@article{566,
abstract = {We consider large random matrices X with centered, independent entries which have comparable but not necessarily identical variances. Girko's circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et. al. [11,12] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of X.
},
author = {Alt, Johannes and Erdös, László and Krüger, Torben H},
journal = {Annals Applied Probability },
number = {1},
pages = {148--203},
publisher = {Institute of Mathematical Statistics},
title = {{Local inhomogeneous circular law}},
doi = {10.1214/17-AAP1302},
volume = {28},
year = {2018},
}
@article{5971,
abstract = {We consider a Wigner-type ensemble, i.e. large hermitian N×N random matrices H=H∗ with centered independent entries and with a general matrix of variances Sxy=𝔼∣∣Hxy∣∣2. The norm of H is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of S that substantially improves the earlier bound 2∥S∥1/2∞ given in [O. Ajanki, L. Erdős and T. Krüger, Universality for general Wigner-type matrices, Prob. Theor. Rel. Fields169 (2017) 667–727]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation.},
author = {Erdös, László and Mühlbacher, Peter},
issn = {2010-3263},
journal = {Random matrices: Theory and applications},
publisher = {World Scientific Publishing},
title = {{Bounds on the norm of Wigner-type random matrices}},
doi = {10.1142/s2010326319500096},
year = {2018},
}
@unpublished{6183,
abstract = {We study the unique solution $m$ of the Dyson equation \[ -m(z)^{-1} = z - a
+ S[m(z)] \] on a von Neumann algebra $\mathcal{A}$ with the constraint
$\mathrm{Im}\,m\geq 0$. Here, $z$ lies in the complex upper half-plane, $a$ is
a self-adjoint element of $\mathcal{A}$ and $S$ is a positivity-preserving
linear operator on $\mathcal{A}$. We show that $m$ is the Stieltjes transform
of a compactly supported $\mathcal{A}$-valued measure on $\mathbb{R}$. Under
suitable assumptions, we establish that this measure has a uniformly
$1/3$-H\"{o}lder continuous density with respect to the Lebesgue measure, which
is supported on finitely many intervals, called bands. In fact, the density is
analytic inside the bands with a square-root growth at the edges and internal
cubic root cusps whenever the gap between two bands vanishes. The shape of
these singularities is universal and no other singularity may occur. We give a
precise asymptotic description of $m$ near the singular points. These
asymptotics generalize the analysis at the regular edges given in the companion
paper on the Tracy-Widom universality for the edge eigenvalue statistics for
correlated random matrices [arXiv:1804.07744] and they play a key role in the
proof of the Pearcey universality at the cusp for Wigner-type matrices
[arXiv:1809.03971,arXiv:1811.04055]. We also extend the finite dimensional band
mass formula from [arXiv:1804.07744] to the von Neumann algebra setting by
showing that the spectral mass of the bands is topologically rigid under
deformations and we conclude that these masses are quantized in some important
cases.},
author = {Alt, Johannes and Erdös, László and Krüger, Torben H},
booktitle = {arXiv},
title = {{The Dyson equation with linear self-energy: Spectral bands, edges and cusps}},
year = {2018},
}
@article{690,
abstract = {We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdős–Rényi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy–Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdős–Rényi graph this establishes the Tracy–Widom fluctuations of the second largest eigenvalue when p is much larger than N−2/3 with a deterministic shift of order (Np)−1.},
author = {Lee, Jii and Schnelli, Kevin},
journal = {Probability Theory and Related Fields},
number = {1-2},
publisher = {Springer},
title = {{Local law and Tracy–Widom limit for sparse random matrices}},
doi = {10.1007/s00440-017-0787-8},
volume = {171},
year = {2018},
}
@article{70,
abstract = {We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by a≥0, which creates a shock in the particle density of order aT−1/3, T the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit lima→∞limT→∞ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order 1. This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several airy processes.},
author = {Nejjar, Peter},
issn = {1980-0436},
journal = {Latin American Journal of Probability and Mathematical Statistics},
number = {2},
pages = {1311--1334},
publisher = {ALEA},
title = {{Transition to shocks in TASEP and decoupling of last passage times}},
doi = {10.30757/ALEA.v15-49},
volume = {15},
year = {2018},
}
@phdthesis{149,
