Setup: The Aztec diamond $\mathcal{A}_n$ of order $n$ consists of all unit squares $[i, i+1] \times [j, j+1]$ with $|i| + |j| < n$. It contains $2n^2$ unit squares and is tiled by $n^2$ dominoes.
Domino Types: Each domino is classified by orientation and parity:
- North (N): vertical domino with black square on top
- South (S): vertical domino with white square on top
- East (E): horizontal domino with black square on right
- West (W): horizontal domino with white square on right
The Shuffling Map $\mathcal{A}_n \to \mathcal{A}_{n+1}$:
- Deletion: Find all $2 \times 2$ blocks containing exactly two dominoes that form either:
- Two horizontal dominoes stacked (one N, one S), or
- Two vertical dominoes side-by-side (one E, one W)
These are called bad blocks. Delete all dominoes in bad blocks.
- Sliding: Each remaining domino slides outward by one unit in its natural direction:
$$\begin{aligned}
\text{N dominoes} &\to \text{slide up (} +y \text{)} \\
\text{S dominoes} &\to \text{slide down (} -y \text{)} \\
\text{E dominoes} &\to \text{slide right (} +x \text{)} \\
\text{W dominoes} &\to \text{slide left (} -x \text{)}
\end{aligned}$$
- Creation: After sliding, empty $2 \times 2$ blocks appear. Each block is filled with exactly one of two choices:
- Two horizontal dominoes (N on top, S on bottom), or
- Two vertical dominoes (W on left, E on right)
For uniform random tilings, each choice is made independently with probability $\tfrac{1}{2}$.
Key Properties:
- The map is a $2^{n+1}$-to-1 correspondence from tilings of $\mathcal{A}_{n+1}$ to tilings of $\mathcal{A}_n$
- Uniform measure on $\mathcal{A}_n$ lifts to uniform measure on $\mathcal{A}_{n+1}$
- Starting from the unique tiling of $\mathcal{A}_0$ and shuffling $n$ times samples uniformly from $\mathcal{A}_n$
Reference: arXiv:math/9201305 — Elkies, Kuperberg, Larsen, Propp, "Alternating-Sign Matrices and Domino Tilings" (1992).
q-Whittaker Deformation (q > 0)
Key Insight: The q-deformation modifies only Step 3 (Creation). Steps 1 and 2 remain unchanged.
In the classical case, block-filling is deterministic: geometric constraints force a unique cascade. In the q-Whittaker case, particles have probabilistic "stickiness"—even when a particle could move, it might stay with probability depending on $q$.
Step 3': q-Weighted Creation
Empty 2×2 blocks form islands—consecutive positions where new dominoes must be placed. For each island $[k, m]$:
- Sample Bernoulli bit: $B \sim \text{Bernoulli}\left(\dfrac{x_i y_j}{1 + x_i y_j}\right)$
- Special case: If $B = 1$ and $k = 0$, all particles jump (fill all blocks with same orientation).
- Otherwise, sample stopping position:
- Compute $f_k = \dfrac{1 - q^{\lambda_k - \bar\nu_k + 1}}{1 - q^{\bar\nu_{k-1} - \bar\nu_k + 1}}$
- Sample $U \sim \text{Uniform}[0,1]$. If $U < f_k$: stop at $k$
- Else, for $s = k+1, \ldots, m$:
- Compute $g_s = 1 - q^{\lambda_s - \mu_s + 1}$
- Sample $U \sim \text{Uniform}[0,1]$. If $U < g_s$: stop at $s$
- If no stop occurs: all particles jump
- Fill blocks based on stopping position $s$:
- Blocks at positions $k, \ldots, s-1$: orientation A (e.g., vertical EW pair)
- Block at position $s$: orientation B (opposite, e.g., horizontal NS pair)
- Blocks at positions $s+1, \ldots, m$: orientation A
Domino Orientation Mapping: The stopped particle creates a domain wall:
Block: k k+1 ... s-1 s s+1 ... m
Fill: [A] [A] ... [A] [B] [A] ... [A]
References:
- arXiv:1407.3764 — Betea, Bouttier, Nejjar, Vuletić, "The free boundary Schur process and applications I"
- arXiv:1504.00666 — Matveev, Petrov, "q-randomized Robinson-Schensted-Knuth correspondences and random polymers"