@article{11332,
abstract = {We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size N converge to the Tracy–Widom laws at a rate O(N^{-1/3+\omega }), as N tends to infinity. For Wigner matrices this improves the previous rate O(N^{-2/9+\omega }) obtained by Bourgade (J Eur Math Soc, 2021) for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős et al. (Adv Math 229(3):1435–1515, 2012) to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein–Uhlenbeck process. Precise estimates on leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and recursive comparisons for correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian invariant ensembles.},
author = {Schnelli, Kevin and Xu, Yuanyuan},
issn = {1432-0916},
journal = {Communications in Mathematical Physics},
pages = {839--907},
publisher = {Springer Nature},
title = {{Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices}},
doi = {10.1007/s00220-022-04377-y},
volume = {393},
year = {2022},
}
@article{11418,
abstract = {We consider the quadratic form of a general high-rank deterministic matrix on the eigenvectors of an N×N
Wigner matrix and prove that it has Gaussian fluctuation for each bulk eigenvector in the large N limit. The proof is a combination of the energy method for the Dyson Brownian motion inspired by Marcinek and Yau (2021) and our recent multiresolvent local laws (Comm. Math. Phys. 388 (2021) 1005–1048).},
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
issn = {2168-894X},
journal = {Annals of Probability},
number = {3},
pages = {984--1012},
publisher = {Institute of Mathematical Statistics},
title = {{Normal fluctuation in quantum ergodicity for Wigner matrices}},
doi = {10.1214/21-AOP1552},
volume = {50},
year = {2022},
}
@article{11732,
abstract = {We study the BCS energy gap Ξ in the high–density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential V on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3, 2022) on the critical temperature Tc at high densities, we prove the universality of the ratio of the energy gap and the critical temperature.},
author = {Henheik, Sven Joscha and Lauritsen, Asbjørn Bækgaard},
issn = {0022-4715},
journal = {Journal of Statistical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
publisher = {Springer Nature},
title = {{The BCS energy gap at high density}},
doi = {10.1007/s10955-022-02965-9},
volume = {189},
year = {2022},
}
@article{11741,
abstract = {Following E. Wigner’s original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix H yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly, we prove universality for a monoparametric family of deformed Wigner matrices H+xA with a deterministic Hermitian matrix A and a fixed Wigner matrix H, just using the randomness of a single scalar real random variable x. Both results constitute quenched versions of bulk universality that has so far only been proven in annealed sense with respect to the probability space of the matrix ensemble.},
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
issn = {1432-2064},
journal = {Probability Theory and Related Fields},
publisher = {Springer Nature},
title = {{Quenched universality for deformed Wigner matrices}},
doi = {10.1007/s00440-022-01156-7},
year = {2022},
}
@article{12110,
abstract = {A recently proposed approach for avoiding the ultraviolet divergence of Hamiltonians with particle creation is based on interior-boundary conditions (IBCs). The approach works well in the non-relativistic case, i.e., for the Laplacian operator. Here, we study how the approach can be applied to Dirac operators. While this has successfully been done already in one space dimension, and more generally for codimension-1 boundaries, the situation of point sources in three dimensions corresponds to a codimension-3 boundary. One would expect that, for such a boundary, Dirac operators do not allow for boundary conditions because they are known not to allow for point interactions in 3D, which also correspond to a boundary condition. Indeed, we confirm this expectation here by proving that there is no self-adjoint operator on a (truncated) Fock space that would correspond to a Dirac operator with an IBC at configurations with a particle at the origin. However, we also present a positive result showing that there are self-adjoint operators with an IBC (on the boundary consisting of configurations with a particle at the origin) that are away from those configurations, given by a Dirac operator plus a sufficiently strong Coulomb potential.},
