TY - JOUR
AB - Only recently has it been possible to construct a self-adjoint Hamiltonian that involves the creation of Dirac particles at a point source in 3d space. Its definition makes use of an interior-boundary condition. Here, we develop for this Hamiltonian a corresponding theory of the Bohmian configuration. That is, we (non-rigorously) construct a Markov jump process $(Q_t)_{t\in\mathbb{R}}$ in the configuration space of a variable number of particles that is $|\psi_t|^2$-distributed at every time t and follows Bohmian trajectories between the jumps. The jumps correspond to particle creation or annihilation events and occur either to or from a configuration with a particle located at the source. The process is the natural analog of Bell's jump process, and a central piece in its construction is the determination of the rate of particle creation. The construction requires an analysis of the asymptotic behavior of the Bohmian trajectories near the source. We find that the particle reaches the source with radial speed 0, but orbits around the source infinitely many times in finite time before absorption (or after emission).
AU - Henheik, Sven Joscha
AU - Tumulka, Roderich
ID - 14421
IS - 44
JF - Journal of Physics A: Mathematical and Theoretical
SN - 1751-8113
TI - Creation rate of Dirac particles at a point source
VL - 56
ER -
TY - JOUR
AB - It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature Ξ
and the critical temperature Tc is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density and high density. The goal of this short note is to extend the universal behavior to lower dimensions d=1,2 and give an exemplary proof in the weak coupling limit.
AU - Henheik, Sven Joscha
AU - Lauritsen, Asbjørn Bækgaard
AU - Roos, Barbara
ID - 14542
JF - Reviews in Mathematical Physics
SN - 0129-055X
TI - Universality in low-dimensional BCS theory
ER -
TY - JOUR
AB - For large dimensional non-Hermitian random matrices X with real or complex independent, identically distributed, centered entries, we consider the fluctuations of f (X) as a matrix where f is an analytic function around the spectrum of X. We prove that for a generic bounded square matrix A, the quantity Tr f (X)A exhibits Gaussian fluctuations as the matrix size grows to infinity, which consists of two independent modes corresponding to the tracial and traceless parts of A. We find a new formula for the variance of the traceless part that involves the Frobenius norm of A and the L2-norm of f on the boundary of the limiting spectrum.
AU - Erdös, László
AU - Ji, Hong Chang
ID - 14667
IS - 4
JF - Annales de l'institut Henri Poincare (B) Probability and Statistics
SN - 0246-0203
TI - Functional CLT for non-Hermitian random matrices
VL - 59
ER -
TY - JOUR
AB - 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].
AU - Cipolloni, Giorgio
AU - Erdös, László
AU - Schröder, Dominik J
ID - 12179
IS - 3
JF - SIAM Journal on Matrix Analysis and Applications
KW - Analysis
SN - 0895-4798
TI - On the condition number of the shifted real Ginibre ensemble
VL - 43
ER -
TY - JOUR
AB - We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalized super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians, which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem also holds for certain perturbations of gapped ground states that close the spectral gap (so it is also an adiabatic theorem for resonances and, in this sense, “generalized”), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called “super-adiabatic”). In addition to the existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations for infinite systems. While we consider the result and its proof as new and interesting in itself, we also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article.
AU - Henheik, Sven Joscha
AU - Teufel, Stefan
ID - 10600
IS - 1
JF - Journal of Mathematical Physics
KW - mathematical physics
KW - statistical and nonlinear physics
SN - 0022-2488
TI - Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap
VL - 63
ER -
TY - JOUR
AB - Based on a result by Yarotsky (J Stat Phys 118, 2005), we prove that localized but otherwise arbitrary perturbations of weakly interacting quantum spin systems with uniformly gapped on-site terms change the ground state of such a system only locally, even if they close the spectral gap. We call this a strong version of the local perturbations perturb locally (LPPL) principle which is known to hold for much more general gapped systems, but only for perturbations that do not close the spectral gap of the Hamiltonian. We also extend this strong LPPL-principle to Hamiltonians that have the appropriate structure of gapped on-site terms and weak interactions only locally in some region of space. While our results are technically corollaries to a theorem of Yarotsky, we expect that the paradigm of systems with a locally gapped ground state that is completely insensitive to the form of the Hamiltonian elsewhere extends to other situations and has important physical consequences.
