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Growth exponent for loop-erased random walk in three dimensions. (English) Zbl 1387.60067
If \(S\) is the simple random walk on \(Z^d\) started at the origin and \(\tau_n\) is its first exit from the ball of radius \(n\) entered at the origin, how does the random walk path \(S[0,\tau_n]\) look like?
It is proved that the expected number of cut points is of order \(n^2\) for \(d>5\), of order \(n^2(\log n)^{-1/2}\) for \(d=4\), and of order \(n^{2-\xi_d}\) for \(d=2,3\) for some \(\xi_d\) which is called the intersection exponent. The value \(\xi_d\) is known for \(d=2\), \(\xi_2=5/4\), and is not known for \(d=3\).
The author proves that for \(d=3\), \[ \lim_{n\to+\infty}\frac{\log E(M_n)}{\log n}=2-\alpha, \] where \(M_n\) is the length of the loop-erasure of \(S[0,\tau_n]\), \(\alpha\in [1/3, 1)\).

60G17 Sample path properties
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
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