author = {Henheik, Sven Joscha and Tumulka, Roderich},
issn = {0022-2488},
journal = {Journal of Mathematical Physics},
number = {12},
publisher = {AIP Publishing},
title = {{Interior-boundary conditions for the Dirac equation at point sources in three dimensions}},
doi = {10.1063/5.0104675},
volume = {63},
year = {2022},
}
@article{12148,
abstract = {We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables.},
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
issn = {2050-5094},
journal = {Forum of Mathematics, Sigma},
keywords = {Computational Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics, Statistics and Probability, Algebra and Number Theory, Theoretical Computer Science, Analysis},
publisher = {Cambridge University Press},
title = {{Rank-uniform local law for Wigner matrices}},
doi = {10.1017/fms.2022.86},
volume = {10},
year = {2022},
}
@article{12184,
abstract = {We review recent results on adiabatic theory for ground states of extended gapped fermionic lattice systems under several different assumptions. More precisely, we present generalized super-adiabatic theorems for extended but finite and infinite systems, assuming either a uniform gap or a gap in the bulk above the unperturbed ground state. The goal of this Review is to provide an overview of these adiabatic theorems and briefly outline the main ideas and techniques required in their proofs.},
author = {Henheik, Sven Joscha and Wessel, Tom},
issn = {0022-2488},
journal = {Journal of Mathematical Physics},
number = {12},
publisher = {AIP Publishing},
title = {{On adiabatic theory for extended fermionic lattice systems}},
doi = {10.1063/5.0123441},
volume = {63},
year = {2022},
}
@article{12179,
abstract = {We derive an accurate lower tail estimate on the lowest singular value σ1(X−z) of a real Gaussian (Ginibre) random matrix X shifted by a complex parameter z. Such shift effectively changes the upper tail behavior of the condition number κ(X−z) from the slower (κ(X−z)≥t)≲1/t decay typical for real Ginibre matrices to the faster 1/t2 decay seen for complex Ginibre matrices as long as z is away from the real axis. This sharpens and resolves a recent conjecture in [J. Banks et al., https://arxiv.org/abs/2005.08930, 2020] on the regularizing effect of the real Ginibre ensemble with a genuinely complex shift. As a consequence we obtain an improved upper bound on the eigenvalue condition numbers (known also as the eigenvector overlaps) for real Ginibre matrices. The main technical tool is a rigorous supersymmetric analysis from our earlier work [Probab. Math. Phys., 1 (2020), pp. 101--146].},
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
issn = {1095-7162},
journal = {SIAM Journal on Matrix Analysis and Applications},
keywords = {Analysis},
number = {3},
pages = {1469--1487},
publisher = {Society for Industrial and Applied Mathematics},
title = {{On the condition number of the shifted real Ginibre ensemble}},
doi = {10.1137/21m1424408},
volume = {43},
year = {2022},
}
@article{12214,
abstract = {Motivated by Kloeckner’s result on the isometry group of the quadratic Wasserstein space W2(Rn), we describe the isometry group Isom(Wp(E)) for all parameters 0 < p < ∞ and for all separable real Hilbert spaces E. In particular, we show that Wp(X) is isometrically rigid for all Polish space X whenever 0 < p < 1. This is a consequence of our more general result: we prove that W1(X) is isometrically rigid if X is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters p > 1, by solving Kloeckner’s problem affirmatively on the existence of mass-splitting isometries. },
author = {Gehér, György Pál and Titkos, Tamás and Virosztek, Daniel},
issn = {1469-7750},
journal = {Journal of the London Mathematical Society},
keywords = {General Mathematics},
number = {4},
pages = {3865--3894},
publisher = {Wiley},
title = {{The isometry group of Wasserstein spaces: The Hilbertian case}},
doi = {10.1112/jlms.12676},
