Random walks and random permutations

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 asymme teric version of the model can be determined to be the same as the scaled d istribution of the eigenvalues at the soft edge of the GUE (random Hermitia n matrices). The scaling of the distribution gives the maximum mean displac ement mu after t time steps as mu = (2t)(1/2) with standard deviation propo rtional to mu (1/3). The exponent 1/3 is typical of a large class of two-di mensional growth problems.