On the Shape of a Random String

Abstract

We define a string of length $m$ to be a sequence $z_0, z_1,\cdots, z_m$ of points in $Z^2$ which are chosen so that $z_{i-1}$ and $z_i$ are adjacent points on the lattice. If we consider the set of all strings from $z_0 = (0, 0)$ to $z_m = (n, 0)$ then we can introduce a probability distribution by assuming that all the strings are equally likely. In this paper we will prove some results which describe the shape of the random string when $m/n\rightarrow \lambda \in (1, \infty)$. The main result states that if we let $(U_k, V_k)$ be the coordinate of $z_k$ then $(U_{\lbrack m\cdot \rbrack}/n, V_{\lbrack m\cdot \rbrack}/n^{\frac{1}{2}})$ converges weakly to $(\cdot, \rho^{\frac{1}{2}}W_0(\cdot))$ where $W_0$ is a Brownian bridge and $\rho = (\lambda^2 - 1)/2\lambda$. A second set of results describes $B_n = \sup\{U_k - U_l: 0 \leqslant k \leqslant l \leqslant m\}$, a quantity which measures the amount of backtracking in the string. We find that this is $0(\log n)$ and there is a sequence of constants $b_n$ so that $B_n - b_n$ approaches a double exponential distribution. The results described above were motivated by, and are related to, results of Abraham and Reed, and Gallovotti on the shape of the interface profile in the two dimensional Ising model.