Dynamics of a bricklayer model: multi-walker realizations of true self-avoiding motion

Abstract

We consider a multi-walker generalization of the true self-avoiding walk: the bricklayer model. We perform stochastic simulations, and solve the partial differential equations that describe the collective motion of $N$ bricklayers/walkers coupled to the contour of an expanding wall. In the large-$N$ limit, the results from simulation agree with the solution of the partial differential equations.

Figures (6)

• Figure 1: Evolution of the density profile with builder number, for fixed time $t=512$ and $N= 1, 4, 16, 512$ bricklayers. The case $N=1$ reproduces the known distribution of the true self-avoiding walk. Curves scaled to unit area and unit variance for comparison.
• Figure 2: Evolution of the interface $\bar{h}(\bar{x})$ with builder number $t=512$, $N= 1, 4, 16, 512$ builders. For large $N$ the distribution has a parabolic form, corresponding to the linear curve for $g(y)$ in Fig. \ref{['fig:compare']}. Curves scaled to unit area and unit variance for comparison.
• Figure 3: Evolution of the density $\rho$ with time for 512 builders, for times $t=2$, $t=32$, $t=512$. For the shortest time (blue) a peak of density remains at the origin. This peak rapidly disappears (red), leaving a central depression in the density distribution. At the longest time (yellow) the density approaches an asymptotic parabolic form, with a rapid fall to zero at finite $\bar{x}$.
• Figure 4: Integration of the coupled equations eq. (\ref{['eq:coupled']}), starting from $f(0)=1$ and $g(0)=0$. The functions $f(y)$ and $g(y)$ do not decrease to zero for large $y$.
• Figure 5: Density distributions for $N$=32, 64, 128, 256, 512, 1024. Plotted on a quadratic scale to show convergence of the distributions to triangular form as $N$ increases. Straight line guide for the eye. This plot implies that the final scaling function $f(y)$ is a simple quadratic function for large $N$ and $t \gg N$.