Random matrices with log-range correlations, and log-Sobolev inequalities

Kemp, Todd; Zimmermann, David

Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 2, pp. 207-232.

Let $X_N$ be a symmetric $N\times N$ random matrix whose $\sqrt{N}$-scaled entries are uniformly square integrable. We prove that if the entries of $X_N$ can be partitioned into independent subsets each of size $o(logN)$, then the empirical eigenvalue distribution of $X_N$, minus its mean, converges weakly to $0$ in probability; hence if the averaged empirical eigenvalue distribution converges to a law, the empirical spectral distribution converges to this limit law weakly in probability. If the entries are bounded, the convergence is almost sure; if the entries are Gaussian, we prove almost sure convergence with larger blocks of size $o(N^2/logN)$. This significantly extends the best previously known results on convergence of eigenvalues for matrices with correlated entries, where the partition subsets are blocks and of size $O\left(1\right)$. We also prove the strongest known convergence results for eigenvalues of band matrices.

We prove these results by developing a new log-Sobolev inequality which generalizes the second author’s introduction of mollified log-Sobolev inequalities: we show that if $\mathbf{Y}$ is a bounded random vector and $\mathbf{Z}$ is a standard normal random vector independent from $\mathbf{Y}$, then the law of $\mathbf{Y}+t^{1/2}\mathbf{Z}$ satisfies a log-Sobolev inequality for all $t>0$, and we give bounds on the optimal log-Sobolev constant.

[1] Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer An introduction to random matrices, Cambridge Studies in Advanced Mathematics, 118, Cambridge University Press, 2010, xiv+492 pages | MR | Zbl

[2] Anderson, Greg W.; Zeitouni, Ofer A CLT for a band matrix model, Probab. Theory Relat. Fields, Volume 134 (2006) no. 2, pp. 283-338 | DOI | MR | Zbl

[3] Arizmendi, Octavio; Nechita, Ion; Vargas, Carlos On the asymptotic distribution of block-modified random matrices, J. Math. Phys., Volume 57 (2016) no. 1, 015216, 25 pages | DOI | MR | Zbl

[4] Bai, Zhi-Dong; Yin, Y. Q. Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix, Ann. Probab., Volume 16 (1988) no. 4, pp. 1729-1741 | MR | Zbl

[5] Bakry, Dominique L’hypercontractivité et son utilisation en théorie des semigroupes, Lectures on probability theory (Saint-Flour, 1992) (Lecture Notes in Mathematics), Volume 1581, Springer, 1994, pp. 1-114 | DOI | MR | Zbl

[6] Bakry, Dominique On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, New trends in stochastic analysis (Charingworth, 1994), World Scientific, 1997, pp. 43-75 | MR

[7] Bakry, Dominique; Barthe, Franck; Cattiaux, Patrick; Guillin, Arnaud A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case, Electron. Commun. Probab., Volume 13 (2008), pp. 60-66 | DOI | MR | Zbl

[8] Bakry, Dominique; Émery, Michel Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84 (Lecture Notes in Mathematics), Volume 1123, Springer, 1985, pp. 177-206 | DOI | Numdam | MR | Zbl

[9] Bakry, Dominique; Ledoux, Michel A logarithmic Sobolev form of the Li-Yau parabolic inequality, Rev. Mat. Iberoam., Volume 22 (2006) no. 2, pp. 683-702 | DOI | MR | Zbl

[10] Banica, Teodor; Nechita, Ion Asymptotic eigenvalue distributions of block-transposed Wishart matrices, J. Theor. Probab., Volume 26 (2013) no. 3, pp. 855-869 | DOI | MR | Zbl

[11] Banica, Teodor; Nechita, Ion Block-modified Wishart matrices and free Poisson laws, Houston J. Math., Volume 41 (2015) no. 1, pp. 113-134 | MR | Zbl

