Giorgio Cipolloni

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• Benigni, L., & Cipolloni, G. (2024). Fluctuations of eigenvector overlaps and the Berry conjecture for Wigner matrices. Electronic Journal of Probability, 29. doi:10.1214/24-EJP1203
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  We prove that any finite collection of quadratic forms (overlaps) of general deterministic matrices and eigenvectors of an N × N Wigner matrix has joint Gaussian fluctuations. This can be viewed as the random matrix analogue of the Berry random wave conjecture.
• Cipolloni, G., & Grometto, N. (2024). The Dissipative Spectral Form Factor for I.I.D. Matrices. Journal of Statistical Physics, 191(2). doi:10.1007/s10955-024-03237-4
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  The dissipative spectral form factor (DSFF), recently introduced in Li et al. (Phys Rev Lett 127(17):170602, 2021) for the Ginibre ensemble, is a key tool to study universal properties of dissipative quantum systems. In this work we compute the DSFF for a large class of random matrices with real or complex entries up to an intermediate time scale, confirming the predictions from Li et al. (Phys Rev Lett 127(17):170602, 2021). The analytic formula for the DSFF in the real case was previously unknown. Furthermore, we show that for short times the connected component of the DSFF exhibits a non-universal correction depending on the fourth cumulant of the entries. These results are based on the central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices Cipolloni et al. (Electron J Prob 26:1–61, 2021) and Cipolloni et al. (Commun Pure Appl Math 76(5): 946–1034, 2023).
• Cipolloni, G., & Kudler-Flam, J. (2024). Non-Hermitian Hamiltonians violate the eigenstate thermalization hypothesis. Physical Review B, 109(2). doi:10.1103/PhysRevB.109.L020201
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  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.
• Cipolloni, G., Erdős, L., & Schröder, D. (2024). Mesoscopic central limit theorem for non-Hermitian random matrices. Probability Theory and Related Fields, 188(3-4). doi:10.1007/s00440-023-01229-1
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  We prove that the mesoscopic linear statistics ∑if(na(σi-z0)) of the eigenvalues {σi}i of large n×n non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any H02-functions f around any point z0 in the bulk of the spectrum on any mesoscopic scale 0-1/2. 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.
• Cipolloni, G., Henheik, J., & Erdős, L. (2024). Out-of-time-ordered correlators for Wigner matrices. Advances in Theoretical and Mathematical Physics, 28(6). doi:10.4310/ATMP.241031013250
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  We consider the time evolution of the out-of-time-ordered correla-tor (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 sepa-rated 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, allow-ing 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 zCiwjqh/EQy6ZOvfPf5Rn1nvpQIAcFJunifies and improves previous local laws involving either the much cruder operator norm (cf. [10]) or the Hilbert-Schmidt norm (cf. [11]).
• Cipolloni, G., Henheik, J., Erdős, L., & Schröder, D. (2024). Optimal lower bound on eigenvector overlaps for non-Hermitian random matrices. Journal of Functional Analysis, 287(4). doi:10.1016/j.jfa.2024.110495
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  We consider large non-Hermitian N×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 [34] and proven for Wigner matrices in [23], 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 [7] and [45]. 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.
• Cipolloni, G., Peled, R., Schenker, J., & Shapiro, J. (2024). Dynamical Localization for Random Band Matrices Up to W≪N1/4. Communications in Mathematical Physics, 405(3). doi:10.1007/s00220-024-04948-1
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  We prove that a large class of N×N Gaussian random band matrices with band width W exhibits dynamical Anderson localization at all energies when W≪N1/4. The proof uses the fractional moment method (Aizenman and Molchanov in Commun Math Phys 157(2):245–278, 1993. https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-157/issue-2/Localizationat-large-disorder-and-at-extreme-energies–an/cmp/1104253939.full) and an adaptive Mermin–Wagner style shift.
• Cipolloni, G., Xu, Y., & Erdős, L. (2024). Precise asymptotics for the spectral radius of a large random matrix. Journal of Mathematical Physics, 65(6). doi:10.1063/5.0209705
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  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.
• Aizenman, M., & Cipolloni, G. (2023). Ruminations on matrix convexity and the strong subadditivity of quantum entropy. Letters in Mathematical Physics, 113(1). doi:10.1007/s11005-023-01638-2
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  The familiar second derivative test for convexity, combined with resolvent calculus, is shown to yield a useful tool for the study of convex matrix-valued functions. We demonstrate the applicability of this approach on a number of theorems in this field. These include convexity principles which play an essential role in the Lieb–Ruskai proof of the strong subadditivity of quantum entropy.