volume = {106},
year = {2022},
}
@article{12232,
abstract = {We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter z as the dimension tends to infinity. For z away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in Cipolloni et al. (Prob Math Phys 1:101–146, 2020). On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles as the shift parameter z becomes genuinely complex; the analogous phenomenon has been well known for eigenvalues. We use the superbosonization formula (Littelmann et al. in Comm Math Phys 283:343–395, 2008) in a regime where the main contribution comes from a three dimensional saddle manifold.},
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
issn = {1424-0661},
journal = {Annales Henri Poincaré},
keywords = {Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics},
number = {11},
pages = {3981--4002},
publisher = {Springer Nature},
title = {{Density of small singular values of the shifted real Ginibre ensemble}},
doi = {10.1007/s00023-022-01188-8},
volume = {23},
year = {2022},
}
@article{12243,
abstract = {We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel distribution and that these extreme eigenvalues form a Poisson point process as the dimension asymptotically tends to infinity. In the complex case, these facts have already been established by Bender [Probab. Theory Relat. Fields 147, 241 (2010)] and in the real case by Akemann and Phillips [J. Stat. Phys. 155, 421 (2014)] even for the more general elliptic ensemble with a sophisticated saddle point analysis. The purpose of this article is to give a very short direct proof in the Ginibre case with an effective error term. Moreover, our estimates on the correlation kernel in this regime serve as a key input for accurately locating [Formula: see text] for any large matrix X with i.i.d. entries in the companion paper [G. Cipolloni et al., arXiv:2206.04448 (2022)]. },
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J and Xu, Yuanyuan},
issn = {1089-7658},
journal = {Journal of Mathematical Physics},
keywords = {Mathematical Physics, Statistical and Nonlinear Physics},
number = {10},
publisher = {AIP Publishing},
title = {{Directional extremal statistics for Ginibre eigenvalues}},
doi = {10.1063/5.0104290},
volume = {63},
year = {2022},
}
@article{12290,
abstract = {We prove local laws, i.e. optimal concentration estimates for arbitrary products of resolvents of a Wigner random matrix with deterministic matrices in between. We find that the size of such products heavily depends on whether some of the deterministic matrices are traceless. Our estimates correctly account for this dependence and they hold optimally down to the smallest possible spectral scale.},
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
issn = {1083-6489},
journal = {Electronic Journal of Probability},
keywords = {Statistics, Probability and Uncertainty, Statistics and Probability},
pages = {1--38},
publisher = {Institute of Mathematical Statistics},
title = {{Optimal multi-resolvent local laws for Wigner matrices}},
doi = {10.1214/22-ejp838},
volume = {27},
year = {2022},
}
@article{10221,
abstract = {We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with very high probability and with an optimal error inversely proportional to the square root of the dimension. Our theorem thus rigorously verifies the Eigenstate Thermalisation Hypothesis by Deutsch (Phys Rev A 43:2046–2049, 1991) for the simplest chaotic quantum system, the Wigner ensemble. In mathematical terms, we prove the strong form of Quantum Unique Ergodicity (QUE) with an optimal convergence rate for all eigenvectors simultaneously, generalizing previous probabilistic QUE results in Bourgade and Yau (Commun Math Phys 350:231–278, 2017) and Bourgade et al. (Commun Pure Appl Math 73:1526–1596, 2020).},
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
issn = {1432-0916},
journal = {Communications in Mathematical Physics},
number = {2},
pages = {1005–1048},
publisher = {Springer Nature},
title = {{Eigenstate thermalization hypothesis for Wigner matrices}},
doi = {10.1007/s00220-021-04239-z},
volume = {388},
year = {2021},
}
@article{10285,
