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Abstract:A compression is a function $F:\mathbb{R}\times[0,1]\to[0,1]$ such that each $F(\cdot,\tau)$ is the distribution function of a measure of mass $\tau$, while each $F(x,\cdot)$ is increasing and $1$-Lipschitz. Compressions are continuum analogues of Gelfand--Tsetlin patterns: if $(t_{k,j})_{ 0 \leq j \leq k \leq n}$ is a Gelfand--Tsetlin pattern, setting $F(t_{k,j},k/n) = j/n$ and interpolating creates a compression. For a differentiable compression $F$, we define the compression entropy \begin{align*} \mathcal{H}[F] :=\int_{-\infty}^\infty\int_0^1 F_x \left\{-\log F_x+\log\sin(\pi F_\tau)+1-\log\pi\right\} \mathrm{d}\tau\mathrm{d}x. \end{align*} If $\mu$ is absolutely continuous and compactly supported, we prove \begin{equation*} \sup\left\{\mathcal{H}[F]:F \text{ compression}, F(\cdot,1)\text{ is the distribution function of }\mu\right\} =\chi[\mu], \end{equation*} where $\chi[\mu]$ is Voiculescu's free entropy. By identifying the Euler--Lagrange equations for $\mathcal{H}[F]$ with a Burgers equation for Cauchy transforms, we show that the supremum is attained uniquely by the free compression of free probability theory.
We also view Gelfand--Tsetlin patterns as Ginzburg--Landau $\nabla\phi$-interface models with a hard-core interaction, and compute the surface tension: \begin{equation*} \sigma(u_1,u_2) =-\log(u_1+u_2)-\log\sin\left(\pi\frac{u_1}{u_1+u_2}\right)-1+\log\pi. \end{equation*} Finally, we prove that uniform $n$-dimensional Gelfand--Tsetlin patterns with deterministic bottom rows converging to $\mu$ satisfy a large deviation principle with speed $n^2$ and rate function \begin{equation*} I_\mu[F]=-\mathcal{H}[F]+\chi[\mu]. \end{equation*} These results resolve a conjecture of Shlyakhtenko and Tao stating that the Euler--Lagrange equations for free compression arise from the statistical mechanics of interlacing point processes.

Submission history

From: Samuel Johnston [view email]
[v1] Mon, 14 Oct 2024 17:26:27 UTC (62 KB)
[v2] Tue, 15 Oct 2024 07:40:23 UTC (62 KB)
[v3] Thu, 23 Jul 2026 18:59:39 UTC (3,275 KB)