Nordenstam's scaling-limit conjecture for the Aztec diamond particle process

Let $(\mathpzc{X}(t))_{t=0,1,\dots}$ be the particle process arising from the shuffling algorithm for random tilings of the Aztec diamond, and let $(\mathbf{X}(t))_{t\geq 0}$ denote Warren's process of interlacing Dyson Brownian motions. For the component process $X^n(t)=(X^n_1(t),\dots,X^n_n(t))$, define

$$\tilde X^n_i(t)=\frac{X^n_i(Nt)-\frac{1}{2}Nt}{\frac{1}{2}\sqrt{N}}$$

with linear interpolation for non-integer values of $Nt$. Nordenstam's scaling-limit conjecture. Consider the rescaled process $\tilde{\mathpzc{X}}(t)$. It converges to Warren's process $\mathbf{X}(t)$ as $N\rightarrow\infty$, in the sense of convergence of finite-dimensional distributions.

This conjecture proposes that Warren's continuous interlacing Brownian-motion process is the scaling limit of the discrete particle dynamics underlying the Aztec-diamond shuffling algorithm. The paper proves the corresponding convergence for each fixed component to Dyson Brownian motion and establishes further partial results supporting the full-process convergence.