abstract = {We study the overlaps between right and left eigenvectors for random matrices of the spherical ensemble, as well as truncated unitary ensembles in the regime where half of the matrix at least is truncated. These two integrable models exhibit a form of duality, and the essential steps of our investigation can therefore be performed in parallel. In every case, conditionally on all eigenvalues, diagonal overlaps are shown to be distributed as a product of independent random variables with explicit distributions. This enables us to prove that the scaled diagonal overlaps, conditionally on one eigenvalue, converge in distribution to a heavy-tail limit, namely, the inverse of a γ2 distribution. We also provide formulae for the conditional expectation of diagonal and off-diagonal overlaps, either with respect to one eigenvalue, or with respect to the whole spectrum. These results, analogous to what is known for the complex Ginibre ensemble, can be obtained in these cases thanks to integration techniques inspired from a previous work by Forrester & Krishnapur.},
author = {Dubach, Guillaume},
issn = {1083-6489},
journal = {Electronic Journal of Probability},
publisher = {Institute of Mathematical Statistics},
title = {{On eigenvector statistics in the spherical and truncated unitary ensembles}},
doi = {10.1214/21-EJP686},
volume = {26},
year = {2021},
}
@article{10405,
abstract = {We consider large non-Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having 2+ϵ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32]. },
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
issn = {1097-0312},
journal = {Communications on Pure and Applied Mathematics},
publisher = {Wiley},
title = {{Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices}},
doi = {10.1002/cpa.22028},
year = {2021},
}
@article{8373,
abstract = {It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted A#H-mean introduced by Kim, Lawson, and Lim.},
author = {Pitrik, József and Virosztek, Daniel},
issn = {0024-3795},
journal = {Linear Algebra and its Applications},
keywords = {Kubo-Ando mean, weighted multivariate mean, barycenter},
pages = {203--217},
publisher = {Elsevier},
title = {{A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means}},
doi = {10.1016/j.laa.2020.09.007},
volume = {609},
year = {2021},
}
@phdthesis{9022,
abstract = {In the first part of the thesis we consider Hermitian random matrices. Firstly, we consider sample covariance matrices XX∗ with X having independent identically distributed (i.i.d.) centred entries. We prove a Central Limit Theorem for differences of linear statistics of XX∗ and its minor after removing the first column of X. Secondly, we consider Wigner-type matrices and prove that the eigenvalue statistics near cusp singularities of the limiting density of states are universal and that they form a Pearcey process. Since the limiting eigenvalue distribution admits only square root (edge) and cubic root (cusp) singularities, this concludes the third and last remaining case of the Wigner-Dyson-Mehta universality conjecture. The main technical ingredients are an optimal local law at the cusp, and the proof of the fast relaxation to equilibrium of the Dyson Brownian motion in the cusp regime.
In the second part we consider non-Hermitian matrices X with centred i.i.d. entries. We normalise the entries of X to have variance N −1. It is well known that the empirical eigenvalue density converges to the uniform distribution on the unit disk (circular law). In the first project, we prove universality of the local eigenvalue statistics close to the edge of the spectrum. This is the non-Hermitian analogue of the TracyWidom universality at the Hermitian edge. Technically we analyse the evolution of the spectral distribution of X along the Ornstein-Uhlenbeck flow for very long time
(up to t = +∞). In the second project, we consider linear statistics of eigenvalues for macroscopic test functions f in the Sobolev space H2+ϵ and prove their convergence to the projection of the Gaussian Free Field on the unit disk. We prove this result for non-Hermitian matrices with real or complex entries. The main technical ingredients are: (i) local law for products of two resolvents at different spectral parameters, (ii) analysis of correlated Dyson Brownian motions.
