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Gelfand–Tsetlin polytopes and random contractions away from the limiting shape.
Benoît Collins1; Anthony Metcalfe2
1 Kyoto University, Department of Mathematics, Kyoto 606-8502 (Japan)
2 Hagavagen 16, 16969, Solna, Sweden
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 3, pp. 423-533.

In this paper, we consider a sequence of selfadjoint matrices A n having a limiting spectral distribution as n→∞, and we consider a sequence of full flags {0≤p 1 (n) ≤...≤p i (n) ≤...≤1 n } chosen at random according to the uniform measure on full flag manifolds. We are interested in the behaviour of the extremal eigenvalues of p i (n) A n p i (n) . This problem is known to be equivalent to the study of uniform probability measures on Gelfand–Tsetlin polytopes. Our main results consist in explicit uniform estimates for extremal eigenvalues, and the fact that an outlier behavior has an exponentially small probability. This problem is of intrinsic interest in random matrix theory, but it was motivated from a problem in Quantum Information Theory, which we discuss. The proofs rely on a reinterpretation of the problem with the help of determinantal point processes and the techniques are based on steepest descent analysis.

Dans cet article, nous nous intéressons à une suite de matrices autoadjointes A n possédant une distribution spectrale lorsque n→∞, et nous étudions une suite de drapeaux complets {0≤p 1 (n) ≤...≤p i (n) ≤...≤1 n } choisis au hasard selon la loi uniforme sur les varietes drapeaux complètes. Nous nous intéressons au comportement des valeurs propres extrêmes de p i (n) A n p i (n) . Il est connu que ce problème est équivalent à l’étude de la mesure de probabilité uniforme sur des polytopes de Gelfand–Tsetlin. Notre résultat principal consiste en des estimées uniformes pour des valeurs propres extrémales, et le fait que les outliers sont de probabilité exponentiellement petite. Ce problème revêt un interêt intrinsèque en matrices aléatoires ; par ailleurs, il trouve une motivation dans des questions d’information quantique que nous évoquons aussi. Les preuves se fonde sur une interpretation du problème a l’aide de processus de points déterminantaux, et les techniques reposent sur de l’analyse de type « steepest descent ».

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1742
Classification: 15B52, 60B20
Keywords: Random contractions, largest eigenvalue, steepest descent
Mot clés : Contractions aléatoires, plus grande valeur propre, steepest descent

Benoît Collins 1; Anthony Metcalfe 2

1 Kyoto University, Department of Mathematics, Kyoto 606-8502 (Japan)
2 Hagavagen 16, 16969, Solna, Sweden
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Benoît Collins; Anthony Metcalfe. Gelfand–Tsetlin polytopes and random contractions away from the limiting shape.. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 3, pp. 423-533. doi : 10.5802/afst.1742. https://afst.centre-mersenne.org/articles/10.5802/afst.1742/

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