AU - Henheik, Sven Joscha
AU - Teufel, Stefan
AU - Wessel, Tom
ID - 10642
IS - 1
JF - Letters in Mathematical Physics
KW - mathematical physics
KW - statistical and nonlinear physics
SN - 0377-9017
TI - Local stability of ground states in locally gapped and weakly interacting quantum spin systems
VL - 112
ER -
TY - JOUR
AB - We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding Gelfand–Naimark–Segal Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite-volume Hamiltonians need not have a spectral gap.
AU - Henheik, Sven Joscha
AU - Teufel, Stefan
ID - 10643
JF - Forum of Mathematics, Sigma
KW - computational mathematics
KW - discrete mathematics and combinatorics
KW - geometry and topology
KW - mathematical physics
KW - statistics and probability
KW - algebra and number theory
KW - theoretical computer science
KW - analysis
TI - Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk
VL - 10
ER -
TY - JOUR
AB - We investigate the BCS critical temperature Tc in the high-density limit and derive an asymptotic formula, which strongly depends on the behavior of the interaction potential V on the Fermi-surface. Our results include a rigorous confirmation for the behavior of Tc at high densities proposed by Langmann et al. (Phys Rev Lett 122:157001, 2019) and identify precise conditions under which superconducting domes arise in BCS theory.
AU - Henheik, Sven Joscha
ID - 10623
IS - 1
JF - Mathematical Physics, Analysis and Geometry
KW - geometry and topology
KW - mathematical physics
SN - 1385-0172
TI - The BCS critical temperature at high density
VL - 25
ER -
TY - JOUR
AB - We compute the deterministic approximation of products of Sobolev functions of large Wigner matrices W and provide an optimal error bound on their fluctuation with very high probability. This generalizes Voiculescu's seminal theorem from polynomials to general Sobolev functions, as well as from tracial quantities to individual matrix elements. Applying the result to eitW for large t, we obtain a precise decay rate for the overlaps of several deterministic matrices with temporally well separated Heisenberg time evolutions; thus we demonstrate the thermalisation effect of the unitary group generated by Wigner matrices.
AU - Cipolloni, Giorgio
AU - Erdös, László
AU - Schröder, Dominik J
ID - 10732
IS - 8
JF - Journal of Functional Analysis
SN - 0022-1236
TI - Thermalisation for Wigner matrices
VL - 282
ER -
TY - JOUR
AB - We consider a correlated NxN Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one.
AU - Reker, Jana
ID - 11135
IS - 4
JF - Random Matrices: Theory and Applications
KW - Discrete Mathematics and Combinatorics
KW - Statistics
KW - Probability and Uncertainty
KW - Statistics and Probability
KW - Algebra and Number Theory
SN - 2010-3263
TI - On the operator norm of a Hermitian random matrix with correlated entries
VL - 11
ER -
TY - JOUR
AB - 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.
AU - Schnelli, Kevin
AU - Xu, Yuanyuan
ID - 11332
JF - Communications in Mathematical Physics
SN - 0010-3616
TI - Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices
VL - 393
ER -
TY - JOUR
AB - 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).
AU - Cipolloni, Giorgio
AU - Erdös, László
AU - Schröder, Dominik J
ID - 11418
IS - 3
JF - Annals of Probability
SN - 0091-1798
TI - Normal fluctuation in quantum ergodicity for Wigner matrices
VL - 50
ER -
TY - JOUR
AB - 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.
AU - Henheik, Sven Joscha
AU - Tumulka, Roderich
ID - 12110
IS - 12
JF - Journal of Mathematical Physics
SN - 0022-2488
TI - Interior-boundary conditions for the Dirac equation at point sources in three dimensions
VL - 63
ER -
TY - JOUR
AB - 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.
