A Borodin--Okounkov--Geronimo--Case identity for tilted Toeplitz minors
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.
Plancherel vs TASEP Fluctuations
Computation and sampling for Schubert specializations
We present computational results related to principal specializations of the Schubert polynomials $\mathfrak{S}_w(1^n)$ for permutations $w\in S_n$. Equivalently, these specializations count reduced pipe dreams (and reduced bumpless pipe dreams - RBPD) with boundary conditions determined by $w$. We find the first counterexample, at $n=17$, to the conjecture of Merzon-Smirnov that the maximal value of $\mathfrak{S}_w(1^n)$ is obtained at a layered permutation. We explore the typical permutation obtained from uniformly random RBPDs, revealing a permuton-like asymptotic behavior similar to the one derived for Grothendieck polynomials.
We implement and compare three recurrence relations for computing $\mathfrak{S}_w(1^n)$: the descent formula of Macdonald, the transition formula of Lascoux-Schützenberger, and the cotransition formula of Knutson. We prove that the global constraint of reducedness breaks the sublattice property of the underlying alternating sign matrix (ASM) lattice, preventing standard monotone Coupling From The Past (CFTP). To bypass this, we develop a highly efficient MCMC sampler augmented with macroscopic “droop” updates to guarantee state space connectivity and accelerate mixing. Our implementations enable computation of $\mathfrak{S}_w(1^n)$ up to $n\sim 20$ on a personal computer, and uniform sampling of reduced bumpless pipe dreams up to $n\sim 100$ on a cluster.
Integrable Probability arXiv Feed
Math Experimental Lab Spring 2026
Colored Interlacing Triangles and Genocchi Medians
Colored interlacing triangles, introduced by Aggarwal-Borodin-Wheeler (2024), provide the combinatorial framework for the Central Limit Theorem for probability measures arising from the Lascoux-Leclerc-Thibon (LLT) polynomials. Colored interlacing triangles depend on two key parameters: the number of colors $n$ and the depth of the triangle $N$. Recent work of Gaetz-Gao (2025) connects these objects to Schubert calculus and resolves the enumeration for $n=3$ and arbitrary depth $N$. However, the enumerative behavior for general $n$ has remained open.
In this paper, we analyze the complementary regime: fixed depth $N=2$ and arbitrary number of colors $n$. We prove that in this setting, colored interlacing triangles are in bijection with Dumont derangements, identifying their enumeration with the Genocchi medians. This connects the probabilistic model to a rich hierarchy of classical combinatorial objects.
Furthermore, we introduce a $q$-deformation of this enumeration arising naturally from the LLT transition energy. This yields new $q$-analogs of the Genocchi medians. Finally, we present computational results and sampling algorithms for colored interlacing triangles with higher $N$ or $n$, which suggests the limits of combinatorial tractability in the $(N,n)$ parameter space.
