@article{6186, abstract = {We prove that the local eigenvalue statistics of real symmetric Wigner-type matrices near the cusp points of the eigenvalue density are universal. Together with the companion paper [arXiv:1809.03971], which proves the same result for the complex Hermitian symmetry class, this completes the last remaining case of the Wigner-Dyson-Mehta universality conjecture after bulk and edge universalities have been established in the last years. We extend the recent Dyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp regime using the optimal local law from [arXiv:1809.03971] and the accurate local shape analysis of the density from [arXiv:1506.05095, arXiv:1804.07752]. We also present a PDE-based method to improve the estimate on eigenvalue rigidity via the maximum principle of the heat flow related to the Dyson Brownian motion.}, author = {Cipolloni, Giorgio and Erdös, László and Krüger, Torben H and Schröder, Dominik J}, issn = {2578-5885}, journal = {Pure and Applied Analysis }, number = {4}, pages = {615–707}, publisher = {MSP}, title = {{Cusp universality for random matrices, II: The real symmetric case}}, doi = {10.2140/paa.2019.1.615}, volume = {1}, year = {2019}, } @article{284, abstract = {Borel probability measures living on metric spaces are fundamental mathematical objects. There are several meaningful distance functions that make the collection of the probability measures living on a certain space a metric space. We are interested in the description of the structure of the isometries of such metric spaces. We overview some of the recent results of the topic and we also provide some new ones concerning the Wasserstein distance. More specifically, we consider the space of all Borel probability measures on the unit sphere of a Euclidean space endowed with the Wasserstein metric W_p for arbitrary p >= 1, and we show that the action of a Wasserstein isometry on the set of Dirac measures is induced by an isometry of the underlying unit sphere.}, author = {Virosztek, Daniel}, issn = {2064-8316}, journal = {Acta Scientiarum Mathematicarum}, number = {1-2}, pages = {65 -- 80}, publisher = {Springer Nature}, title = {{Maps on probability measures preserving certain distances - a survey and some new results}}, doi = {10.14232/actasm-018-753-y}, volume = {84}, year = {2018}, } @article{556, abstract = {We investigate the free boundary Schur process, a variant of the Schur process introduced by Okounkov and Reshetikhin, where we allow the first and the last partitions to be arbitrary (instead of empty in the original setting). The pfaffian Schur process, previously studied by several authors, is recovered when just one of the boundary partitions is left free. We compute the correlation functions of the process in all generality via the free fermion formalism, which we extend with the thorough treatment of “free boundary states.” For the case of one free boundary, our approach yields a new proof that the process is pfaffian. For the case of two free boundaries, we find that the process is not pfaffian, but a closely related process is. We also study three different applications of the Schur process with one free boundary: fluctuations of symmetrized last passage percolation models, limit shapes and processes for symmetric plane partitions and for plane overpartitions.}, author = {Betea, Dan and Bouttier, Jeremie and Nejjar, Peter and Vuletic, Mirjana}, issn = {1424-0637}, journal = {Annales Henri Poincare}, number = {12}, pages = {3663--3742}, publisher = {Springer Nature}, title = {{The free boundary Schur process and applications I}}, doi = {10.1007/s00023-018-0723-1}, volume = {19}, 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}}, doi = {10.48550/arXiv.1804.07752}, 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 = {Instituto Nacional de Matematica Pura e Aplicada}, title = {{Transition to shocks in TASEP and decoupling of last passage times}}, doi = {10.30757/ALEA.v15-49}, volume = {15}, year = {2018}, } @article{181, abstract = {We consider large random matrices X with centered, independent entries but possibly di erent variances. We compute the normalized trace of f(X)g(X∗) for f, g functions analytic on the spectrum of X. We use these results to compute the long time asymptotics for systems of coupled di erential equations with random coe cients. We show that when the coupling is critical, the norm squared of the solution decays like t−1/2.}, author = {Erdös, László and Krüger, Torben H and Renfrew, David T}, journal = {SIAM Journal on Mathematical Analysis}, number = {3}, pages = {3271 -- 3290}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Power law decay for systems of randomly coupled differential equations}}, doi = {10.1137/17M1143125}, volume = {50}, 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 = {1073-7928}, 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{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 of 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}, } @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}, issn = {2663-337X}, pages = {456}, publisher = {Institute of Science and Technology Austria}, title = {{Dyson equation and eigenvalue statistics of random matrices}}, doi = {10.15479/AT:ISTA:TH_1040}, 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 = {2010-3271}, 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}, } @article{1207, abstract = {The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.}, author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin}, issn = {0010-3616}, journal = {Communications in Mathematical Physics}, number = {3}, pages = {947 -- 990}, publisher = {Springer}, title = {{Local law of addition of random matrices on optimal scale}}, doi = {10.1007/s00220-016-2805-6}, volume = {349}, year = {2017}, } @article{1337, abstract = {We consider the local eigenvalue distribution of large self-adjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\mathbbmE|hij|2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),…,mN(z)) solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.