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.
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.
