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On the limit of random hives with GUE boundary conditions

Hariharan Narayanan

TL;DR

This work establishes a sharp continuum limit for random hives with independent GUE boundary conditions on two sides, normalized so edge eigenvalues are asymptotically constant. Building on concentration results from NST and incorporating Fefferman’s quantitative differentiation and Tao’s minor-bead universality, it proves that the normalized hive height at a bulk edge converges in probability to a single continuum hive. The limit is characterized variationally: the edge value equals the supremum of a functional $S_v$ over asymptotic height functions $\mathtt{AHT}_v^\infty$, integrating a surface-tension $\sigma_{\diamond}$ that encodes local tiling energetics. The result provides a rigorous link between random hives, lozenge tilings, and the Horn problem via a continuum variational principle, with broad implications for large-$n$ limits in the Horn/free-convolution context. The methods combine probabilistic concentration, geometric-analytic tools (CKP-type height-function theory), and microlocal control of GUE minor processes to yield a complete limit theorem without subsequences.

Abstract

We show that hives chosen at random with independent GUE boundary conditions on two sides, weighted by a Vandermonde factor depending on the third side (which is necessary in the context of the randomized Horn problem), when normalized so that the eigenvalues at the edge are asymptotically constant, converge in probability to a continuum hive as $n \rightarrow \infty.$ It had previously been shown in joint work with Sheffield and Tao \cite{NST} that the variance of these scaled random hives tends to $0$ and consequently, from compactness, that they converge in probability subsequentially. In the present paper, building on \cite{NST}, we prove convergence in probability to a single continuum hive, without having to pass to a subsequence. We moreover show that the value at a given point $v$ of this continuum hive equals the supremum of a certain functional acting on asymptotic height functions of lozenge tilings.

On the limit of random hives with GUE boundary conditions

TL;DR

This work establishes a sharp continuum limit for random hives with independent GUE boundary conditions on two sides, normalized so edge eigenvalues are asymptotically constant. Building on concentration results from NST and incorporating Fefferman’s quantitative differentiation and Tao’s minor-bead universality, it proves that the normalized hive height at a bulk edge converges in probability to a single continuum hive. The limit is characterized variationally: the edge value equals the supremum of a functional over asymptotic height functions , integrating a surface-tension that encodes local tiling energetics. The result provides a rigorous link between random hives, lozenge tilings, and the Horn problem via a continuum variational principle, with broad implications for large- limits in the Horn/free-convolution context. The methods combine probabilistic concentration, geometric-analytic tools (CKP-type height-function theory), and microlocal control of GUE minor processes to yield a complete limit theorem without subsequences.

Abstract

We show that hives chosen at random with independent GUE boundary conditions on two sides, weighted by a Vandermonde factor depending on the third side (which is necessary in the context of the randomized Horn problem), when normalized so that the eigenvalues at the edge are asymptotically constant, converge in probability to a continuum hive as It had previously been shown in joint work with Sheffield and Tao \cite{NST} that the variance of these scaled random hives tends to and consequently, from compactness, that they converge in probability subsequentially. In the present paper, building on \cite{NST}, we prove convergence in probability to a single continuum hive, without having to pass to a subsequence. We moreover show that the value at a given point of this continuum hive equals the supremum of a certain functional acting on asymptotic height functions of lozenge tilings.

Paper Structure

This paper contains 32 sections, 44 theorems, 228 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Background on hives
  4. Large deviations for hives
  5. Concentration and convergence of GUE hives
  6. Lozenge tilings of $\hexagon_v^n$
  7. Height functions of lozenge tilings
  8. Log-concavity
  9. Marked Bead Model
  10. Key definitions
  11. Key propositions and theorems
  12. Bead model
  13. Height functions
  14. Surface tension
  15. Quantitative differentiation
  16. ...and 17 more sections

Key Result

Proposition 1.1

Let $\lambda \in \mathtt{Spec}^\circ$.

Figures (25)

  • Figure 1.1: The mean and variance of a random hive corresponding to two GUE distributions with $n = 250$ from GangNar.
  • Figure 1.2: Values taken at interior vertices in the hive model satisfy rhombus inequalities as shown above. A function which satisfies all possible rhombus inequalities is called rhombus concave.
  • Figure 1.3: An $n=4$ Gelfand--Tsetlin pattern. Each number $\lambda_{i,j}$ in the pattern is greater than or equal to numbers immediately to the northeast or southeast of the pattern; in particular, every row of the pattern is decreasing. Note that such patterns are sometimes depicted as inverted pyramids instead of pyramids in the literature.
  • Figure 1.4: A Gelfand--Tsetlin pattern with boundary $\mathrm{diag}(15,5,-5,-15) \to (3,4,3,-10)$.
  • Figure 1.5: A schematic depiction of an augmented hive in $\mathtt{AUGHIVE}_{\mathrm{diag}(\lambda \boxplus \mu \to \nu) \to a}$, where we artificially shift by a tuple $\Lambda$ with large gaps in order to create two hives, instead of a hive and a Gelfand--Tsetlin pattern.
  • ...and 20 more figures

Theorems & Definitions (117)

  • Definition 1: Discrete Hessian and the ${{\Delta}}_i$ on $\mathbb{T}_n$
  • Definition 2: Rhombus concavity
  • Definition 3: Hive
  • Proposition 1.1
  • Theorem 1.2
  • Definition 4: $\psi_1$ norm
  • Corollary 1.3
  • Definition 5: Lozenges and border triangles
  • Definition 6: Octahedron recurrence
  • Definition 7
  • ...and 107 more