Random walks and random fixed-point free involutions

Abstract

A bijection is given between fixed point free involutions of $\\{1,2,...,2N\\}$\nwith maximum decreasing subsequence size $2p$ and two classes of vicious\n(non-intersecting) random walker configurations confined to the half line\nlattice points $l \\ge 1$. In one class of walker configurations the maximum\ndisplacement of the right most walker is $p$. Because the scaled distribution\nof the maximum decreasing subsequence size is known to be in the soft edge GOE\n(random real symmetric matrices) universality class, the same holds true for\nthe scaled distribution of the maximum displacement of the right most walker.\n