Title: Random Sorting
Speaker: Professor Alexander Holroyd
Speaker Info: University of British Columbia
See http://www.math.ubc.ca/~holroyd/sort for pictures.Date: Saturday, October 21, 2006
Joint work with Omer Angel, Dan Romik and Balint Virag.
Sorting a list of items is perhaps the most celebrated problem in mathematical 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.