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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.24976 |
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| _version_ | 1866910252458311680 |
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| author | Petrov, Leonid |
| author_facet | Petrov, Leonid |
| contents | We prove a Fredholm determinantal identity for the tilted Toeplitz minor $$ D_{N}^{ξ,θ}(φ):= \det\bigl[(θ_{i}ξ_{j}φ)_{i-j}\bigr]_{i,j=1}^{N}, $$ generalizing the Borodin-Okounkov-Geronimo-Case (BOGC) identity to oblique splittings of the Hardy space. The tilts $ξ_{j},θ_{i}$ enter only through an oblique projection that multiplies the trace-class kernel $K$ inside the Fredholm determinant; the BOGC operator $A=I-K$ constructed from $φ$ is unchanged.
Baik-Liao-Liu (arXiv:2603.01964) and Liu-Tripathi (arXiv:2604.24747) have recently shown that the same tilted Toeplitz minor admits a contour Fredholm-determinantal representation, in connection with the periodic Totally Asymmetric Simple Exclusion Process (TASEP). In the periodic TASEP application of Baik-Liao-Liu, the formula plays an important role in identifying the periodic KPZ fixed point with general initial data. Our formula is a companion to their Fredholm determinant and readily reduces to the original BOGC identity.
The one-sided tilted Toeplitz minor (that is, when all $θ_i=1$) admits a bialternant form recovering Schur and Grothendieck polynomials as special cases. A Cauchy-Binet expansion realizes $D_{N}^{ξ,θ}$ as a restricted sum over partitions of products of Jacobi-Trudi type determinants, generalizing Gessel's theorem. In the pure-shift setting this specializes to a skew Schur expansion. Finally, for finite Laurent exponential symbols, we record explicit resolvent-block flow identities and formulate the associated finite-dimensional closure problem. We also illustrate a possible asymptotic application leading to finite-rank perturbations of the Airy kernel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24976 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Borodin-Okounkov-Geronimo-Case identity for tilted Toeplitz minors Petrov, Leonid Functional Analysis Classical Analysis and ODEs Probability 47B35, 15B05, 05E05, 37K10 We prove a Fredholm determinantal identity for the tilted Toeplitz minor $$ D_{N}^{ξ,θ}(φ):= \det\bigl[(θ_{i}ξ_{j}φ)_{i-j}\bigr]_{i,j=1}^{N}, $$ generalizing the Borodin-Okounkov-Geronimo-Case (BOGC) identity to oblique splittings of the Hardy space. The tilts $ξ_{j},θ_{i}$ enter only through an oblique projection that multiplies the trace-class kernel $K$ inside the Fredholm determinant; the BOGC operator $A=I-K$ constructed from $φ$ is unchanged. Baik-Liao-Liu (arXiv:2603.01964) and Liu-Tripathi (arXiv:2604.24747) have recently shown that the same tilted Toeplitz minor admits a contour Fredholm-determinantal representation, in connection with the periodic Totally Asymmetric Simple Exclusion Process (TASEP). In the periodic TASEP application of Baik-Liao-Liu, the formula plays an important role in identifying the periodic KPZ fixed point with general initial data. Our formula is a companion to their Fredholm determinant and readily reduces to the original BOGC identity. The one-sided tilted Toeplitz minor (that is, when all $θ_i=1$) admits a bialternant form recovering Schur and Grothendieck polynomials as special cases. A Cauchy-Binet expansion realizes $D_{N}^{ξ,θ}$ as a restricted sum over partitions of products of Jacobi-Trudi type determinants, generalizing Gessel's theorem. In the pure-shift setting this specializes to a skew Schur expansion. Finally, for finite Laurent exponential symbols, we record explicit resolvent-block flow identities and formulate the associated finite-dimensional closure problem. We also illustrate a possible asymptotic application leading to finite-rank perturbations of the Airy kernel. |
| title | A Borodin-Okounkov-Geronimo-Case identity for tilted Toeplitz minors |
| topic | Functional Analysis Classical Analysis and ODEs Probability 47B35, 15B05, 05E05, 37K10 |
| url | https://arxiv.org/abs/2605.24976 |