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

Abstract

We prove a Fredholm determinantal identity for the tilted Toeplitz minor $$ D_{N}^{\xi,\theta}(\varphi):= \det\bigl[(\theta_{i}\xi_{j}\varphi)_{i-j}\bigr]_{i,j=1}^{N}, $$ generalizing the Borodin-Okounkov-Geronimo-Case (BOGC) 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.