Perturbed Beta Corners Process
We introduce and study the perturbed $\beta$-corners process, a deformation of the classical $\beta$-corners process. The latter is a probability measure on interlacing arrays of real numbers that, for the classical values $\beta=1,2,4$, describes the joint distribution of eigenvalues of principal submatrices of a Gaussian random matrix with the corresponding GOE/GUE/GSE symmetry.
For classical $\beta$, the perturbed process arises from adding a deterministic diagonal matrix $A=\mathrm{diag}(a_1,\ldots,a_N)$ to a GOE/GUE/GSE matrix. The eigenvalues of the resulting random matrix depend symmetrically on $a_1,\ldots,a_N$, but this symmetry does not extend to the whole corners process.
We extend the construction to all $\beta>0$ via multivariate Bessel functions, and analyze the crystallization ($\beta\to\infty$) of the resulting interlacing array, proving a law of large numbers and a central limit theorem in two regimes. For fixed perturbation, the eigenvalue repulsion dominates, and the array crystallizes on the same lattice of polynomial-derivative roots, with the same discrete Gaussian free field fluctuations, as in the unperturbed case treated by Gorin and Marcus (arXiv:1706.07393). New phenomena appear in the second regime, where the $a_i$’s grow linearly in $\beta$. Then the external source competes with the repulsion at leading order. Here the array freezes on a deformed lattice characterized by a coupled system of optimality equations. The fluctuations are governed by the same discrete Gaussian free field, now attached to the deformed lattice.
The deformed lattice equations decouple in the special case of a single spike in the last coordinate. Then the deformed lattice is obtained explicitly by applying one shifted derivative $D_c f = f^{\prime} + cf$ followed by iterated ordinary derivatives.
From Yang-Baxter to Robinson-Schensted-Knuth
Random Surfaces from Stacking Cubes: A Visual Journey (web app slides; 1920×1080)
Abstract. How does a corner of a crystal (like sugar cube) get its rounded shape? This seemingly simple question leads to beautiful mathematics at the intersection of probability and geometry.
We explore random lozenge tilings — ways of covering regions with diamond-shaped tiles chosen uniformly at random from astronomically many possibilities. These tilings can be viewed as discrete 3D surfaces built from unit cubes, and the central question is: what does a “typical” random surface look like? The answer reveals a striking phenomenon: as the system grows large, randomness gives way to order, and the random surface concentrates around a deterministic “limit shape,” with sharp boundaries separating frozen crystalline regions from disordered liquid regions — a phase transition you can see with your eyes.
This probabilistic story connects to surprising areas of mathematics: algebraic combinatorics (symmetric functions), algebraic geometry (limit shapes are algebraic curves), statistical mechanics (exactly solvable models), and operations research (Markov chain Monte Carlo, coupling from the past for perfect sampling). At the same time, it provides a rich source of visual inspiration which, naturally, can even be 3D printed.
Breaking Universality in Dimer Models (web app slides; 1920×1080)
Abstract. Dimer models (random lozenge or domino tilings) on large planar domains exhibit universality behavior: local convergence to translation-invariant Gibbs measures, global fluctuations described by the Gaussian Free Field (GFF), and Airy line ensemble at the edges. In this talk, I discuss two mechanisms that break this universality while preserving some exactly solvable structure. First, applying a strong double-well potential parallel to one of the triangular lattice directions induces a new “waterfall” phase in lozenge tilings, where the 2D Gibbs structure collapses into a new 1D process with an emergent period-two structure. The exact solvability is powered by the q-Racah orthogonal polynomials. Second, randomizing edge weights in Aztec domino tilings (in a diagonally layered manner) deforms limit shapes. Moreover, it leads to non-GFF Brownian motion-like fluctuations living on root-N scale or the same constant scale as the GFF, depending on the variance scaling in the random edge weights. The exact solvability here comes from explicit annealed Schur generating functions.
Based on joint works with Knizel and Bufetov-Zografos.
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.