In the third and final part we discuss the mathematically rigorous application of supersymmetric techniques (SUSY ) to give a lower tail estimate of the lowest singular value of X − z, with z ∈ C. More precisely, we use superbosonisation formula to give an integral representation of the resolvent of (X − z)(X − z)∗ which reduces to two and three contour integrals in the complex and real case, respectively. The rigorous analysis of these integrals is quite challenging since simple saddle point analysis cannot be applied (the main contribution comes from a non-trivial manifold). Our result
improves classical smoothing inequalities in the regime |z| ≈ 1; this result is essential to prove edge universality for i.i.d. non-Hermitian matrices.},
author = {Cipolloni, Giorgio},
issn = {2663-337X},
pages = {380},
publisher = {IST Austria},
title = {{Fluctuations in the spectrum of random matrices}},
doi = {10.15479/AT:ISTA:9022},
year = {2021},
}
@article{9036,
abstract = {In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.},
author = {Virosztek, Daniel},
issn = {0001-8708},
journal = {Advances in Mathematics},
keywords = {General Mathematics},
number = {3},
publisher = {Elsevier},
title = {{The metric property of the quantum Jensen-Shannon divergence}},
doi = {10.1016/j.aim.2021.107595},
volume = {380},
year = {2021},
}
@article{9412,
abstract = {We extend our recent result [22] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [49] or the first four moments of the matrix elements match the real Gaussian [59, 44]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [22] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.},
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
issn = {10836489},
journal = {Electronic Journal of Probability},
publisher = {Institute of Mathematical Statistics},
title = {{Fluctuation around the circular law for random matrices with real entries}},
doi = {10.1214/21-EJP591},
volume = {26},
year = {2021},
}
@article{9550,
abstract = {We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices. },
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
issn = {20505094},
journal = {Forum of Mathematics, Sigma},
publisher = {Cambridge University Press},
title = {{Equipartition principle for Wigner matrices}},
doi = {10.1017/fms.2021.38},
volume = {9},
year = {2021},
}
@article{9912,
abstract = {In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via 𝑁≪𝑀 channels, the density 𝜌 of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio 𝜙:=𝑁/𝑀≤1; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit 𝜙→0, we recover the formula for the density 𝜌 that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker’s formula persists for any 𝜙<1 but in the borderline case 𝜙=1 an anomalous 𝜆−2/3 singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries.},
author = {Erdös, László and Krüger, Torben H and Nemish, Yuriy},
issn = {1424-0661},
journal = {Annales Henri Poincaré },
pages = {4205–4269},
publisher = {Springer Nature},
title = {{Scattering in quantum dots via noncommutative rational functions}},
doi = {10.1007/s00023-021-01085-6},
volume = {22},
year = {2021},
}
@unpublished{9230,
abstract = {We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length (log T)θ, where θ is either fixed or tends to zero at a suitable rate.
It is shown that the deterministic level of the maximum interpolates smoothly between the ones
of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition ‘from
3/4 to 1/4’ in the second order. This provides a natural context where extreme value statistics of
log-correlated variables with time-dependent variance and rate occur. A key ingredient of the
proof is a precise upper tail tightness estimate for the maximum of the model on intervals of
size one, that includes a Gaussian correction. This correction is expected to be present for the
Riemann zeta function and pertains to the question of the correct order of the maximum of
the zeta function in large intervals.},
author = {Arguin, Louis-Pierre and Dubach, Guillaume and Hartung, Lisa},
booktitle = {arXiv},
title = {{Maxima of a random model of the Riemann zeta function over intervals of varying length}},
doi = {10.48550/arXiv.2103.04817},
year = {2021},
}
@unpublished{9281,
abstract = {We comment on two formal proofs of Fermat's sum of two squares theorem, written using the Mathematical Components libraries of the Coq proof assistant. The first one follows Zagier's celebrated one-sentence proof; the second follows David Christopher's recent new proof relying on partition-theoretic arguments. Both formal proofs rely on a general property of involutions of finite sets, of independent interest. The proof technique consists for the most part of automating recurrent tasks (such as case distinctions and computations on natural numbers) via ad hoc tactics.},
author = {Dubach, Guillaume and Mühlböck, Fabian},
