Eden Model in Many Dimensions

Abstract

The Eden model in many dimensions is studied by an exact enumeration method. Our result including the first-order $\frac{1}{d}$ correction has the asymptotic behavior $〈{R_n}^2〉\sim 2 \ln \mathrm{}\left(n\right)\mathrm{}(1+\frac{3}{2d})$, which does not agree with the naive expectation $R\sim n^{\frac{1}{d}}$ [or $d(n^{\frac{1}{d}}-1)$]. This suggests that when $n$ is large, $〈{R_n}^2〉$ might be a singular function of $d$. We also present a result on the Cayley tree.