## Random Sorting Networks

- Date: 12/04/2006

Alexander Holroyd (University of British Columbia)

University of British Columbia

See http://www.math.ubc.ca/~holroyd/sort for pictures.

Joint work with Omer Angel, Dan Romik and Balint Virag.

Sorting a list of items is perhaps the most celebrated problem in

computer science. If one must do this by swapping neighbouring pairs,

the worst initial condition is when the n items are in reverse order,

in which case n choose 2 swaps are needed. A sorting network is any

sequence of n choose 2 swaps which achieves this.

We investigate the behaviour of a uniformly random n-item sorting

network as n->infinity. We prove a law of large numbers for the

space-time process of swaps. Exact simulations and heuristic arguments

have led us to astonishing conjectures. For example, the half-time

permutation matrix appears to be circularly symmetric, while the

trajectories of individual items appear to converge to a famous family

of smooth curves. We prove the more modest results that,

asymptotically, the support of the matrix lies within a certain

octagon, while the trajectories are Holder-1/2. A key tool is a

connection with Young tableaux.

Probability Seminar 2006