AU - Cipolloni, Giorgio
AU - Erdös, László
AU - Schröder, Dominik J
ID - 12148
JF - Forum of Mathematics, Sigma
KW - Computational Mathematics
KW - Discrete Mathematics and Combinatorics
KW - Geometry and Topology
KW - Mathematical Physics
KW - Statistics and Probability
KW - Algebra and Number Theory
KW - Theoretical Computer Science
KW - Analysis
SN - 2050-5094
TI - Rank-uniform local law for Wigner matrices
VL - 10
ER -
TY - JOUR
AB - 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.
AU - Henheik, Sven Joscha
AU - Wessel, Tom
ID - 12184
IS - 12
JF - Journal of Mathematical Physics
SN - 0022-2488
TI - On adiabatic theory for extended fermionic lattice systems
VL - 63
ER -
TY - JOUR
AB - 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.
AU - Gehér, György Pál
AU - Titkos, Tamás
AU - Virosztek, Daniel
ID - 12214
IS - 4
JF - Journal of the London Mathematical Society
KW - General Mathematics
SN - 0024-6107
TI - The isometry group of Wasserstein spaces: The Hilbertian case
VL - 106
ER -
TY - JOUR
AB - 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.
AU - Cipolloni, Giorgio
AU - Erdös, László
AU - Schröder, Dominik J
ID - 12232
IS - 11
JF - Annales Henri Poincaré
KW - Mathematical Physics
KW - Nuclear and High Energy Physics
KW - Statistical and Nonlinear Physics
SN - 1424-0637
TI - Density of small singular values of the shifted real Ginibre ensemble
VL - 23
ER -
TY - JOUR
AB - 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)].
AU - Cipolloni, Giorgio
AU - Erdös, László
AU - Schröder, Dominik J
AU - Xu, Yuanyuan
ID - 12243
IS - 10
JF - Journal of Mathematical Physics
KW - Mathematical Physics
KW - Statistical and Nonlinear Physics
SN - 0022-2488
TI - Directional extremal statistics for Ginibre eigenvalues
VL - 63
ER -
TY - JOUR
AB - 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.
AU - Cipolloni, Giorgio
AU - Erdös, László
AU - Schröder, Dominik J
ID - 12290
JF - Electronic Journal of Probability
KW - Statistics
KW - Probability and Uncertainty
KW - Statistics and Probability
TI - Optimal multi-resolvent local laws for Wigner matrices
VL - 27
ER -
TY - JOUR
AB - 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.
AU - Henheik, Sven Joscha
AU - Lauritsen, Asbjørn Bækgaard
ID - 11732
JF - Journal of Statistical Physics
KW - Mathematical Physics
KW - Statistical and Nonlinear Physics
SN - 0022-4715
TI - The BCS energy gap at high density
VL - 189
ER -
TY - JOUR
AB - 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.
AU - Dubach, Guillaume
ID - 10285
JF - Electronic Journal of Probability
TI - On eigenvector statistics in the spherical and truncated unitary ensembles
VL - 26
ER -
TY - GEN
AB - 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.
AU - Arguin, Louis-Pierre
AU - Dubach, Guillaume
AU - Hartung, Lisa
ID - 9230
T2 - arXiv
TI - Maxima of a random model of the Riemann zeta function over intervals of varying length
ER -
TY - GEN
AB - 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.
AU - Dubach, Guillaume
AU - Mühlböck, Fabian
ID - 9281
T2 - arXiv
TI - Formal verification of Zagier's one-sentence proof
ER -
TY - JOUR
AB - 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.
AU - Pitrik, József
AU - Virosztek, Daniel
ID - 8373
JF - Linear Algebra and its Applications
KW - Kubo-Ando mean
KW - weighted multivariate mean
KW - barycenter
SN - 0024-3795
TI - A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means
VL - 609
ER -
TY - JOUR
AB - 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.
