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.
]]>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.
]]>We prove a law of large numbers and exponential concentration, showing that the random tilings converge to a deterministic waterfall profile. We further conjecture an explicit correlation kernel of the one-dimensional barcode process arising in the limit. Remarkably, the limit is invariant under shifts by $2\mathbb{Z}$ but not by $\mathbb{Z}$, exhibiting an emergent period-two structure absent from the original weights. Our conjectures are supported by extensive numerical evidence and perfect sampling simulations. The kernel is built from a family of functions orthogonal in both spaces $\ell^{2}(\mathbb{Z})$ and $\ell^{2}(\mathbb{Z}+\frac12)$, that may be of independent interest.
Our proofs adapt the spectral projection method of Borodin–Gorin–Rains (2009) to the regime with fixed $q$. The resulting asymptotic analysis is substantially more involved, and leads to non-self-adjoint operators. We overcome these challenges in the exponential concentration result by a separate argument based on sharp bounds for the ratios of probabilities under the $q$-Racah orthogonal polynomial ensemble.
]]>4-7 • Washington, DC • Joint Mathematics Meetings, organizing a special session on Random Tilings, Random Permutations, and Particle Systems (January 7)
23 • Fairfax, VA • George Mason University, Colloquium
19 • Farmville, VA • Longwood University, Colloquium
13 • Charlottesville, VA • Blue Ridge Probability Day
16-17 • Stanford University, Palo Alto, CA • Probability seminar
18 • UC Berkeley, Berkeley, CA • Probability seminar
6 • Vassar College, Poughkeepsie, NY • Colloquium
20-24 • College Park, MD • The Kardar-Parisi-Zhang Universality Class & Related Topics workshop at the Brin Mathematics Research Center
1-5 • Edinburgh, UK • Universality in the KPZ and RMT workshop at ICMS
5-18 • Stony Brook, NY • 4th Simons Math Summer Workshop: Algebraic methods in probability at the Simons Center for Geometry and Physics
27-31 • Djursholm, Sweden • Solvable lattice models, representation theory of quantum groups and algebraic combinatorics at Institut Mittag-Leffler
14-Oct 10 • Vienna, Austria • Statistical Mechanics and Combinatorics of Discrete Planar Structures thematic programme at the Erwin Schrödinger Institute
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