The University of Michigan Combinatorics Seminar
Fall 2000
October 27, 4:10-5:00, 3866 East Hall


Loop-erased walks and total positivity

Sergey Fomin

University of Michigan


Abstract

Now one day Pooh and Piglet were playing Poohsticks together. They had dropped their sticks in on one side of the bridge, and were waiting to see them come out on the other side. Suppose that

  • the sticks are indistinguishable material points;
  • their movements are independent realizations of the same Markov process;
  • they move in a simply-connected planar domain whose boundary is a simple closed Jordan curve;
  • two sides of the bridge are two disjoint connected pieces of the boundary;
  • the process is stationary in time but not necessarily isotropic in space;
  • the trajectories are continuous with probability 1;
  • a stick will never show up again at the drop-off side of the bridge (so this side is repelling);
  • once a stick shows up one the other side, it does not disappear again (so the other side is absorbing);
  • as Pooh and Piglet themselves rush to the other side of the bridge, their paths do not cross. Let us say, Pooh stays on the left (looking from upstream).
We now ask the perennial Bayesian question: given the locations at which the two sticks emerged on the other side of the bridge, but not knowing the times at which those events happened, is it more likely that Pooh's stick is the left one, and Piglet's is the right one, or could it be that the switch-over was more likely to have occured?

I will discuss this question in the context of more general total positivity phenomena associated with 2-dimensional Brownian motion, directed planar graphs, and electric networks.

The paper under the same title is available.