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 study a problem at the interface of enumerative algebraic geometry, algebraic combinatorics, and the statistical mechanics of vertex models with long-range interactions. We develop methods to explore the values $\mathfrak{S}_w(1^n)$ of Schubert polynomials (called principal specializations), which count reduced bumpless pipe dreams (RBPDs) with boundary permutation $w$. We exhibit counterexamples to the conjecture of Merzon and Smirnov that $\mathfrak{S}_w(1^n)$ is maximized on layered permutations; the smallest such example we find is at $n=17$. Despite the counterexamples, we present bounds that support a refined conjecture that the maximum of $\mathfrak{S}_w(1^n)$ over all $w\in S_n$ grows at the same asymptotic rate as the maximum over layered permutations (found by Morales, Pak, and Panova).
We implement and benchmark four recurrences for Schubert specializations: the descent formula of Macdonald, the transition formula of Lascoux-Schützenberger, the cotransition formula of Knutson, and the formula of Billey, Jockusch, and Stanley. We also explore typical random permutations drawn with probability proportional to $\mathfrak{S}_w(1^n)$, equivalently from uniformly random RBPDs. Our simulations reveal a permuton-like limit analogous to the Grothendieck setting studied by Morales, Panova, Petrov, and Yeliussizov. For sampling, we show that the standard monotone Coupling From The Past framework does not apply to RBPDs, leaving exact uniform sampling open. To bypass this, we develop an efficient Markov chain Monte Carlo sampler augmented with macroscopic “droop” updates that preserve connectivity and accelerate mixing. Our publicly available fast implementations reach $n=20$ for exact computation of $\mathfrak{S}_w(1^n)$ and $n=100$ for Markov chain Monte Carlo sampling using a chain whose stationary distribution is uniform on RBPDs (run 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.