booktitle = {arXiv},
title = {{Formal verification of Zagier's one-sentence proof}},
doi = {10.48550/arXiv.2103.11389},
year = {2021},
}
@article{10862,
abstract = {We consider the sum of two large Hermitian matrices A and B with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of A and B as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [4], [5] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
issn = {0022-1236},
journal = {Journal of Functional Analysis},
keywords = {Analysis},
number = {7},
publisher = {Elsevier},
title = {{Spectral rigidity for addition of random matrices at the regular edge}},
doi = {10.1016/j.jfa.2020.108639},
volume = {279},
year = {2020},
}
@article{6185,
abstract = {For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner–Dyson–Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also the key input in the companion paper (Cipolloni et al. in Pure Appl Anal, 2018. arXiv:1811.04055) where the cusp universality for real symmetric Wigner-type matrices is proven. The novel cusp fluctuation mechanism is also essential for the recent results on the spectral radius of non-Hermitian random matrices (Alt et al. in Spectral radius of random matrices with independent entries, 2019. arXiv:1907.13631), and the non-Hermitian edge universality (Cipolloni et al. in Edge universality for non-Hermitian random matrices, 2019. arXiv:1908.00969).},
author = {Erdös, László and Krüger, Torben H and Schröder, Dominik J},
issn = {1432-0916},
journal = {Communications in Mathematical Physics},
pages = {1203--1278},
publisher = {Springer Nature},
title = {{Cusp universality for random matrices I: Local law and the complex Hermitian case}},
doi = {10.1007/s00220-019-03657-4},
volume = {378},
year = {2020},
}
@article{6184,
abstract = {We prove edge universality for a general class of correlated real symmetric or complex Hermitian Wigner matrices with arbitrary expectation. Our theorem also applies to internal edges of the self-consistent density of states. In particular, we establish a strong form of band rigidity which excludes mismatches between location and label of eigenvalues close to internal edges in these general models.},
author = {Alt, Johannes and Erdös, László and Krüger, Torben H and Schröder, Dominik J},
journal = {Annals of Probability},
number = {2},
pages = {963--1001},
publisher = {Project Euclid},
title = {{Correlated random matrices: Band rigidity and edge universality}},
volume = {48},
year = {2020},
}
@article{6488,
abstract = {We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix W˜ and its minor W. We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of W˜ and W. Our result identifies the fluctuation of the spatial derivative of the approximate Gaussian field in the recent paper by Dumitru and Paquette. Unlike in a similar result for Wigner matrices, for sample covariance matrices, the fluctuation may entirely vanish.},
author = {Cipolloni, Giorgio and Erdös, László},
issn = {20103271},
journal = {Random Matrices: Theory and Application},
number = {3},
publisher = {World Scientific Publishing},
title = {{Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices}},
doi = {10.1142/S2010326320500069},
volume = {9},
year = {2020},
}
@article{7389,
abstract = {Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W_2(R^n). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute Isom(W_p(R)), the isometry group of the Wasserstein space
W_p(R) for all p \in [1,\infty) \setminus {2}. We show that W_2(R) is also exceptional regarding the
parameter p: W_p(R) is isometrically rigid if and only if p is not equal to 2. Regarding the underlying
space, we prove that the exceptionality of p = 2 disappears if we replace R by the compact
interval [0,1]. Surprisingly, in that case, W_p([0,1]) is isometrically rigid if and only if
p is not equal to 1. Moreover, W_1([0,1]) admits isometries that split mass, and Isom(W_1([0,1]))
cannot be embedded into Isom(W_1(R)).},
author = {Geher, Gyorgy Pal and Titkos, Tamas and Virosztek, Daniel},
issn = {10886850},
journal = {Transactions of the American Mathematical Society},
keywords = {Wasserstein space, isometric embeddings, isometric rigidity, exotic isometry flow},
number = {8},
pages = {5855--5883},
publisher = {American Mathematical Society},
title = {{Isometric study of Wasserstein spaces - the real line}},
doi = {10.1090/tran/8113},
volume = {373},
year = {2020},
}
@article{7512,
abstract = {We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these classes of ensembles. In particular, we generalize a similar result of Anderson for anticommutator. For more general polynomials our conditions are effectively checkable numerically.},
author = {Erdös, László and Krüger, Torben H and Nemish, Yuriy},
issn = {10960783},
journal = {Journal of Functional Analysis},
number = {12},
publisher = {Elsevier},
title = {{Local laws for polynomials of Wigner matrices}},
