Random walks and random permutations

Abstract

A connection is made between the random-turns model of vicious walkers and random permutations indexed by their increasing subsequences. Consequently the scaled distribution of the maximum displacements in a particular asymmeteric version of the model can be determined to be the same as the scaled distribution of the eigenvalues at the soft edge of the GUE (random Hermitian matrices). The scaling of the distribution gives the maximum mean displacement μ after t time steps as μ = (2t)1/2 with standard deviation proportional to μ1/3. The exponent 1/3 is typical of a large class of two-dimensional growth problems.