[12] Bardet, Jean-Baptiste; Gozlan, Nathaël; Malrieu, Florent; Zitt, Pierre-André Functional inequalities for Gaussian convolutions of compactly supported measures: explicit bounds and dimension dependence, Bernoulli, Volume 24 (2018) no. 1, pp. 333-353 | DOI | MR | Zbl

[13] Bobkov, Sergey G.; Houdré, Christian Some connections between isoperimetric and Sobolev-type inequalities, Mem. Am. Math. Soc., Volume 129 (1997) no. 616, p. viii+111 | DOI | MR | Zbl

[14] Bobkov, Sergey G.; Ledoux, Michel From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal., Volume 10 (2000) no. 5, pp. 1028-1052 | DOI | MR | Zbl

[15] Bobkov, Sergey G.; Tetali, Prasad Modified logarithmic Sobolev inequalities in discrete settings, J. Theor. Probab., Volume 19 (2006) no. 2, pp. 289-336 | DOI | MR | Zbl

[16] Bogachev, Leonid V.; Molchanov, Stanislav A.; Pastur, Leonid A. On the density of states of random band matrices, Mat. Zametki, Volume 50 (1991) no. 6, pp. 31-42 | DOI | MR | Zbl

[17] Bryc, Włodzimierz; Dembo, Amir; Jiang, Tiefeng Spectral measure of large random Hankel, Markov and Toeplitz matrices, Ann. Probab., Volume 34 (2006) no. 1, pp. 1-38 | DOI | MR | Zbl

[18] Casati, Giulio; Girko, Vyacheslav Wigner’s semicircle law for band random matrices, Random Oper. Stoch. Equ., Volume 1 (1993) no. 1, pp. 15-21 | DOI | MR | Zbl

[19] Cattiaux, Patrick; Guillin, Arnaud; Wu, Li-Ming A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality, Probab. Theory Relat. Fields, Volume 148 (2010) no. 1-2, pp. 285-304 | DOI | MR | Zbl

[20] Davies, Edward B. Explicit constants for Gaussian upper bounds on heat kernels, Am. J. Math., Volume 109 (1987) no. 2, pp. 319-333 | DOI | MR | Zbl

[21] Davies, Edward B. Heat kernels and spectral theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, 1990, x+197 pages | MR

[22] Davies, Edward B.; Simon, Barry Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal., Volume 59 (1984) no. 2, pp. 335-395 | DOI | MR | Zbl

[23] Diaconis, Persi; Saloff-Coste, Laurent Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab., Volume 6 (1996) no. 3, pp. 695-750 | DOI | MR

[24] Elliott Smith, R.; Huang, H.; Kemp, T.; Ling, Y.; Luo, X.; Lybrand, E.; Wang, J. Random Matrices with Independent Diagonals (in preparation)

[25] Friesen, Olga; Löwe, Matthias A phase transition for the limiting spectral density of random matrices, Electron. J. Probab., Volume 18 (2013), 17, 17 pages | DOI | MR | Zbl

[26] Gross, Leonard Logarithmic Sobolev inequalities, Am. J. Math., Volume 97 (1975) no. 4, pp. 1061-1083 | DOI | MR

[27] Gross, Leonard; Rothaus, Oscar Herbst inequalities for supercontractive semigroups, J. Math. Kyoto Univ., Volume 38 (1998) no. 2, pp. 295-318 | DOI | MR | Zbl

[28] Guionnet, Alice Large random matrices: lectures on macroscopic asymptotics, Lecture Notes in Mathematics, 1957, Springer, 2009, xii+294 pages (Lectures from the 36th Probability Summer School held in Saint-Flour, 2006) | DOI | MR | Zbl

[29] Guionnet, Alice; Zegarliński, Bogusław Lectures on logarithmic Sobolev inequalities, Séminaire de Probabilités, XXXVI (Lecture Notes in Mathematics), Volume 1801, Springer, 2003, pp. 1-134 | DOI | MR

[30] Hoffman, Alan J.; Wielandt, Helmut W. The variation of the spectrum of a normal matrix, Duke Math. J., Volume 20 (1953), pp. 37-39 | DOI | MR | Zbl