doi = {10.1016/j.jfa.2020.108507},
volume = {278},
year = {2020},
}
@article{7618,
abstract = {This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. (Lett Math Phys 109:1777–1804, 2019) with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences that are of the form ϕ(A,B)=Tr((1−c)A+cB−AσB), where σ is an arbitrary Kubo–Ando mean, and c∈(0,1) is the weight of σ. We note that these divergences belong to the family of maximal quantum f-divergences, and hence are jointly convex, and satisfy the data processing inequality. We derive a characterization of the barycenter of finitely many positive definite operators for these generalized quantum Hellinger divergences. We note that the characterization of the barycenter as the weighted multivariate 1/2-power mean, that was claimed in Bhatia et al. (2019), is true in the case of commuting operators, but it is not correct in the general case. },
author = {Pitrik, Jozsef and Virosztek, Daniel},
issn = {1573-0530},
journal = {Letters in Mathematical Physics},
number = {8},
pages = {2039--2052},
publisher = {Springer Nature},
title = {{Quantum Hellinger distances revisited}},
doi = {10.1007/s11005-020-01282-0},
volume = {110},
year = {2020},
}
@article{8601,
abstract = {We consider large non-Hermitian real or complex random matrices X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of X are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy–Widom distribution at the spectral edges of the Wigner ensemble.},
author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
issn = {14322064},
journal = {Probability Theory and Related Fields},
publisher = {Springer Nature},
title = {{Edge universality for non-Hermitian random matrices}},
doi = {10.1007/s00440-020-01003-7},
year = {2020},
}
@article{9104,
abstract = {We consider the free additive convolution of two probability measures μ and ν on the real line and show that μ ⊞ v is supported on a single interval if μ and ν each has single interval support. Moreover, the density of μ ⊞ ν is proven to vanish as a square root near the edges of its support if both μ and ν have power law behavior with exponents between −1 and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [5].},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
issn = {15658538},
journal = {Journal d'Analyse Mathematique},
pages = {323--348},
publisher = {Springer Nature},
title = {{On the support of the free additive convolution}},
doi = {10.1007/s11854-020-0135-2},
volume = {142},
year = {2020},
}
@article{10879,
abstract = {We study effects of a bounded and compactly supported perturbation on multidimensional continuum random Schrödinger operators in the region of complete localisation. Our main emphasis is on Anderson orthogonality for random Schrödinger operators. Among others, we prove that Anderson orthogonality does occur for Fermi energies in the region of complete localisation with a non-zero probability. This partially confirms recent non-rigorous findings [V. Khemani et al., Nature Phys. 11 (2015), 560–565]. The spectral shift function plays an important role in our analysis of Anderson orthogonality. We identify it with the index of the corresponding pair of spectral projections and explore the consequences thereof. All our results rely on the main technical estimate of this paper which guarantees separate exponential decay of the disorder-averaged Schatten p-norm of χa(f(H)−f(Hτ))χb in a and b. Here, Hτ is a perturbation of the random Schrödinger operator H, χa is the multiplication operator corresponding to the indicator function of a unit cube centred about a∈Rd, and f is in a suitable class of functions of bounded variation with distributional derivative supported in the region of complete localisation for H.},
author = {Dietlein, Adrian M and Gebert, Martin and Müller, Peter},
issn = {1664-039X},
journal = {Journal of Spectral Theory},
keywords = {Random Schrödinger operators, spectral shift function, Anderson orthogonality},
number = {3},
pages = {921--965},
publisher = {European Mathematical Society Publishing House},
title = {{Perturbations of continuum random Schrödinger operators with applications to Anderson orthogonality and the spectral shift function}},
doi = {10.4171/jst/267},
volume = {9},
year = {2019},
}
@article{429,
abstract = {We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.},
author = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H},
issn = {14322064},
journal = {Probability Theory and Related Fields},
number = {1-2},
pages = {293–373},
publisher = {Springer},
title = {{Stability of the matrix Dyson equation and random matrices with correlations}},
doi = {10.1007/s00440-018-0835-z},
volume = {173},
year = {2019},
}
@article{405,
