The inhomogeneous $t$-PushTASEP and Macdonald polynomials

Abstract

We study a multispecies $t$-PushTASEP system on a finite ring of $n$ sites with site-dependent rates $x_1,\dots,x_n$. Let $λ=(λ_1,\dots,λ_n)$ be a partition whose parts represent the species of the $n$ particles on the ring. We show that for each composition $η$ obtained by permuting the parts of $λ$, the stationary probability of being in state $η$ is proportional to the ASEP polynomial $F_η(x_1,\dots,x_n; q,t)$ at $q=1$; the normalizing constant (or partition function) is the Macdonald polynomial $P_λ(x_1,\dots,x_n;q,t)$ at $q=1$. Our approach involves new relations between the families of ASEP polynomials and of non-symmetric Macdonald polynomials at $q=1$. We also use multiline diagrams, showing that a single jump of the PushTASEP system is closely related to the operation of moving from one line to the next in a multiline diagram. We derive symmetry properties for the system under permutation of its jump rates, as well as a formula for the current of a single-species system.

Figures (4)

• Figure 1: Let $\eta = (2, 4, 3, 0,2, 4, 1, 3)$ with $n=8$ and $s=4$. If the bell rings at site $3$, some particles will move -- the table shows the possible destination configurations, along with the rate of the jump to each one. In each case the particles which moved are underlined. The transition corresponding to the $4$th line of the table is illustrated on the right. Site $1$ is shown at the top of the ring, and site $3$ where the bell rings is on the extreme right.
• Figure 2: The transition graph of the multispecies $t$-PushTASEP for $\mathbf{m} = (1,1,1)$.
• Figure 3: A multiline diagram $D$ with $n=6$ columns and $s=5$ rows, with content $\lambda=(5,4,3,1,0,0)$ and bottom row $\rho^{(1)}(D)=(4,0,1,5,3,0)\in S_\lambda$. It has weight $\mathop{\mathrm{wt}}\nolimits(D)=\mathop{\mathrm{wt}}\nolimits_x(D) \mathop{\mathrm{wt}}\nolimits_t(D)=x_1^3 x_3^2 x_4^4 x_5^2 x_6^2\, t^2$.
• Figure 4: On the left, a multiline diagram $D$ with content $\lambda=(4,3,2,0)$. On the right, a multiline diagram $D$ with content $\phi(\lambda)=(3,2,1,0)$ (where $\phi$ is defined as in the proof of \ref{['lem:bottom two lines']}). The configurations $\rho^{(2)}(D)$ and $\rho^{(1)}(D)$ (the two lowest rows of $D$) have the same distribution, and the distribution of $\phi(\rho^{(1)}(D))$ and of $\phi(\rho^{(2)}(D))$ is the same as that of $\rho^{(1)}(D')$.

Theorems & Definitions (65)

• Theorem 1.1
• Example 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• Proposition 2.4