Gaussian behavior of loop-erased self-avoiding random walk in four dimensions
Let \(0,S_ 1,S_ 2,..\). be a (nearest-neighbor, symmetric) random walk on \({\mathbb{Z}}^ d\). A self-avoiding walk is formed by ``loop-erasure''. That is, let \(\phi:=\min \{j:\) there exists an \(i<j\) with \(S_ i=S_ j\}\); if \(\tau <\phi\) is the unique \(i<\phi\) with \(S_ i=S_{\phi}\), form a path \(S'_ 0,S'_ 1,..\). with \(S'_ i=S_ i\) for \(i\leq \tau\), \(S'_ i=S_{i+(\phi -\tau)}\) for \(i\geq \tau\). Repeat this procedure until we get a path \(\hat S_ 0,\hat S_ 1,..\). without double points. For \(d\geq 3\), the random walk \(\{S_ n\}\) is transient and \(\{\) \(\hat S_ n\}\) is a nontrivial path. For \(d\geq 5\), the author [ibid. 47, 655- 693 (1980; Zbl 0445.60058)] proved weak convergence of \(Y_ n(t):=n^{- 1/2}\hat S_{[nt]}\) to a d-dimensional Brownian motion. Here the author considers the case \(d=4\). He shows that, for a suitable sequence \(\{c_ n\}\), \(Z_ n(t):=(nc_ n)^{-1/2}\hat S_{[nt]}\) converges weakly to Brownian motion. He conjectures that \(c_ n\) behaves like (log n)\({}^{1/3}\), but only proves that \[ (1/2)\leq \liminf (\log c_ n/\log \log n)\leq \limsup (\log c_ n/\log \log n)\leq (1/3). \]
scientific article; zbMATH DE number 4086684