Multiparticle random walks

Original · EN

An overview is presented of recent work on some statistical problems on multiparticle random walks. We consider a Euclidean, deterministic fractal or disordered lattice and N >> 1 independent random walkers initially (t=0) placed onto the same site of the substrate. Three classes of problems are considered: (i) the evaluation of the average number <Sₙ(t)> of distinct sites visited (territory explored) up to time t by the N random walkers, (ii) the statistical description of the first passage time tⱼ,ₙ to a given distance of the first j random walkers (order statistics of exit times), and (iii) the statistical description of the time tⱼ,ₙ elapsed until the first j random walkers are trapped when a Euclidean lattice is randomly occupied by a concentration c of traps (order statistics of the trapping problem). Although these problems are very different in nature, their solutions share the same form of a series in ln⁻ⁿ(N) ᵐ (N) (with n>=1 and 0<=m<=n) for N>>1. These corrective terms contribute substantially to the statistical quantities even for relatively large values of N.