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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2410.10754 |
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| author | Johnston, Samuel G. G. Prochno, Joscha |
| author_facet | Johnston, Samuel G. G. Prochno, Joscha |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10754 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The macroscopic shape of Gelfand-Tsetlin patterns and free probability Johnston, Samuel G. G. Prochno, Joscha Probability 82B41, 82B20, 46L54, 15A42, 60G55, 60F10, 49Q20 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. |
| title | The macroscopic shape of Gelfand-Tsetlin patterns and free probability |
| topic | Probability 82B41, 82B20, 46L54, 15A42, 60G55, 60F10, 49Q20 |
| url | https://arxiv.org/abs/2410.10754 |