. 20, No. 4, 2117--2140 (1992; Zbl 0762.60096)]NEWLINENEWLINETheorem. Assume \(d\geq 2.\) There is a positive constant \(A_d\) such that NEWLINE\[NEWLINE\operatorname{P}(\exists n(\omega):\forall n\geq n(\omega)\delta_I(n)\leq A_d\log(n))=1NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\operatorname{P}(\exists n(\omega):\forall n\geq n(\omega)\delta_O(n)\leq A_d\log^2(n))=1,NEWLINE\]NEWLINE where \(B(x,r)=\{y\in \mathbb{R}^d: ||y-x||<r\}\) and \(\mathbb{B}(x,r)=B(x,r)\cap \mathbb{Z}^d.\)