The inhomogeneous $t$-PushTASEP and Macdonald polynomials
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Ayyer, Arvind Martin, James Williams, Lauren |
| author_facet | Ayyer, Arvind Martin, James Williams, Lauren |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10485 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The inhomogeneous $t$-PushTASEP and Macdonald polynomials Ayyer, Arvind Martin, James Williams, Lauren Combinatorics Statistical Mechanics Mathematical Physics Probability 05A10, 05A19, 05A05, 05E05, 05E10, 33D52, 60J10, 60K35 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. |
| title | The inhomogeneous $t$-PushTASEP and Macdonald polynomials |
| topic | Combinatorics Statistical Mechanics Mathematical Physics Probability 05A10, 05A19, 05A05, 05E05, 05E10, 33D52, 60J10, 60K35 |
| url | https://arxiv.org/abs/2403.10485 |