• Cipolloni, G., & Kudler-Flam, J. (2023). Entanglement Entropy of Non-Hermitian Eigenstates and the Ginibre Ensemble. Physical Review Letters, 130(1). doi:10.1103/PhysRevLett.130.010401
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  Entanglement entropy is a powerful tool in characterizing universal features in quantum many-body systems. In quantum chaotic Hermitian systems, typical eigenstates have near maximal entanglement with very small fluctuations. Here, we show that for Hamiltonians displaying non-Hermitian many-body quantum chaos, modeled by the Ginibre ensemble, the entanglement entropy of typical eigenstates is greatly suppressed. The entropy does not grow with the Hilbert space dimension for sufficiently large systems, and the fluctuations are of equal order. We derive the novel entanglement spectrum that has infinite support in the complex plane and strong energy dependence. We provide evidence of universality, and similar behavior is found in the non-Hermitian Sachdev-Ye-Kitaev model, indicating the general applicability of the Ginibre ensemble to dissipative many-body quantum chaos.
• Cipolloni, G., Cipolloni, G., Erdős, L., Erdős, L., Schröder, D., & Schröder, D. (2023). On the Spectral Form Factor for Random Matrices. Communications in Mathematical Physics, 401(2). doi:10.1007/s00220-023-04692-y
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  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 in J Stat Phys 183:33, 2021. https://doi.org/10.1007/s10955-021-02767-5, Commun Math Phys 387:215–235, 2021. https://doi.org/10.1007/s00220-021-04193-w). 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, supplementing the recently proven Wigner–Dyson universality (Cipolloni et al. in Probab Theory Relat Fields, 2021. https://doi.org/10.1007/s00440-022-01156-7) to 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.
• Cipolloni, G., Erdos, L., & Schröder, D. (2023). FUNCTIONAL CENTRAL LIMIT THEOREMS FOR WIGNER MATRICES. Annals of Applied Probability, 33(1). doi:10.1214/22-AAP1820
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  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, Tr f (W), which is equivalent to the customary linear statistics of eigenvalues, we show that Tr f (W)A is asymptotically normal for any nontrivial 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. As a main motivation to study CLT in such generality on small mesoscopic scales, we determine the fluctuations in the eigenstate thermalization hypothesis (Phys. Rev. A 43 (1991) 2046–2049), that is, prove that the eigenfunction overlaps with any deterministic matrix are asymptotically Gaussian after a small spectral averaging. Finally, in the macroscopic regime our result also generalizes (Zh. Mat. Fiz. Anal. Geom. 9 (2013) 536–581, 611, 615) to complex W and to all crossover ensembles in between. The main technical inputs are the recent multiresolvent local laws with traceless deterministic matrices from the companion paper (Comm. Math. Phys. 388 (2021) 1005–1048).
• Cipolloni, G., Erdos, L., Henheik, J., & Kolupaiev, O. (2023). Gaussian fluctuations in the equipartition principle for Wigner matrices. Forum of Mathematics, Sigma, 11. doi:10.1017/fms.2023.70
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  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.
• Cipolloni, G., Erdős, L., & Schröder, D. (2023). Central Limit Theorem for Linear Eigenvalue Statistics of Non-Hermitian Random Matrices. Communications on Pure and Applied Mathematics, 76(5). doi:10.1002/cpa.22028
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  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 (Formula presented.) derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32]. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.
• Cipolloni, G., Erdős, L., & Schröder, D. (2023). Quenched universality for deformed Wigner matrices. Probability Theory and Related Fields, 185(3-4). doi:10.1007/s00440-022-01156-7
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  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.
• Cipolloni, G., Xu, Y., Erdo˝s, L., & Schröder, D. (2023). ON THE RIGHTMOST EIGENVALUE OF NON-HERMITIAN RANDOM MATRICES. Annals of Probability, 51(6). doi:10.1214/23-AOP1643
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  We establish a precise three-term asymptotic expansion, with an optimal estimate of the error term, for the rightmost eigenvalue of an n × n random matrix with independent identically distributed complex entries as n tends to infinity. All terms in the expansion are universal.
• Cipolloni, G. (2022). Fluctuations in the spectrum of non-Hermitian i.i.d. matrices. Journal of Mathematical Physics, 63(5). doi:10.1063/5.0089089
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  We consider large non-Hermitian random matrices X with independent identically distributed real or complex entries. In this paper, we review recent results about the eigenvalues of X: (i) universality of local eigenvalue statistics close to the edge of the spectrum of X [Cipolloni et al., "Edge universality for non-Hermitian random matrices,"Probab. Theory Relat. Fields 179, 1-28 (2021)], which is the non-Hermitian analog of the celebrated Tracy-Widom universality; (ii) central limit theorem for linear eigenvalue statistics of macroscopic test functions having 2 + ϵ derivatives [Cipolloni et al., "Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices,"Commun. Pure Appl. Math. (published online) (2021) and Cipolloni et al., "Fluctuation around the circular law for random matrices with real entries,"Electron. J. Probab. 26, 1-61 (2021)]. The main novel ingredients in the proof of these results are local laws for products of two resolvents of the Hermitization of X at two different spectral parameters, coupling of weakly dependent Dyson Brownian motions, and the lower tail estimate for the smallest singular value of X - z in the transitional regime |z| ≈ 1 [Cipolloni et al., "Optimal lower bound on the least singular value of the shifted Ginibre ensemble,"Probab. Math. Phys. 1, 101-146 (2020)].