}, author = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H}, issn = {0178-8051}, journal = {Probability Theory and Related Fields}, number = {3-4}, pages = {667 -- 727}, publisher = {Springer}, title = {{Universality for general Wigner-type matrices}}, doi = {10.1007/s00440-016-0740-2}, volume = {169}, year = {2017}, } @article{1528, abstract = {We consider N×N Hermitian random matrices H consisting of blocks of size M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H−z)−1 satisfy the local semicircle law with spectral parameter z=E+iη down to the real axis for any η≫N−1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫M−1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.}, author = {Bao, Zhigang and Erdös, László}, issn = {0178-8051}, journal = {Probability Theory and Related Fields}, number = {3-4}, pages = {673 -- 776}, publisher = {Springer}, title = {{Delocalization for a class of random block band matrices}}, doi = {10.1007/s00440-015-0692-y}, volume = {167}, year = {2017}, } @article{447, abstract = {We consider last passage percolation (LPP) models with exponentially distributed random variables, which are linked to the totally asymmetric simple exclusion process (TASEP). The competition interface for LPP was introduced and studied in Ferrari and Pimentel (2005a) for cases where the corresponding exclusion process had a rarefaction fan. Here we consider situations with a shock and determine the law of the fluctuations of the competition interface around its deter- ministic law of large number position. We also study the multipoint distribution of the LPP around the shock, extending our one-point result of Ferrari and Nejjar (2015).}, author = {Ferrari, Patrik and Nejjar, Peter}, journal = {Revista Latino-Americana de Probabilidade e Estatística}, pages = {299 -- 325}, publisher = {Instituto Nacional de Matematica Pura e Aplicada}, title = {{Fluctuations of the competition interface in presence of shocks}}, doi = {10.30757/ALEA.v14-17}, volume = {9}, year = {2017}, } @article{550, abstract = {For large random matrices X with independent, centered entries but not necessarily identical variances, the eigenvalue density of XX* is well-approximated by a deterministic measure on ℝ. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [5] to the vicinity of these singularities.}, author = {Alt, Johannes}, issn = {1083-589X}, journal = {Electronic Communications in Probability}, publisher = {Institute of Mathematical Statistics}, title = {{Singularities of the density of states of random Gram matrices}}, doi = {10.1214/17-ECP97}, volume = {22}, year = {2017}, } @book{567, abstract = {This book is a concise and self-contained introduction of recent techniques to prove local spectral universality for large random matrices. Random matrix theory is a fast expanding research area, and this book mainly focuses on the methods that the authors participated in developing over the past few years. Many other interesting topics are not included, and neither are several new developments within the framework of these methods. The authors have chosen instead to present key concepts that they believe are the core of these methods and should be relevant for future applications. They keep technicalities to a minimum to make the book accessible to graduate students. With this in mind, they include in this book the basic notions and tools for high-dimensional analysis, such as large deviation, entropy, Dirichlet form, and the logarithmic Sobolev inequality. }, author = {Erdös, László and Yau, Horng}, isbn = {9-781-4704-3648-3}, pages = {226}, publisher = {American Mathematical Society}, title = {{A Dynamical Approach to Random Matrix Theory}}, doi = {10.1090/cln/028}, volume = {28}, year = {2017}, } @article{615, abstract = {We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion of the eigenvalues hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from Wigner's semicircle law.}, author = {Erdös, László and Schnelli, Kevin}, issn = {0246-0203}, journal = {Annales de l'institut Henri Poincare (B) Probability and Statistics}, number = {4}, pages = {1606 -- 1656}, publisher = {Institute of Mathematical Statistics}, title = {{Universality for random matrix flows with time dependent density}}, doi = {10.1214/16-AIHP765}, volume = {53}, year = {2017}, } @article{1010, abstract = {We prove a local law in the bulk of the spectrum for random Gram matrices XX∗, a generalization of sample covariance matrices, where X is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of XX∗. }, author = {Alt, Johannes and Erdös, László and Krüger, Torben H}, issn = {1083-6489}, journal = {Electronic Journal of Probability}, publisher = {Institute of Mathematical Statistics}, title = {{Local law for random Gram matrices}}, doi = {10.1214/17-EJP42}, volume = {22}, year = {2017}, } @article{1023, abstract = {We consider products of independent square non-Hermitian random matrices. More precisely, let X1,…, Xn be independent N × N random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1/N. Soshnikov-O’Rourke [19] and Götze-Tikhomirov [15] showed that the empirical spectral distribution of the product of n random matrices with iid entries converges to (equation found). We prove that if the entries of the matrices X1,…, Xn are independent (but not necessarily identically distributed) and satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of X1,…, Xn to (0.1) holds up to the scale N–1/2+ε.}, author = {Nemish, Yuriy}, issn = {1083-6489}, journal = {Electronic Journal of Probability}, publisher = {Institute of Mathematical Statistics}, title = {{Local law for the product of independent non-Hermitian random matrices with independent entries}}, doi = {10.1214/17-EJP38}, volume = {22}, year = {2017}, }