abstract = {We investigate the quantum Jensen divergences from the viewpoint of joint convexity. It turns out that the set of the functions which generate jointly convex quantum Jensen divergences on positive matrices coincides with the Matrix Entropy Class which has been introduced by Chen and Tropp quite recently.},
author = {Virosztek, Daniel},
journal = {Linear Algebra and Its Applications},
pages = {67--78},
publisher = {Elsevier},
title = {{Jointly convex quantum Jensen divergences}},
doi = {10.1016/j.laa.2018.03.002},
volume = {576},
year = {2019},
}
@article{6086,
abstract = {We show that linear analytic cocycles where all Lyapunov exponents are negative infinite are nilpotent. For such one-frequency cocycles we show that they can be analytically conjugated to an upper triangular cocycle or a Jordan normal form. As a consequence, an arbitrarily small analytic perturbation leads to distinct Lyapunov exponents. Moreover, in the one-frequency case where the th Lyapunov exponent is finite and the st negative infinite, we obtain a simple criterion for domination in which case there is a splitting into a nilpotent part and an invertible part.},
author = {Sadel, Christian and Xu, Disheng},
journal = {Ergodic Theory and Dynamical Systems},
number = {4},
pages = {1082--1098},
publisher = {Cambridge University Press},
title = {{Singular analytic linear cocycles with negative infinite Lyapunov exponents}},
doi = {10.1017/etds.2017.52},
volume = {39},
year = {2019},
}
@article{6182,
abstract = {We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields 173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is
a systematic diagrammatic control of a multivariate cumulant expansion.},
author = {Erdös, László and Krüger, Torben H and Schröder, Dominik J},
issn = {20505094},
journal = {Forum of Mathematics, Sigma},
publisher = {Cambridge University Press},
title = {{Random matrices with slow correlation decay}},
doi = {10.1017/fms.2019.2},
volume = {7},
year = {2019},
}
@phdthesis{6179,
abstract = {In the first part of this thesis we consider large random matrices with arbitrary expectation and a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent in the bulk and edge regime. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.
In the second part we consider Wigner-type matrices and show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are uni- versal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner- Dyson-Mehta universality conjecture for the last remaining universality type. Our analysis holds not only for exact cusps, but approximate cusps as well, where an ex- tended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp, and extend the fast relaxation to equilibrium of the Dyson Brow- nian motion to the cusp regime.
In the third and final part we explore the entrywise linear statistics of Wigner ma- trices and identify the fluctuations for a large class of test functions with little regularity. This enables us to study the rectangular Young diagram obtained from the interlacing eigenvalues of the random matrix and its minor, and we find that, despite having the same limit, the fluctuations differ from those of the algebraic Young tableaux equipped with the Plancharel measure.},
author = {Schröder, Dominik J},
pages = {375},
publisher = {IST Austria},
title = {{From Dyson to Pearcey: Universal statistics in random matrix theory}},
doi = {10.15479/AT:ISTA:th6179},
year = {2019},
}
@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{6240,
abstract = {For a general class of large non-Hermitian random block matrices X we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of X as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from (Probab. Theory Related Fields (2018)) offers a unified treatment of many structured matrix ensembles.},
author = {Alt, Johannes and Erdös, László and Krüger, Torben H and Nemish, Yuriy},
issn = {02460203},
journal = {Annales de l'institut Henri Poincare},
number = {2},
pages = {661--696},
publisher = {Institut Henri Poincaré},
title = {{Location of the spectrum of Kronecker random matrices}},
doi = {10.1214/18-AIHP894},
volume = {55},
year = {2019},
}
@article{6511,
abstract = {Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Σ be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189–1217] asserts that the empirical eigenvalue distribution of the matrix X:=UΣV∗ converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in ℂ. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N−1/2+ε and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N).},
author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
issn = {00911798},
journal = {Annals of Probability},
number = {3},
pages = {1270--1334},
publisher = {Institute of Mathematical Statistics},
title = {{Local single ring theorem on optimal scale}},
doi = {10.1214/18-AOP1284},
volume = {47},
year = {2019},
}
@article{6843,
abstract = {The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space Wp(X), where S is a countable discrete metric space and 0