Hideki Tanemura, Nobuo Yoshida
Localization transition of (d+1)-friendly walkers
(118K, dvi)

ABSTRACT.  Friendly walkers is a stochastic model obtained from 
independent one-dimensional simple random walks 
$\{ \tl{S}^k_n \}_{n\ge 0}$, $k=1,2,\dots, d+1$ by 
introducing ``non-crossing condition'': $\tl{S}^1_j \ge \tl{S}^2_j 
\ge \ldots \ge \tl{S}^{d+1}_j, 
j=1,2,\dots, n$ and 
``reward for collisions'' characterized by parameters 
$\b_1, \ldots, \b_d \ge 0$. 
Here, the reward for collisions is 
described as follows. If there are exactly $m$ 
collisions at time $j$, i.e., 
$m=\sharp \{1\le k \le d:\tl{S}^k_j = \tl{S}^{k+1}_j \} \ge 1$, 
then the probabilistic weight for the walkers increases 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. In particular, we prove the existence of the critical 
surface in the $d$-dimensional space for the parameters 
$(\b_1, \ldots, \b_d)$.