Giorgio Cipolloni
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Giorgio Cipolloni has a total of 13 co-authors. Here are the top co-authors by number of shared publications:
- László Erdős
Institute of Science and Technology Austria, Klosterneuburg, Austria
30 shared publications - Dominik Schröder
Department of Mathematics ETH Zurich 8006 Zürich Switzerland
25 shared publications - Jonah Kudler-Flam
Institute for Advanced Study
4 shared publications - Joscha Henheik
Institute of Science and Technology Austria, Klosterneuburg, Austria
4 shared publications - Yuanyuan Xu
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, P. R. China
4 shared publications - Michael Aizenman
Departments of Physics and Mathematics, Princeton University, Princeton, USA
3 shared publications
Latest Publications
- Open Access
- 8 August 2024
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 114 (4) , 1-3
Correction to: Ruminations on matrix convexity and the strong subadditivity of quantum entropy
- 8 August 2024
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 114 (4) , 1-3
Open AccessCorrection to: Ruminations on matrix convexity and the strong subadditivity of quantum entropy
- 1 June 2024
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 65 (6)
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 Cipolloni et al., Ann. Probab. 51(6), 2192–2242 (2023). 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.
- 1 June 2024
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 65 (6)
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 Cipolloni et al., Ann. Probab. 51(6), 2192–2242 (2023). 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.
- Open Access
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.Open AccessQuenched 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. - Open Access
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.Open AccessPrecise 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. - Open Access
Mesoscopic Central Limit Theorem for non-Hermitian Random Matrices
We prove that the mesoscopic linear statistics $\sum_i f(n^a(\sigma_i-z_0))$ of the eigenvalues $\{\sigma_i\}_i$ of large $n\times n$ non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any $H^{2}_0$-functions $f$ around any point $z_0$ in the bulk of the spectrum on any mesoscopic scale $0<a<1/2$. This extends our previous result [arXiv:1912.04100], that was valid on the macroscopic scale, $a=0$, to cover the entire mesoscopic regime. The main novelty is a local law for the product of resolvents for the Hermitization of $X$ at spectral parameters $z_1, z_2$ with an improved error term in the entire mesoscopic regime $|z_1-z_2|\gg n^{-1/2}$. The proof is dynamical; it relies on a recursive tandem of the characteristic flow method and the Green function comparison idea combined with a separation of the unstable mode of the underlying stability operator.Open AccessMesoscopic Central Limit Theorem for non-Hermitian Random Matrices
We prove that the mesoscopic linear statistics $\sum_i f(n^a(\sigma_i-z_0))$ of the eigenvalues $\{\sigma_i\}_i$ of large $n\times n$ non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any $H^{2}_0$-functions $f$ around any point $z_0$ in the bulk of the spectrum on any mesoscopic scale $0<a<1/2$. This extends our previous result [arXiv:1912.04100], that was valid on the macroscopic scale, $a=0$, to cover the entire mesoscopic regime. The main novelty is a local law for the product of resolvents for the Hermitization of $X$ at spectral parameters $z_1, z_2$ with an improved error term in the entire mesoscopic regime $|z_1-z_2|\gg n^{-1/2}$. The proof is dynamical; it relies on a recursive tandem of the characteristic flow method and the Green function comparison idea combined with a separation of the unstable mode of the underlying stability operator. - Open Access
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]).Open AccessOut-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]). - 15 February 2024
- Published by Springer Nature in Journal of Statistical Physics
- Vol. 191 (2) , 1-14
The Dissipative Spectral Form Factor for I.I.D. Matrices
- 15 February 2024
- Published by Springer Nature in Journal of Statistical Physics
- Vol. 191 (2) , 1-14
The Dissipative Spectral Form Factor for I.I.D. Matrices
- Open Access
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.Open AccessOptimal 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. - 3 January 2024
- Published by American Physical Society (APS) in Physical Review B
- Vol. 109 (2) , L020201
Non-Hermitian Hamiltonians violate the eigenstate thermalization hypothesis
The eigenstate thermalization hypothesis (ETH) represents a cornerstone in the theoretical understanding of the emergence of thermal behavior in closed quantum systems. The ETH asserts that expectation values of simple observables in energy eigenstates are accurately described by smooth functions of the thermodynamic parameters, with fluctuations and off-diagonal matrix elements exponentially suppressed in the entropy. We investigate to what extent the ETH holds in non-Hermitian many-body systems and come to the surprising conclusion that the fluctuations between eigenstates are of equal order to the average, indicating no eigenstate thermalization. We support this conclusion with mathematically rigorous results in the Ginibre ensemble and numerical results in other random matrix ensembles, the non-Hermitian Sachdev-Ye-Kitaev model, and a local non-Hermitian spin chain, indicating universality in a broad class of chaotic non-Hermitian quantum systems.
- 3 January 2024
- Published by American Physical Society (APS) in Physical Review B
- Vol. 109 (2) , L020201
Non-Hermitian Hamiltonians violate the eigenstate thermalization hypothesis
The eigenstate thermalization hypothesis (ETH) represents a cornerstone in the theoretical understanding of the emergence of thermal behavior in closed quantum systems. The ETH asserts that expectation values of simple observables in energy eigenstates are accurately described by smooth functions of the thermodynamic parameters, with fluctuations and off-diagonal matrix elements exponentially suppressed in the entropy. We investigate to what extent the ETH holds in non-Hermitian many-body systems and come to the surprising conclusion that the fluctuations between eigenstates are of equal order to the average, indicating no eigenstate thermalization. We support this conclusion with mathematically rigorous results in the Ginibre ensemble and numerical results in other random matrix ensembles, the non-Hermitian Sachdev-Ye-Kitaev model, and a local non-Hermitian spin chain, indicating universality in a broad class of chaotic non-Hermitian quantum systems.
- Open Access
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.
Open AccessGaussian 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.