Random walks and random fixed-point free involutions

A bijection is given between fixed-point free involutions of {1, 2,...,2N} with maximum decreasing subsequence size 2p and two classes of vicious (non -intersecting) random walker configurations confined to the half-line latti ce points l greater than or equal to 1. In one class of walker configuratio ns the maximum displacement of the rightmost walker is p. Because the scale d distribution of the maximum decreasing subsequence size is known to be in the soft edge GOE (random real symmetric matrices) universality class, the same holds true for the scaled distribution of the maximum displacement of the rightmost walker.