Localization transition of d-friendly walkers,
Abstract
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
$(\b_2, \ldots, \b_d)$.
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