Abstract: In the symmetric group of order N, start from the identity permutation and repeatedly choose a uniform random integer 0< k< N, then apply to the permutation the adjacent transposition (k,k+1) if the permutation that results has more inversions than the current one. This is a finite random walk that ends after exactly N(N-1)/2 steps, when one reaches the permutation (N,N-1,...2,1) with the maximal number of inversions. I will describe a surprising analysis of the limiting behavior of this random walk when N goes to infinity, using results from the theory of interacting particle systems (prior knowledge of this theory will not be assumed).

The talk is based on joint work with Omer Angel and Alexander Holroyd.