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The macroscopic shape of Gelfand-Tsetlin patterns and free probability

Samuel G. G. Johnston, Joscha Prochno

Abstract

A Gelfand-Tsetlin function is a real-valued function $φ:C \to \mathbb{R}$ defined on a finite subset $C$ of the lattice $\mathbb{Z}^2$ with the property that $φ(x) \leq φ(y)$ for every edge $\langle x,y \rangle$ directed north or east between two elements of $C$. We study the statistical physics properties of random Gelfand-Tsetlin functions from the perspective of random surfaces, showing in particular that the surface tension of Gelfand-Tsetlin functions at gradient $u = (u_1,u_2) \in \mathbb{R}_{>0}^2$ is given by \begin{align*} σ(u_1,u_2) = - \log (u_1 + u_2 ) - \log \sin (πu_1/(u_1+u_2)) -1 + \log π. \end{align*} A Gelfand-Tsetlin pattern is a Gelfand-Tsetlin function defined on the triangle $T_n = \{(x_1,x_2) \in \mathbb{Z}^2 : 1 \leq x_2 \leq x_1 \leq n \}$. We show that after rescaling, a sequence of random Gelfand-Tsetlin patterns with fixed diagonal heights approximating a probability measure $μ$ satisfies a large deviation principle with speed $n^2$ and rate functional of the form \begin{align*} \mathcal{E}[ψ] := \int_{\blacktriangle} σ(\nabla ψ)\, \mathrm{d}s \,\mathrm{d}t - χ[μ] \end{align*} where $χ[μ]$ is Voiculescu's free entropy. We show that the Euler-Lagrange equations satisfied by the minimiser of the rate functional agree with those governing the free compression operation in free probability, thereby resolving a recent conjecture of Shlyakhtenko and Tao.

The macroscopic shape of Gelfand-Tsetlin patterns and free probability

Abstract

A Gelfand-Tsetlin function is a real-valued function defined on a finite subset of the lattice with the property that for every edge directed north or east between two elements of . We study the statistical physics properties of random Gelfand-Tsetlin functions from the perspective of random surfaces, showing in particular that the surface tension of Gelfand-Tsetlin functions at gradient is given by \begin{align*} σ(u_1,u_2) = - \log (u_1 + u_2 ) - \log \sin (πu_1/(u_1+u_2)) -1 + \log π. \end{align*} A Gelfand-Tsetlin pattern is a Gelfand-Tsetlin function defined on the triangle . We show that after rescaling, a sequence of random Gelfand-Tsetlin patterns with fixed diagonal heights approximating a probability measure satisfies a large deviation principle with speed and rate functional of the form \begin{align*} \mathcal{E}[ψ] := \int_{\blacktriangle} σ(\nabla ψ)\, \mathrm{d}s \,\mathrm{d}t - χ[μ] \end{align*} where is Voiculescu's free entropy. We show that the Euler-Lagrange equations satisfied by the minimiser of the rate functional agree with those governing the free compression operation in free probability, thereby resolving a recent conjecture of Shlyakhtenko and Tao.

Paper Structure

This paper contains 43 sections, 39 theorems, 311 equations, 8 figures.

Table of Contents

  1. Introduction and overview of main results
  2. Background
  3. An overview of our results
  4. Overview
  5. Further discussion
  6. Related work on the bead model and interlacing patterns
  7. Literature on random surfaces
  8. The hive model
  9. Convex geometry tools
  10. Calculus of variations and free probability
  11. Free compression and the Cauchy transform
  12. The inviscid Burgers' equation for $G_\nu(z)$
  13. Behavior of Cauchy transforms on real line
  14. PDE for quantile functions of free compression
  15. The total energy of the minimizer and Voiculescu's free entropy
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $A$ be an $n\times n$ unitarily invariant random matrix with eigenvalues $s_1 \leq \cdots \leq s_n$. Then the eigenvalue process of $A$ is uniformly distributed on the set $\mathrm{GT}(s_1,\ldots,s_n)$ of Gelfand--Tsetlin patterns with bottom row $s_1 \leq \cdots \leq s_n$.

Figures (8)

  • Figure 1: A hive is a concave function defined on a triangular lattice. Knutson and Tao KT showed that the height changes on the boundary of a hive may be associated with the possible eigenvalues of Hermitian matrices satisfying $A+B+C=0$.
  • Figure 2: A bead configuration on $n = 5$ strings with $k = 3$ beads per string. A particular bead $y_i$ is highlighted, as are the respective distances $p_i$ and $q_i$ to the next bead on the same string, and the next bead on the next string. For topological reasons, the average of the $q_i$ divided by the average of the $p_i$ takes the form $\ell/n$ for some integer $1 \leq \ell \leq n-1$.
  • Figure 3: A bead configuration on $n = 5$ strings of length $n$ with $n = 5$ beads per string and tilt $2/5$. We attempt to construct a square within the torus. To do this, we extend, setting $X_{0,5} = X_{2,2}$ and $X_{5,5} = n + X_{0,0}$
  • Figure 4: If we condition on the event that $X_{0,0}$ lies at zero, and thereafter that $X_{0,i}$, $X_{i,n}$ and $X_{i,i}$ all lie at their expected positions, we can rearrange the configuration to create a bead configuration on a square with fixed points along the diagonal.
  • Figure 5: We have $\Delta_n$ in gray and $\Delta_{n+1}'$ in blue.
  • ...and 3 more figures

Theorems & Definitions (75)

  • Theorem 1.1: Baryshnikov baryshnikov
  • Theorem 1.2: Voiculescu Vcon3, Nica and Speicher NS
  • Conjecture 1.3: Shlyakhtenko and Tao ST
  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 2.1: The Prékopa--Leindler inequalities for random surfaces
  • Conjecture 2.2: The sine process has log-concave marginals
  • Lemma 3.1: Theorem 1.7 of ST
  • proof
  • ...and 65 more