A Borodin--Okounkov--Geronimo--Case identity for tilted Toeplitz minors

2026/05/22

We prove a Fredholm determinantal identity for the tilted Toeplitz minor [ D_{N}^{\xi,\theta}(\varphi)\coloneqq \det\bigl[(\theta_{i}\xi_{j}\varphi){i-j}\bigr]{i,j=1}^{N}, ] generalizing the Borodin–Okounkov–Geronimo–Case identity to oblique splittings of the Hardy space. The tilts $\xi_j,\theta_i$ 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 $\varphi$ 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 $\theta_i=1$) admits a bialternant form recovering Schur and Grothendieck polynomials as special cases. A Cauchy–Binet expansion realizes $D_N^{\xi,\theta}$ 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.