# Domino Shuffling and Warren's process

In this paper by Nordenstam, it is shown that a certain interlacing particle process that arises from uniformly random Aztec diamond tilings is amazingly similar to Warren's process. One of the results (theorem 1.1) is that each level converges to a Dyson Brownian. The main conjecture is that the full process converges to Warren's process.

I have two questions concerning this.

Question 1: What is the status of of the main conjecture in Nordenstam's paper?

Question 2: This is an extremely basic misunderstanding on my part: The Aztec diamond particle process of section 2 of Nordenstam's paper is defined in such a way that each line of particles $X^k(t)=(X_1^k(t),\cdots, X_k^k(t))$ either stays still, or jumps forward one step at any given time. Given that the motion is only in one direction, how is it possible that this becomes Dyson brownian motion, which can go both up and down?

-

## 1 Answer

(Q1) I think it is still open though I don't know with complete certainty. Roughly a year ago, Nordenstam told me it was still open; he has since left academic mathematics. I'd add, though, that several related proccesses, including Nordenstam's process and a couple of processes of Warren's, are treated in Borodin-Ferrari. It is possible that the convergence is proven implicitly there, or in a sequel.

(Q2) Note that in Nordenstam's coordinates, one corner of the Aztec diamond stays fixed, which is not the way you want to scale things in the limit you want. But remember that the Aztec diamond is growing, too. So if you fix the center of the line of particles instead, then particles either jump a half unit "up" or a half unit "down" at each step.

-