Gelfand–Tsetlin polytopes and random contractions away from the limiting shape.

Collins, Benoît; Metcalfe, Anthony

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 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 $n\to \infty$, and we consider a sequence of full flags $\{0\le p_1^{\left(n\right)}\le ...\le p_i^{\left(n\right)}\le ...\le 1_n\}$ chosen at random according to the uniform measure on full flag manifolds. We are interested in the behaviour of the extremal eigenvalues of $p_i^{\left(n\right)}{A}_n{p}_i^{\left(n\right)}$. 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 $n\to \infty$, et nous étudions une suite de drapeaux complets $\{0\le p_1^{\left(n\right)}\le ...\le p_i^{\left(n\right)}\le ...\le 1_n\}$ choisis au hasard selon la loi uniforme sur les varietes drapeaux complètes. Nous nous intéressons au comportement des valeurs propres extrêmes de $p_i^{\left(n\right)}{A}_n{p}_i^{\left(n\right)}$. 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 ».

Classification : 15B52, 60B20
Keywords: Random contractions, largest eigenvalue, steepest descent
Mots-clés : Contractions aléatoires, plus grande valeur propre, steepest descent

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[20] Pastur, Leonid; Shcherbina, Mariya 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] Vũ, Văn Sharp Concentration of Random Polytopes, Geom. Funct. Anal., Volume 15 (2005) no. 6, pp. 1284-1318