2024/10/14 by Samuel G. G. Johnston, Joscha Prochno, Johnston, Samuel G. G. +1 · 1 citation
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR
paper · pdf · doi:10.48550/arxiv.2410.10754
openalex publication_date 2024/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
A compression is a function F:ℝ×[0,1]→[0,1] such that each F(⋅,τ) is the distribution function of a measure of mass τ, while each F(x,⋅) is increasing and 1-Lipschitz. Compressions are continuum analogues of Gelfand--Tsetlin patterns: if (tk,j) 0 ≤ j ≤ k ≤ n is a Gelfand--Tsetlin pattern, setting F(tk,j,k/n) = j/n and interpolating creates a compression. For a differentiable compression F, we define the compression entropy H[F] :=∫-∞^∞∫01 Fx \-log Fx+logsin(πFτ)+1-logπ\ dτdx. If μ is absolutely continuous and compactly supported, we prove sup\H[F]:F compression, F(⋅,1) is the distribution function of μ\ =χ[μ], where χ[μ] is Voiculescu's free entropy. By identifying the Euler--Lagrange equations for 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 ∇ϕ-interface models with a hard-core interaction, and compute the surface tension: σ(u1,u2) =-log(u1+u2)-logsin(π(u1)/(u1+u2))-1+logπ. Finally, we prove that uniform n-dimensional Gelfand--Tsetlin patterns with deterministic bottom rows converging to μ satisfy a large deviation principle with speed n2 and rate function Iμ[F]=-H[F]+χ[μ]. 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.