arXiv subjects

László Erdős

Publications and source records attributed to László Erdős.

At least 37 records · Page 2Linked to original sources

arXiv2024

Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case

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 used in the companion paper [arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type matrices is proven.

math.PR↗

arXiv2024

Loschmidt Echo for Deformed Wigner Matrices

We consider two Hamiltonians that are close to each other, $H_1 \approx H_2 $, and analyze the time-decay of the corresponding Loschmidt echo $\mathfrak{M}(t) := |\langle ψ_0, \mathrm{e}^{\mathrm{i} t H_2} \mathrm{e}^{-\mathrm{i} t H_1} ψ_0 \rangle|^2$ that expresses the effect of an imperfect time reversal on the initial state $ψ_0$. Our model Hamiltonians are deformed Wigner matrices that do not share a common eigenbasis. The main tools for our results are two-resolvent laws for such $H_1$ and $H_2$.

math-ph↗

arXiv2024

Quenched universality for deformed Wigner matrices

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.

math-ph↗

arXiv2024

Eigenstate Thermalization Hypothesis for Wigner-type Matrices

We prove the Eigenstate Thermalization Hypothesis for general Wigner-type matrices in the bulk of the self-consistent spectrum, with optimal control on the fluctuations for observables of arbitrary rank. As the main technical ingredient, we prove rank-uniform optimal local laws for one and two resolvents of a Wigner-type matrix with regular observables.

math.PR↗

arXiv2024

Precise asymptotics for the spectral radius of a large random matrix

We consider the spectral radius of a large random matrix $X$ with independent, identically distributed entries. We show that its typical size is given by a precise three-term asymptotics with an optimal error term beyond the radius of the celebrated circular law. The coefficients in this asymptotics are universal but they differ from a similar asymptotics recently proved for the rightmost eigenvalue of $X$ in [29]. To access the more complicated spectral radius, we need to establish a new decorrelation mechanism for the low-lying singular values of $X-z$ for different complex shift parameters $z$ using the Dyson Brownian Motion.

math.PR↗

arXiv2024

Out-of-time-ordered correlators for Wigner matrices

We consider the time evolution of the out-of-time-ordered correlator (OTOC) of two general observables $A$ and $B$ in a mean field chaotic quantum system described by a random Wigner matrix as its Hamiltonian. We rigorously identify three time regimes separated by the physically relevant scrambling and relaxation times. The main feature of our analysis is that we express the error terms in the optimal Schatten (tracial) norms of the observables, allowing us to track the exact dependence of the errors on their rank. In particular, for significantly overlapping observables with low rank the OTOC is shown to exhibit a significant local maximum at the scrambling time, a feature that may not have been noticed in the physics literature before. Our main tool is a novel multi-resolvent local law with Schatten norms that unifies and improves previous local laws involving either the much cruder operator norm (cf. [G. Cipolloni, L. Erdős, D. Schröder. Elect. J. Prob. 27, 1-38, 2022]) or the Hilbert-Schmidt norm (cf. [G. Cipolloni, L. Erdős, D. Schröder. Forum Math., Sigma 10, E96, 2022]).

math-ph↗

arXiv2024

Fluctuation Around the Circular Law for Random Matrices with Real Entries

We extend our recent result [Cipolloni, Erdős, Schröder 2019] 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 [O'Rourke, Renfrew 2016] or the first four moments of the matrix elements match the real Gaussian [Tao, Vu 2015; Kopel 2015]. 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 [Cipolloni, Erdős, Schröder 2019] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.

math.PR↗

arXiv2024

Optimal Lower Bound on Eigenvector Overlaps for non-Hermitian Random Matrices

We consider large non-Hermitian $N\times N$ matrices with an additive independent, identically distributed (i.i.d.) noise for each matrix elements. We show that already a small noise of variance $1/N$ completely thermalises the bulk singular vectors, in particular they satisfy the strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence. In physics terms, we thus extend the Eigenstate Thermalisation Hypothesis, formulated originally by [Deutsch 1991] and proven for Wigner matrices in [Cipolloni, Erdős, Schröder 2020], to arbitrary non-Hermitian matrices with an i.i.d. noise. As a consequence we obtain an optimal lower bound on the diagonal overlaps of the corresponding non-Hermitian eigenvectors. This quantity, also known as the (square of the) eigenvalue condition number measuring the sensitivity of the eigenvalue to small perturbations, has notoriously escaped rigorous treatment beyond the explicitly computable Ginibre ensemble apart from the very recent upper bounds given in [arXiv:2005.08930] and [arXiv:2005.08908]. As a key tool, we develop a new systematic decomposition of general observables in random matrix theory that governs the size of products of resolvents with deterministic matrices in between.

math.PR↗

arXiv2023

Prethermalization for Deformed Wigner Matrices

We prove that a class of weakly perturbed Hamiltonians of the form $H_λ= H_0 + λW$, with $W$ being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by $H_λ$ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order $λ^{-2}$. Moreover, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent law for the deformed Wigner matrix $H_λ$.

math-ph↗

arXiv2023

Gaussian fluctuations in the Equipartition Principle for Wigner matrices

The total energy of an eigenstate in a composite quantum system tends to be distributed equally among its constituents. We identify the quantum fluctuation around this equipartition principle in the simplest disordered quantum system consisting of linear combinations of Wigner matrices. As our main ingredient, we prove the Eigenstate Thermalisation Hypothesis and Gaussian fluctuation for general quadratic forms of the bulk eigenvectors of Wigner matrices with an arbitrary deformation.

