Vertex operators, infinite wedge representations, and correlation functions of the t-Schur measure
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Greaves, Gary Jing, Naihuan Zhu, Haoran |
| author_facet | Greaves, Gary Jing, Naihuan Zhu, Haoran |
| contents | We study the $t$-Schur measure on partitions, defined by $ \mathbb{P}(λ)=Z^{-1}S_λ(x;t)s_λ(y) $, where $S_λ(x;t)$ denotes the $t$-Schur symmetric functions and $s_λ(y)$ the ordinary Schur functions, and $Z$ is the normalising constant. Using vertex operator calculus, we realise $S_λ(x;t)$ in the charged free-fermion Fock space, yielding a $t$-deformation of the classical boson-fermion correspondence. These realisations give vertex-algebraic proofs of the $t$-Cauchy identities and $t$-Gessel identity. Building on this framework, we compute the correlation functions of the $t$-Schur measure and show that the associated point process is determinantal, with an explicit correlation kernel. The Poissonised $t$-Plancherel measure appears as a specialisation of our construction, so its correlation functions follow as a corollary. As an application, we derive the limiting distribution for the length of the longest ascent pair in a random permutation. Our results interpolate the Schur case at $t=0$, connect to the Schur-$Q$ theory at $t=-1$, and provide a probabilistic interpretation of a natural $t$-refinement of increasing subsequences via a generalised RSK correspondence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14190 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vertex operators, infinite wedge representations, and correlation functions of the t-Schur measure Greaves, Gary Jing, Naihuan Zhu, Haoran Mathematical Physics Combinatorics Probability Quantum Algebra We study the $t$-Schur measure on partitions, defined by $ \mathbb{P}(λ)=Z^{-1}S_λ(x;t)s_λ(y) $, where $S_λ(x;t)$ denotes the $t$-Schur symmetric functions and $s_λ(y)$ the ordinary Schur functions, and $Z$ is the normalising constant. Using vertex operator calculus, we realise $S_λ(x;t)$ in the charged free-fermion Fock space, yielding a $t$-deformation of the classical boson-fermion correspondence. These realisations give vertex-algebraic proofs of the $t$-Cauchy identities and $t$-Gessel identity. Building on this framework, we compute the correlation functions of the $t$-Schur measure and show that the associated point process is determinantal, with an explicit correlation kernel. The Poissonised $t$-Plancherel measure appears as a specialisation of our construction, so its correlation functions follow as a corollary. As an application, we derive the limiting distribution for the length of the longest ascent pair in a random permutation. Our results interpolate the Schur case at $t=0$, connect to the Schur-$Q$ theory at $t=-1$, and provide a probabilistic interpretation of a natural $t$-refinement of increasing subsequences via a generalised RSK correspondence. |
| title | Vertex operators, infinite wedge representations, and correlation functions of the t-Schur measure |
| topic | Mathematical Physics Combinatorics Probability Quantum Algebra |
| url | https://arxiv.org/abs/2602.14190 |