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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 $$S[0,\tau_n]$$, $$\alpha\in [1/3, 1)$$.

##### MSC:
 60G17 Sample path properties 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60D05 Geometric probability and stochastic geometry
##### Keywords:
loop-erased random walk; simple random walk; ergodic theory
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