Aztec boundary curve via shuffling vs Bernoulli q-PushTASEP     TASEPs

Boundary observable. For the $t$-deformed Aztec diamond of rank $n$, let $\lambda(m)$ denote the section partition at distance $m \in \{1,\ldots,n\}$ from the SW side. Both samplers produce the first-column length $$B_m \;=\; \lambda(m)'_1 \;=\; \#\{i : \lambda(m)_i \ge 1\},$$ which is the variable controlling the KPZ-edge asymptotics of the AZTEC paper.

Sampler A — Bernoulli $q$-PushTASEP (Matveev–Petrov §6.3, arXiv:1504.00666). State: $R_j(t) = \lambda_1^{(j)}(t)$ for $j=1,\ldots,N$, $N=n$, step IC $R_j(0) = 0$. Independent $V_j(t) \sim \mathrm{Bernoulli}\!\left(\tfrac{\alpha\beta}{1+\alpha\beta}\right)$. Let $\xi_j(t) \in \{0,1\}$ denote whether particle $j$ moves at time $t$; sequentially for $j=1,\ldots,N$, $$ \xi_j(t) \;=\; V_j(t) \;\;\lor\;\; \mathbf{1}_{\{\xi_{j-1}(t)=1\}}\cdot \mathrm{Bernoulli}\!\left(q^{\,R_j(t-1) - R_{j-1}(t-1)}\right), $$ with the push probability evaluated at the old gap (positions at time $t-1$, before any update of the current sweep). Then $R_j(t) = R_j(t-1) + \xi_j(t)$. By Matveev–Petrov, this is the projection of the $\hat\beta$-row dynamics on the $q$-Whittaker process onto the rightmost particles. The boundary curve is the space-time anti-diagonal of one trajectory: $$B_m^{\mathrm{qpush}} \;=\; R_m(n+1-m), \qquad m=1,\ldots,n.$$

Sampler B — $q$-RSK Aztec shuffler (Matveev–Petrov §5). Growth-diagram cascade on the $n \times n$ staircase, Bernoulli inputs of intensity $\alpha\beta/(1+\alpha\beta)$, $q$-Whittaker VH bijection. The cascade returns partitions $\mathtt{parts}[d]$ at anti-diagonals $d=0,1,\ldots,2n$. In Matveev–Petrov's convention, the partition stored at $d=2m-1$ is the $q$-Whittaker level-$m$ partition $\eta^{(m)}$; Macdonald duality identifies this with the conjugate of the AZTEC paper's section partition, $$\mathtt{parts}[2m-1] \;=\; \eta^{(m)} \;=\; \lambda(m)'.$$ (Note: $\mathtt{parts}[d]$ is the $q$-RSK / $q$-Whittaker output, not the AZTEC draft's physical $(\lambda^k, \mu^k)$ chain. In the draft's notation, the corresponding section is the transposed/reindexed one whose marginal carries $m$ of the $b$-parameters and $n+1-m$ of the $a$-parameters.)

Consequently the AZTEC observable $\lambda(m)'_1$ equals the largest part $\eta^{(m)}_1$ of the stored partition, not its number of parts: $$B_m^{\mathrm{shuf}} \;=\; \eta^{(m)}_1 \;=\; \lambda(m)'_1.$$ At $q=0$, the slip is hard to see because Schur transposition symmetries in the homogeneous symmetric setup partially mask it (the Schur measure is invariant under $\lambda \leftrightarrow \lambda'$ jointly with the swap of the two rectangle directions). For $q>0$, $\#\eta^{(m)}$ and $\eta^{(m)}_1$ are genuinely different observables of the $q$-Whittaker measure and respond to $q$ in opposite directions.

Equality of distributions. Both $B_m^{\mathrm{qpush}}$ and $B_m^{\mathrm{shuf}}$ are functionals of the Macdonald measure at parameters $(q, t=0)$ with $(n+1-m)$ $\alpha$-variables on the $P$-side and $m$ $\beta$-variables on the $Q$-side; they have identical one-point distributions for every $(n, m, q, \alpha, \beta)$.