@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 đâȘđ channels, the density đ 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 đ:=đ/đâ€1; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit đâ0, we recover the formula for the density đ that Beenakker (Rev Mod Phys 69:731â808, 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 Beenakkerâs formula persists for any đ<1 but in the borderline case đ=1 an anomalous đâ2/3 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 = {ErdĂ¶s, LĂĄszlĂł and KrĂŒger, Torben H and Nemish, Yuriy},
issn = {1424-0661},
journal = {Annales Henri PoincarĂ© },
pages = {4205â4269},
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@unpublished{9230,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)Îž, where Îž 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 âfrom
3/4 to 1/4â 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 MĂŒhlbĂ¶ck, 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 ErdĂ¶s, LĂĄszlĂł 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 WignerâDysonâMehta 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 = {ErdĂ¶s, LĂĄszlĂł and KrĂŒger, Torben H and SchrĂ¶der, 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 ErdĂ¶s, LĂĄszlĂł and KrĂŒger, Torben H and SchrĂ¶der, 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 WË 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 WË 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 ErdĂ¶s, LĂĄszlĂł},
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 = {ErdĂ¶s, LĂĄszlĂł and KrĂŒger, 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 109:1777â1804, 2019) with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences that are of the form Ï(A,B)=Tr((1âc)A+cBâAÏB), where Ï is an arbitrary KuboâAndo mean, and câ(0,1) is the weight of Ï. 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 TracyâWidom distribution at the spectral edges of the Wigner ensemble.},
author = {Cipolloni, Giorgio and ErdĂ¶s, LĂĄszlĂł and SchrĂ¶der, 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 ÎŒ and Îœ on the real line and show that ÎŒ â v is supported on a single interval if ÎŒ and Îœ each has single interval support. Moreover, the density of ÎŒ â Îœ is proven to vanish as a square root near the edges of its support if both ÎŒ and Îœ have power law behavior with exponents between â1 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 ErdĂ¶s, LĂĄszlĂł 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 SchrĂ¶dinger operators in the region of complete localisation. Our main emphasis is on Anderson orthogonality for random SchrĂ¶dinger 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), 560â565]. 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 Ïa(f(H)âf(HÏ))Ïb in a and b. Here, HÏ is a perturbation of the random SchrĂ¶dinger operator H, Ïa is the multiplication operator corresponding to the indicator function of a unit cube centred about aâRd, 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 MĂŒller, Peter},
issn = {1664-039X},
journal = {Journal of Spectral Theory},
keywords = {Random SchrĂ¶dinger operators, spectral shift function, Anderson orthogonality},
number = {3},
pages = {921--965},
publisher = {European Mathematical Society Publishing House},
title = {{Perturbations of continuum random SchrĂ¶dinger 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 ErdĂ¶s, LĂĄszlĂł and KrĂŒger, Torben H},
issn = {14322064},
journal = {Probability Theory and Related Fields},
number = {1-2},
pages = {293â373},
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. [âStability of the matrix Dyson equation and random matrices with correlationsâ, Probab. Theory Related Fields 173(1â2) (2019), 293â373] to allow slow correlation decay and arbitrary expectation. The main novel tool is
a systematic diagrammatic control of a multivariate cumulant expansion.},
author = {ErdĂ¶s, LĂĄszlĂł and KrĂŒger, Torben H and SchrĂ¶der, 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 = {SchrĂ¶der, 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 ErdĂ¶s, LĂĄszlĂł and KrĂŒger, Torben H and SchrĂ¶der, Dominik J},
issn = {2578-5885},
journal = {Pure and Applied Analysis },
number = {4},
pages = {615â707},
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 ErdĂ¶s, LĂĄszlĂł and KrĂŒger, Torben H and Nemish, Yuriy},
issn = {02460203},
journal = {Annales de l'institut Henri Poincare},
number = {2},
pages = {661--696},
publisher = {Institut Henri PoincarĂ©},
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 ÎŁ be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189â1217] asserts that the empirical eigenvalue distribution of the matrix X:=UÎŁVâ converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in â. 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 Nâ1/2+Î” 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 ErdĂ¶s, LĂĄszlĂł 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