• Cipolloni, G., Erdo, S., & Schröder, D. (2022). NORMAL FLUCTUATION IN QUANTUM ERGODICITY FOR WIGNER MATRICES. Annals of Probability, 50(3). doi:10.1214/21-AOP1552
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  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
• Cipolloni, G., Erdos, L., Xu, Y., & Schröder, D. (2022). Directional extremal statistics for Ginibre eigenvalues. Journal of Mathematical Physics, 63(10). doi:10.1063/5.0104290
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  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 maxRSpec(X) for any large matrix X with i.i.d. entries in the companion paper [G. Cipolloni et al., arXiv:2206.04448 (2022)].
• Cipolloni, G., Erdå'S, L., & Schröder, D. (2022). Rank-uniform local law for Wigner matrices. Forum of Mathematics, Sigma, 10. doi:10.1017/fms.2022.86
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  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.
• Cipolloni, G., Erdős, L., & Schröder, D. (2022). Density of Small Singular Values of the Shifted Real Ginibre Ensemble. Annales Henri Poincare, 23(11). doi:10.1007/s00023-022-01188-8
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  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.
• Cipolloni, G., Erdős, L., & Schröder, D. (2022). ON THE CONDITION NUMBER OF THE SHIFTED REAL GINIBRE ENSEMBLE. SIAM Journal on Matrix Analysis and Applications, 43(3). doi:10.1137/21M1424408
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  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].
• Cipolloni, G., Erdős, L., & Schröder, D. (2022). Optimal multi-resolvent local laws for Wigner matrices. Electronic Journal of Probability, 27. doi:10.1214/22-EJP838
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  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.
• Cipolloni, G., Erdős, L., & Schröder, D. (2022). Thermalisation for Wigner matrices. Journal of Functional Analysis, 282(8). doi:10.1016/j.jfa.2022.109394
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  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 [36] 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.
• Cipolloni, G., Erdős, L., & Schröder, D. (2021). Edge universality for non-Hermitian random matrices. Probability Theory and Related Fields, 179(1-2). doi:10.1007/s00440-020-01003-7
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  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.
• Cipolloni, G., Erdős, L., & Schröder, D. (2021). Eigenstate Thermalization Hypothesis for Wigner Matrices. Communications in Mathematical Physics, 388(2). doi:10.1007/s00220-021-04239-z
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  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).
• Cipolloni, G., Erdős, L., & Schröder, D. (2021). Fluctuation around the circular law for random matrices with real entries. Electronic Journal of Probability, 26. doi:10.1214/21-EJP591
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  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.
• Cipolloni, G., & Erdos, L. (2020). Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices. Random Matrices: Theory and Application, 9(3). doi:10.1142/S2010326320500069
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  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.
• Cipolloni, G., Erdös, L., & Schröder, D. (2020). OPTIMAL LOWER BOUND ON THE LEAST SINGULAR VALUE OF THE SHIFTED GINIBRE ENSEMBLE. Probability and Mathematical Physics, 1(1). doi:10.2140/pmp.2020.1.101
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  We consider the least singular value of a large random matrix with real or complex i.i.d. Gaussian entries shifted by a constant z 2 C. We prove an optimal lower tail estimate on this singular value in the critical regime where z is around the spectral edge, thus improving the classical bound of Sankar, Spielman and Teng (SIAM J. Matrix Anal. Appl. 28:2 (2006), 446-476) for the particular shift-perturbation in the edge regime. Lacking Brézin-Hikami formulas in the real case, we rely on the superbosonization formula.
• Cipolloni, G., Erdős, L., Krüger, T., & Schröder, D. (2019). CUSP UNIVERSALITY FOR RANDOM MATRICES II: THE REAL SYMMETRIC CASE. Pure and Applied Analysis, 1(4). doi:10.2140/paa.2019.1.615
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  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 by Erd˝os et al. (2018, 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 by Landon and Yau (2017, arXiv:1712.03881) to the cusp regime using the optimal local law by Erd˝os et al. (2018, arXiv:1809.03971) and the accurate local shape analysis of the density by Ajanki et al. (2015, arXiv:1506.05095) and Alt et al. (2018, arXiv:1804.07752). We also present a novel PDE-based method to improve the estimate on eigenvalue rigidity via the maximum principle of the heat flow related to the Dyson Brownian motion.