Publication

Friendly walkers is a stochastic model obtained from independent one-dimensional simple random walks $\{ S^k_j \}_{j\ge 0}$, $k=1,2,\dots, d$ by introducing ``non-crossing condition'': $S^1_j \le S^2_j \le \ldots \le S^{d}_j, j=1,2,\dots, n$ and ``reward for collisions'' characterized by parameters $\b_2, \ldots, \b_d \ge 0$. Here, the reward for collisions is described as follows. If, at a given time $n$, a site in $\Z$ is occupied by exactly $m \ge 2$ walkers, then the site increases the probabilistic weight for the walkers by multiplicative factor $\exp (\b_m )\ge 1$. We study the localization transition of this model in terms of the positivity of the free energy and describe the location and the shape of the critical surface in the $(d-1)$-dimensional space for the parameters $(\b_2, \ldots, \b_d)$.