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 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. The scaling of the distribution gives the maximum mean displacement after t time steps as = (2t) 1=2 with standard deviation proportional to 1=3. The exponent 1=3 is typical of a large class of two-dimensional growth problems. Non-intersecting (vicious) random walkers were introduced into statistical mechanics 6] as models of domain walls and wetting in two-dimensional lattice systems, and have also received attention as exactly solvable systems 7, 8, 9, 4, 3]. They can be viewed as directed lattice paths which start at sites say on the x-axis and nish on sites on the line y = n, with the additional constraint that the paths do not touch or overlap. Alternatively, vicious random walkers can be described as the stochastic evolution of particles on a one-dimensional lattice, which at each tick of the clock move to the left or to the right with a certain probability, subject to the constraint that no two particles can occupy the one lattice site. Our interest is in the random turns vicious walker model 6, 9]. Here, in the stochastic evolution picture, at each time step t (t = 1; 2; : : :) one particle is selected at random and moved one lattice site to the left with probability w ?1 , or one lattice site to the right with probability w 1 (w ?1 + w 1 = 1). However, if the lattice site to the left (right) is already occupied, then the chosen walker moves to the right (left) with probability one unless this lattice site too is occupied. In the latter situation another walker is selected at random and the procedure repeated until one walker has been moved. The move of precisely one walker so determines the state at time interval t. An example of some typical conngurations in the directed paths picture is given in Figure 1. We remark that the random turns vicious walker model can also be regarded as a particular asymmetric exclusion process 17]. Two aspects of the theory of the random turns vicious walker model are the subject …