abstract = {The eigenvalue density of many large random matrices is well approximated by a deterministic measure, the self-consistent density of states. In the present work, we show this behaviour for several classes of random matrices. In fact, we establish that, in each of these classes, the self-consistent density of states approximates the eigenvalue density of the random matrix on all scales slightly above the typical eigenvalue spacing. For large classes of random matrices, the self-consistent density of states exhibits several universal features. We prove that, under suitable assumptions, random Gram matrices and Hermitian random matrices with decaying correlations have a 1/3-Hölder continuous self-consistent density of states ρ on R, which is analytic, where it is positive, and has either a square root edge or a cubic root cusp, where it vanishes. We, thus, extend the validity of the corresponding result for Wigner-type matrices from [4, 5, 7]. We show that ρ is determined as the inverse Stieltjes transform of the normalized trace of the unique solution m(z) to the Dyson equation −m(z) −1 = z − a + S[m(z)] on C N×N with the constraint Im m(z) ≥ 0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of C N×N and S is a positivity-preserving operator on C N×N encoding the first two moments of the random matrix. In order to analyze a possible limit of ρ for N → ∞ and address some applications in free probability theory, we also consider the Dyson equation on infinite dimensional von Neumann algebras. We present two applications to random matrices. We first establish that, under certain assumptions, large random matrices with independent entries have a rotationally symmetric self-consistent density of states which is supported on a centered disk in C. Moreover, it is infinitely often differentiable apart from a jump on the boundary of this disk. Second, we show edge universality at all regular (not necessarily extreme) spectral edges for Hermitian random matrices with decaying correlations.},
author = {Alt, Johannes},
pages = {456},
publisher = {IST Austria},
title = {{Dyson equation and eigenvalue statistics of random matrices}},
doi = {10.15479/AT:ISTA:TH_1040},
year = {2018},
}
@article{1012,
abstract = {We prove a new central limit theorem (CLT) for the difference of linear eigenvalue statistics of a Wigner random matrix H and its minor H and find that the fluctuation is much smaller than the fluctuations of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of H and H. In particular, our theorem identifies the fluctuation of Kerov's rectangular Young diagrams, defined by the interlacing eigenvalues ofH and H, around their asymptotic shape, the Vershik'Kerov'Logan'Shepp curve. Young diagrams equipped with the Plancherel measure follow the same limiting shape. For this, algebraically motivated, ensemble a CLT has been obtained in Ivanov and Olshanski [20] which is structurally similar to our result but the variance is different, indicating that the analogy between the two models has its limitations. Moreover, our theorem shows that Borodin's result [7] on the convergence of the spectral distribution of Wigner matrices to a Gaussian free field also holds in derivative sense.},
author = {Erdös, László and Schröder, Dominik J},
issn = {10737928},
journal = {International Mathematics Research Notices},
number = {10},
pages = {3255--3298},
publisher = {Oxford University Press},
title = {{Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues}},
doi = {10.1093/imrn/rnw330},
volume = {2018},
year = {2018},
}
@article{721,
abstract = {Let S be a positivity-preserving symmetric linear operator acting on bounded functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex upper half-plane ℍ has a unique solution m with values in ℍ. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ℝ. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3. Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles: (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation-invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or cubic root cusps; no other singularities occur.},
author = {Ajanki, Oskari H and Krüger, Torben H and Erdös, László},
issn = {00103640},
journal = {Communications on Pure and Applied Mathematics},
number = {9},
pages = {1672 -- 1705},
publisher = {Wiley-Blackwell},
title = {{Singularities of solutions to quadratic vector equations on the complex upper half plane}},
doi = {10.1002/cpa.21639},
volume = {70},
year = {2017},
}
@article{733,
abstract = {Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H = A + UBU∗ converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate in the bulk of the spectrum.},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
journal = {Advances in Mathematics},
pages = {251 -- 291},
publisher = {Academic Press},
title = {{Convergence rate for spectral distribution of addition of random matrices}},
doi = {10.1016/j.aim.2017.08.028},
volume = {319},
year = {2017},
}
@article{483,
abstract = {We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, W ~ N. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.},
author = {Bourgade, Paul and Erdös, László and Yau, Horng and Yin, Jun},
issn = {10950761},
journal = {Advances in Theoretical and Mathematical Physics},
number = {3},
pages = {739 -- 800},
publisher = {International Press},
title = {{Universality for a class of random band matrices}},
doi = {10.4310/ATMP.2017.v21.n3.a5},
volume = {21},
year = {2017},
}