[31] Holley, Richard; Stroock, Daniel Logarithmic Sobolev inequalities and stochastic Ising models, J. Stat. Phys., Volume 46 (1987) no. 5-6, pp. 1159-1194 | DOI | MR | Zbl

[32] Kargin, Vladislav Spectrum of random Toeplitz matrices with band structure, Electron. Commun. Probab., Volume 14 (2009), pp. 412-421 | DOI | MR | Zbl

[33] Ledoux, Michel Isoperimetry and Gaussian analysis, Lectures on probability theory and statistics (Saint-Flour, 1994) (Lecture Notes in Mathematics), Volume 1648, Springer, 1996, pp. 165-294 | DOI | MR | Zbl

[34] Ledoux, Michel The concentration of measure phenomenon, Mathematical Surveys and Monographs, 89, American Mathematical Society, 2001, x+181 pages | DOI | MR | Zbl

[35] Ledoux, Michel A remark on hypercontractivity and tail inequalities for the largest eigenvalues of random matrices, Séminaire de Probabilités XXXVII (Lecture Notes in Mathematics), Volume 1832, Springer, 2003, pp. 360-369 | DOI | MR | Zbl

[36] Massey, Adam; Miller, Steven J.; Sinsheimer, John Distribution of eigenvalues of real symmetric palindromic Toeplitz matrices and circulant matrices, J. Theor. Probab., Volume 20 (2007) no. 3, pp. 637-662 | DOI | MR | Zbl

[37] Rashidi Far, Reza; Oraby, Tamer; Bryc, Wlodek; Speicher, Roland On slow-fading MIMO systems with nonseparable correlation, IEEE Trans. Inf. Theory, Volume 54 (2008) no. 2, pp. 544-553 | DOI | MR | Zbl

[38] Shlyakhtenko, Dimitri Random Gaussian band matrices and freeness with amalgamation, Int. Math. Res. Not., Volume 1996 (1996) no. 20, pp. 1013-1025 | DOI | MR | Zbl

[39] Stam, Aart J. Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inform. and Control, Volume 2 (1959), pp. 101-112 | DOI | MR | Zbl

[40] Villani, Cédric Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, 2003, xvi+370 pages | DOI | MR | Zbl

[41] Wang, Feng-Yu; Wang, Jian Functional inequalities for convolution probability measures, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 52 (2016) no. 2, pp. 898-914 | DOI | MR | Zbl

[42] Wigner, Eugene P. Characteristic vectors of bordered matrices with infinite dimensions, Ann. Math., Volume 62 (1955), pp. 548-564 | DOI | MR | Zbl

[43] Wigner, Eugene P. On the distribution of the roots of certain symmetric matrices, Ann. Math., Volume 67 (1958), pp. 325-327 | DOI | MR | Zbl

[44] Yau, Horng-Tzer Logarithmic Sobolev inequality for lattice gases with mixing conditions, Commun. Math. Phys., Volume 181 (1996) no. 2, pp. 367-408 | MR | Zbl

[45] Yau, Horng-Tzer Logarithmic Sobolev inequality for generalized simple exclusion processes, Probab. Theory Relat. Fields, Volume 109 (1997) no. 4, pp. 507-538 | DOI | MR | Zbl

[46] Zegarliński, Bogusław Dobrushin uniqueness theorem and logarithmic Sobolev inequalities, J. Funct. Anal., Volume 105 (1992) no. 1, pp. 77-111 | DOI | MR | Zbl

[47] Zimmermann, David Logarithmic Sobolev inequalities for mollified compactly supported measures, J. Funct. Anal., Volume 265 (2013) no. 6, pp. 1064-1083 | DOI | MR | Zbl

[48] Zimmermann, David Elementary proof of logarithmic Sobolev inequalities for Gaussian convolutions on $ℝ$, Ann. Math. Blaise Pascal, Volume 23 (2016) no. 1, pp. 129-140 | DOI | MR | Zbl