AU - Virosztek, Daniel
ID - 9036
IS - 3
JF - Advances in Mathematics
KW - General Mathematics
SN - 0001-8708
TI - The metric property of the quantum Jensen-Shannon divergence
VL - 380
ER -
TY - JOUR
AB - 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.
AU - Cipolloni, Giorgio
AU - Erdös, László
AU - Schröder, Dominik J
ID - 9412
JF - Electronic Journal of Probability
TI - Fluctuation around the circular law for random matrices with real entries
VL - 26
ER -
TY - JOUR
AB - 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.
AU - Bao, Zhigang
AU - Erdös, László
AU - Schnelli, Kevin
ID - 9550
JF - Forum of Mathematics, Sigma
TI - Equipartition principle for Wigner matrices
VL - 9
ER -
TY - JOUR
AB - 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.
AU - Erdös, László
AU - Krüger, Torben H
AU - Nemish, Yuriy
ID - 9912
JF - Annales Henri Poincaré
SN - 1424-0637
TI - Scattering in quantum dots via noncommutative rational functions
VL - 22
ER -
TY - JOUR
AB - 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).
AU - Cipolloni, Giorgio
AU - Erdös, László
AU - Schröder, Dominik J
ID - 10221
IS - 2
JF - Communications in Mathematical Physics
SN - 0010-3616
TI - Eigenstate thermalization hypothesis for Wigner matrices
VL - 388
ER -
TY - THES
AB - 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.
AU - Cipolloni, Giorgio
ID - 9022
SN - 2663-337X
TI - Fluctuations in the spectrum of random matrices
ER -
TY - JOUR
AB - 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)).
AU - Geher, Gyorgy Pal
AU - Titkos, Tamas
AU - Virosztek, Daniel
ID - 7389
IS - 8
JF - Transactions of the American Mathematical Society
KW - Wasserstein space
KW - isometric embeddings
KW - isometric rigidity
KW - exotic isometry flow
SN - 00029947
TI - Isometric study of Wasserstein spaces - the real line
VL - 373
ER -
TY - JOUR
AB - 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.
AU - Erdös, László
AU - Krüger, Torben H
AU - Nemish, Yuriy
ID - 7512
IS - 12
JF - Journal of Functional Analysis
SN - 00221236
TI - Local laws for polynomials of Wigner matrices
VL - 278
ER -
TY - JOUR
AB - 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.
AU - Pitrik, Jozsef
AU - Virosztek, Daniel
ID - 7618
IS - 8
JF - Letters in Mathematical Physics
SN - 0377-9017
TI - Quantum Hellinger distances revisited
VL - 110
ER -
TY - JOUR
AB - 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.
AU - Cipolloni, Giorgio
AU - Erdös, László
AU - Schröder, Dominik J
ID - 8601
JF - Probability Theory and Related Fields
SN - 01788051
TI - Edge universality for non-Hermitian random matrices
ER -
TY - JOUR
AB - 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].
AU - Bao, Zhigang
AU - Erdös, László
AU - Schnelli, Kevin
ID - 9104
JF - Journal d'Analyse Mathematique
SN - 00217670
TI - On the support of the free additive convolution
VL - 142
ER -
TY - JOUR
AB - 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.
AU - Bao, Zhigang
AU - Erdös, László
AU - Schnelli, Kevin
ID - 10862
IS - 7
JF - Journal of Functional Analysis
KW - Analysis
SN - 0022-1236
TI - Spectral rigidity for addition of random matrices at the regular edge
VL - 279
ER -
TY - JOUR
AB - 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.
AU - Cipolloni, Giorgio
AU - Erdös, László
ID - 6488
IS - 3
JF - Random Matrices: Theory and Application
SN - 20103263
TI - Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices
VL - 9
ER -
TY - JOUR
AB - 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).
