logo AFST
no. 3
Gelfand–Tsetlin polytopes and random contractions away from the limiting shape.
[Polytopes de Gelfand-Tsetlin et contractions aléatoires loin de la forme limite.]
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, Série 6, Tome 32 (2023) no. 3, pp. 423-533.

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 ».

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.

Reçu le :
Accepté le :
Publié le :
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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AFST_2023_6_32_3_423_0,
     author = {Beno{\^\i}t Collins and Anthony Metcalfe},
     title = {Gelfand{\textendash}Tsetlin polytopes and random contractions away from the limiting shape.},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {423--533},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 32},
     number = {3},
     year = {2023},
     doi = {10.5802/afst.1742},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1742/}
}
TY  - JOUR
AU  - Benoît Collins
AU  - Anthony Metcalfe
TI  - Gelfand–Tsetlin polytopes and random contractions away from the limiting shape.
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2023
SP  - 423
EP  - 533
VL  - 32
IS  - 3
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1742/
DO  - 10.5802/afst.1742
LA  - en
ID  - AFST_2023_6_32_3_423_0
ER  - 
%0 Journal Article
%A Benoît Collins
%A Anthony Metcalfe
%T Gelfand–Tsetlin polytopes and random contractions away from the limiting shape.
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2023
%P 423-533
%V 32
%N 3
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1742/
%R 10.5802/afst.1742
%G en
%F AFST_2023_6_32_3_423_0
Benoît Collins; Anthony Metcalfe. Gelfand–Tsetlin polytopes and random contractions away from the limiting shape.. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 32 (2023) no. 3, pp. 423-533. doi : 10.5802/afst.1742. https://afst.centre-mersenne.org/articles/10.5802/afst.1742/

[1] Greg W. Anderson; Alice Guionnet; Ofer Zeitouni An introduction to random matrices, Cambridge Studies in Advanced Mathematics, 118, Cambridge University Press, 2010 | Zbl

[2] Guillaume Aubrun; Stanisław Szarek; Elisabeth Werner Hastings’s additivity counterexample via Dvoretzky’s theorem, Commun. Math. Phys., Volume 305 (2011) no. 1, pp. 85-97

[3] Yuliy Baryshnikov GUEs and queues, Probab. Theory Relat. Fields, Volume 119 (2001) no. 2, pp. 256-274

[4] Serban T. Belinschi; Benoît Collins; Ion Nechita Eigenvectors and eigenvalues in a random subspace of a tensor product, Invent. Math., Volume 190 (2012) no. 3, pp. 647-697

[5] Serban T. Belinschi; Benoît Collins; Ion Nechita Almost one bit violation for the additivity of the minimum output entropy, Commun. Math. Phys., Volume 341 (2016) no. 3, pp. 885-909

[6] Fernando G. S. L. Brandão; Michał Horodecki On Hastings’ counterexamples to the minimum output entropy additivity conjecture, Open Syst. Inf. Dyn., Volume 17 (2010) no. 1, pp. 31-52

[7] Benoît Collins Product of random projections, Jacobi ensembles and universality problems arising from free probability, Probab. Theory Relat. Fields, Volume 133 (2005) no. 3, pp. 315-344

[8] Benoît Collins; Camille Male The strong asymptotic freeness of Haar and deterministic matrices, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014) no. 1, pp. 147-163

[9] Benoît Collins; Ion Nechita Random matrix techniques in quantum information theory, J. Math. Phys., Volume 57 (2016) no. 1, 015215, 34 pages

[10] Manon Defosseux Orbit measures, random matrix theory and interlaced determinantal processes, Ann. Inst. Henri Poincaré, Volume 46 (2010) no. 1, pp. 209-249

[11] Erik Duse; Kurt Johansson; Anthony Metcalfe The Cusp-Airy Process, Electron. J. Probab., Volume 21 (2016), 57, 50 pages

[12] Erik Duse; Anthony Metcalfe Asymptotic geometry of discrete interlaced patterns: Part I, Int. J. Math., Volume 26 (2015) no. 11, 1550093, 66 pages

[13] Erik Duse; Anthony Metcalfe Universal edge fluctuations of discrete interlaced particle systems, Ann. Math. Blaise Pascal, Volume 25 (2017) no. 1, pp. 75-197

[14] Laszlo Erdős Universality of Wigner random matrices: A survey of recent results, Russ. Math. Surv., Volume 66 (2011) no. 3, pp. 507-626

[15] Motohisa Fukuda; Christopher King; David K. Moser Comments on Hastings’ additivity counterexamples, Commun. Math. Phys., Volume 296 (2010) no. 1, pp. 111-143

[16] Matthew B. Hastings Superadditivity of communication capacity using entangled inputs, Nat. Phys., Volume 5 (2009), pp. 255-257

[17] Patrick Hayden; Andreas Winter Counterexamples to the maximal p-norm multiplicity conjecture for all p>1, Commun. Math. Phys., Volume 284 (2008) no. 1, pp. 263-280

[18] Kurt Johansson Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices, Commun. Math. Phys., Volume 215 (2001) no. 3, pp. 683-705

[19] Anthony P. Metcalfe Universality properties of Gelfand-Tsetlin patterns, Probab. Theory Relat. Fields, Volume 155 (2013) no. 1-2, pp. 303-346

[20] Leonid Pastur; Mariya Shcherbina Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Stat. Phys., Volume 86 (1997) no. 1-2, pp. 109-147

[21] Văn Vũ Sharp Concentration of Random Polytopes, Geom. Funct. Anal., Volume 15 (2005) no. 6, pp. 1284-1318

Cité par Sources :