Statistical properties of trajectories of friendly walkers on spatio-temporal plane

Friendly walkers are the non-crossing random walkers on a lattice with attr active interactions. We characterize each trajectory of friendly walkers by the number of walkers nz, the time interval of observation t and the total length of trajectory r. A new algorithm to generate trajectories on a spat io-temporal plane is proposed and the distribution function of number of di stinct trajectories characterized by (m, f, r), f(m,t)(r), is estimated by a random sampling method. The variance, the skewness and the kurtosis of f( m,t)(r) converge to finite values without scaling as t --> infinity for eac h m. The distribution is asymmetric and its tails are expressed by stretche d exponential functions. We consider the canonical distribution of m friend ly walkers by introducing a parameter p which plays the same role of the Bo ltzmann factor e(-beta) in the usual equilibrium systems. We calculate the mean and variance of r in the canonical distribution as a function of p for each m at t. It is observed that the variance of r per unit time interval has a peak at a certain value of p for each m = 2, 3, 4 and 5. We discuss t he possibility that the peak indicates the phase transition of trajectories of friendly walkers realized on a spatio-temporal plane.