AU - Erdös, László
AU - Krüger, Torben H
AU - Schröder, Dominik J
ID - 6185
JF - Communications in Mathematical Physics
SN - 0010-3616
TI - Cusp universality for random matrices I: Local law and the complex Hermitian case
VL - 378
ER -
TY - JOUR
AB - 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.
AU - Alt, Johannes
AU - Erdös, László
AU - Krüger, Torben H
AU - Schröder, Dominik J
ID - 6184
IS - 2
JF - Annals of Probability
SN - 0091-1798
TI - Correlated random matrices: Band rigidity and edge universality
VL - 48
ER -
TY - CONF
AB - The aim of this short note is to expound one particular issue that was discussed during the talk [10] given at the symposium ”Researches on isometries as preserver problems and related topics” at Kyoto RIMS. That is, the role of Dirac masses by describing the isometry group of various metric spaces of probability measures. This article is of survey character, and it does not contain any essentially new results.From an isometric point of view, in some cases, metric spaces of measures are similar to C(K)-type function spaces. Similarity means here that their isometries are driven by some nice transformations of the underlying space. Of course, it depends on the particular choice of the metric how nice these transformations should be. Sometimes, as we will see, being a homeomorphism is enough to generate an isometry. But sometimes we need more: the transformation must preserve the underlying distance as well. Statements claiming that isometries in questions are necessarily induced by homeomorphisms are called Banach-Stone-type results, while results asserting that the underlying transformation is necessarily an isometry are termed as isometric rigidity results.As Dirac masses can be considered as building bricks of the set of all Borel measures, a natural question arises:Is it enough to understand how an isometry acts on the set of Dirac masses? Does this action extend uniquely to all measures?In what follows, we will thoroughly investigate this question.
AU - Geher, Gyorgy Pal
AU - Titkos, Tamas
AU - Virosztek, Daniel
ID - 7035
T2 - Kyoto RIMS Kôkyûroku
TI - Dirac masses and isometric rigidity
VL - 2125
ER -
TY - CONF
AB - We study edge asymptotics of poissonized Plancherel-type measures on skew Young diagrams (integer partitions). These measures can be seen as generalizations of those studied by Baik--Deift--Johansson and Baik--Rains in resolving Ulam's problem on longest increasing subsequences of random permutations and the last passage percolation (corner growth) discrete versions thereof. Moreover they interpolate between said measures and the uniform measure on partitions. In the new KPZ-like 1/3 exponent edge scaling limit with logarithmic corrections, we find new probability distributions generalizing the classical Tracy--Widom GUE, GOE and GSE distributions from the theory of random matrices.
AU - Betea, Dan
AU - Bouttier, Jérémie
AU - Nejjar, Peter
AU - Vuletíc, Mirjana
ID - 8175
T2 - Proceedings on the 31st International Conference on Formal Power Series and Algebraic Combinatorics
TI - New edge asymptotics of skew Young diagrams via free boundaries
ER -
TY - JOUR
AB - 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.
AU - Virosztek, Daniel
ID - 405
JF - Linear Algebra and Its Applications
TI - Jointly convex quantum Jensen divergences
VL - 576
ER -
TY - JOUR
AB - 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.
AU - Ajanki, Oskari H
AU - Erdös, László
AU - Krüger, Torben H
ID - 429
IS - 1-2
JF - Probability Theory and Related Fields
SN - 01788051
TI - Stability of the matrix Dyson equation and random matrices with correlations
VL - 173
ER -
TY - JOUR
AB - 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.
AU - Sadel, Christian
AU - Xu, Disheng
ID - 6086
IS - 4
JF - Ergodic Theory and Dynamical Systems
TI - Singular analytic linear cocycles with negative infinite Lyapunov exponents
VL - 39
ER -
TY - JOUR
AB - 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).
AU - Bao, Zhigang
AU - Erdös, László
AU - Schnelli, Kevin
ID - 6511
IS - 3
JF - Annals of Probability
SN - 00911798
TI - Local single ring theorem on optimal scale
VL - 47
ER -
TY - JOUR
AB - 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