math-ph↗

arXiv2023

Central Limit Theorem for Linear Eigenvalue Statistics of non-Hermitian Random Matrices

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 [Rider, Silverstein 2006], or the distribution of the matrix elements needed to be Gaussian [Rider, Virág 2007], or at least match the Gaussian up to the first four moments [Tao, Vu 2016; Kopel 2015]. 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 [Cipolloni, Erdős, Schröder 2019].

math.PR↗

arXiv2023

Rank-uniform local law for Wigner matrices

We prove a general local law for Wigner matrices which optimally handles observables of arbitrary rank and thus it unifies the well-known averaged and isotropic local laws. As an application, we prove that the quadratic forms of a general deterministic matrix $A$ on the bulk eigenvectors of a Wigner matrix has approximately Gaussian fluctuation. For the bulk spectrum, we thus generalize our previous result [arXiv:2103.06730] valid for test matrices $A$ of large rank as well as the result of Benigni and Lopatto [arXiv:2103.12013] valid for specific small rank observables.

math.PR↗

arXiv2023

On the rightmost eigenvalue of non-Hermitian random matrices

We establish a precise three-term asymptotic expansion, with an optimal estimate of the error term, for the rightmost eigenvalue of an $n\times n$ random matrix with independent identically distributed complex entries as $n$ tends to infinity. All terms in the expansion are universal.

math.PR↗

arXiv2023

Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices

We consider $N\times N$ non-Hermitian random matrices of the form $X+A$, where $A$ is a general deterministic matrix and $\sqrt{N}X$ consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, i.e. that the local density of eigenvalues is bounded by $N^{1+o(1)}$ and (ii) that the expected condition number of any bulk eigenvalue is bounded by $N^{1+o(1)}$; both results are optimal up to the factor $N^{o(1)}$. The latter result complements the very recent matching lower bound obtained in [15] (arXiv:2301.03549) and improves the $N$-dependence of the upper bounds in [5,6,32] (arXiv:1906.11819, arXiv:2005.08930, arXiv:2005.08908). Our main ingredient, a near-optimal lower tail estimate for the small singular values of $X+A-z$, is of independent interest.

math.PR↗

arXiv2023

Functional Central Limit Theorems for Wigner Matrices

We consider the fluctuations of regular functions $f$ of a Wigner matrix $W$ viewed as an entire matrix $f(W)$. Going beyond the well studied tracial mode, $\mathrm{Tr}[f(W)]$, which is equivalent to the customary linear statistics of eigenvalues, we show that $\mathrm{Tr}[f(W)]$ is asymptotically normal for any non-trivial bounded deterministic matrix $A$. We identify three different and asymptotically independent modes of this fluctuation, corresponding to the tracial part, the traceless diagonal part and the off-diagonal part of $f(W)$ in the entire mesoscopic regime, where we find that the off-diagonal modes fluctuate on a much smaller scale than the tracial mode. In addition, we determine the fluctuations in the Eigenstate Thermalisation Hypothesis [Deutsch 1991], i.e. prove that the eigenfunction overlaps with any deterministic matrix are asymptotically Gaussian after a small spectral averaging. In particular, in the macroscopic regime our result generalises [Lytova 2013] to complex $W$ and to all crossover ensembles in between. The main technical inputs are the recent multi-resolvent local laws with traceless deterministic matrices from the companion paper [Cipolloni, Erdős, Schröder 2020].

math.PR↗

arXiv2023

Eigenstate Thermalisation Hypothesis for Translation Invariant Spin Systems

We prove the Eigenstate Thermalisation Hypothesis (ETH) for local observables in a typical translation invariant system of quantum spins with mean field interaction. This mathematically verifies the observation made in [L.Santos and M.Rigol, Phys.Rev.E 82, 031130 (2010, https://journals.aps.org/pre/abstract/10.1103/PhysRevE.82.031130)] that ETH may hold for systems with additional translation symmetries for a naturally restricted class of observables. We also present numerical support for the same phenomenon for Hamiltonians with local interactions.

cond-mat.stat-mech↗

arXiv2023

On the Spectral Form Factor for Random Matrices

In the physics literature the spectral form factor (SFF), the squared Fourier transform of the empirical eigenvalue density, is the most common tool to test universality for disordered quantum systems, yet previous mathematical results have been restricted only to two exactly solvable models [Forrester 2020]. We rigorously prove the physics prediction on SFF up to an intermediate time scale for a large class of random matrices using a robust method, the multi-resolvent local laws. Beyond Wigner matrices we also consider the monoparametric ensemble and prove that universality of SFF can already be triggered by a single random parameter, extending the recently proven Wigner-Dyson universality [Cipolloni, Erdős, Schröder 2021] to some larger spectral scales. Remarkably, extensive numerics indicates that our formulas correctly predict the SFF in the entire slope-dip-ramp regime, as customarily called in physics.

math-ph↗

arXiv2023

Dynamics of a rank-one perturbation of a Hermitian matrix

We study the eigenvalue trajectories of a time dependent matrix $ G_t = H+i t vv^*$ for $t \geq 0$, where $H$ is an $N \times N$ Hermitian random matrix and $v$ is a unit vector. In particular, we establish that with high probability, an outlier can be distinguished at all times $t>1+N^{-1/3+ε}$, for any $ε>0$. The study of this natural process combines elements of Hermitian and non-Hermitian analysis, and illustrates some aspects of the intrinsic instability of (even weakly) non-Hermitian matrices.

math.PR↗