Self-Trapping Self-Repelling Random Walks

Although the title seems self-contradictory, it does not contain a misprint. The model we study is a seemingly minor modification of the “true self-avoiding walk” model of Amit, Parisi, and Peliti in two dimensions. The walks in it are self-repelling up to a characteristic time $T^{*}$ (which depends on various parameters), but spontaneously (i.e., without changing any control parameter) become self-trapping after that. For free walks, $T^{*}$ is astronomically large, but on finite lattices the transition is easily observable. In the self-trapped regime, walks are subdiffusive and intermittent, spending longer and longer times in small areas until they escape and move rapidly to a new area. In spite of this, these walks are extremely efficient in covering finite lattices, as measured by average cover times.

Figure 1

Log-linear plots of statistics of TSAWs with $\beta =\infty$ on square lattices of size $L\times L$ with helical boundary conditions. The upper curve (open squares and left-hand $y$ axis) shows the average cover times, divided by $L^2$. The lower curve (filled squares and right-hand $y$ axis) shows the asymptotic height variances of the debris field. In both cases, error bars are much smaller than symbol sizes.

Figure 2

Plot of $\sigma (T)-a_{\sigma }(\beta =\infty ,\epsilon )\ln L$ against the average debris height $⟨h⟩$, for $\epsilon =1/2$. According to Eq. (4), these curves should all approach the same horizontal line for $T\to \infty$. The fact that they do this very slowly and in a nonmonotonic way seems to be a peculiarity of the $\beta =\infty$ limit on the square lattice. It is neither seen for finite $\beta$ nor on the triangular lattice.

Figure 3

Log-log plot of $\sigma (T)$ against the average debris height $⟨h⟩$, for square lattices with $L=512$. Each curve corresponds to a different value of $\epsilon$. They are roughly horizontal up to a characteristic debris height $h^{*}$ that increases roughly as an inverse power of $\epsilon =1/2$, but deviations from such a power law are much larger than statistical errors.

Figure 4

(a) Plot of $\sigma (T)$ against $⟨h⟩$, but for $\beta =1.0$. (b) Same data, but plotted against $(\epsilon -1/2{)}^{2.27}⟨h⟩$.

Figure 5

Plots of $\sigma (T)$ against $(\epsilon -1/2{)}^{\gamma }⟨h⟩$, for triangular lattices. Panel (a) is for $\beta =\infty$ and $\gamma =1.0$, while (b) is for $\beta =1.0$ and $\gamma =1.17$.

Figure 6

Squared average end-to-end distances (divided by $T$) of the $T$ last steps of walks with total length $t\in [t_i,t_{i+1}]$, $0\le i<10$, $t_0=0$, $t_{i+1}=2t_i+4L^2$. The top curve is for $i=0$; the bottom one is for $i=9$. Parameters are $L=16384$, $\beta =\infty$, and $\epsilon =0.7$.