Localization transition of d-friendly walkers

Friendly walkers is a stochastic model obtained from independent one-dimensional simple random walks {S_j^k}_j≥0, k = 1, 2,..., d by introducing "non-crossing condition": S_j¹ ≤ S_j² ≤ ... ≤ S_j^d, j = 1,2, ..., n and "reward for collisions" characterized by parameters β₂, ..., β_d ≥ 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 ≥ 2 walkers, then the site increases the probabilistic weight for the walkers by multiplicative factor exp(β_m) ≥ 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 (β₂,..., β_d).