Dominik Schröder

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Schröder

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Dominik

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0000-0002-2904-1856

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Publications 1 - 10 of 23

Central Limit Theorem for Linear Eigenvalue Statistics of non-Hermitian Random Matrices
Cipolloni, Giorgio; Erdős, László; Schröder, Dominik (2023)
Communications on Pure and Applied Mathematics
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 urn:x-wiley:00103640:media:cpa22028:cpa22028-math-0001 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].
Normal fluctuation in quantum ergodicity for Wigner matrices
Cipolloni, Giorgio; Erdős, László; Schröder, Dominik (2022)
The Annals of Probability
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).
On the rightmost eigenvalue of non-Hermitian random matrices
Cipolloni, Giorgio; Erdős, László; Schröder, Dominik; et al. (2023)
The Annals of Probability
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.
On the Spectral Form Factor for Random Matrices
Cipolloni, Giorgio; Erdős, László; Schröder, Dominik (2023)
Communications in Mathematical Physics
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.
Eigenstate Thermalization Hypothesis for Wigner Matrices
Cipolloni, Giorgio; Erdős, László; Schröder, Dominik (2021)
Communications in Mathematical Physics
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 Thermalization Hypothesis by Deutsch [Deutsch 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, Yau 2017 ] and [Bourgade, Yau, Yin 2020 ] .
Mesoscopic central limit theorem for non-Hermitian random matrices
Cipolloni, Giorgio; Erdős, László; Schröder, Dominik (2024)
Probability Theory and Related Fields
We prove that the mesoscopic linear statistics Sigma(i)f (n(a)(sigma(i) - z(0))) of the eigenvalues {sigma(i)}(i) of large nxn non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any H-0(2) -functions f around any point z0 in the bulk of the spectrum on any mesoscopic scale 0 < a < 1/2. This extends our previous result (Cipolloni et al. in Commun Pure Appl Math, 2019. 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)| >> 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.
Deterministic equivalent and error universality of deep random features learning
Schröder, Dominik; Cui, Hugo; Dmitriev, Daniil; et al. (2024)
Journal of Statistical Mechanics: Theory and Experiment
This manuscript considers the problem of learning a random Gaussian network function using a fully connected network with frozen intermediate layers and trainable readout layer. This problem can be seen as a natural generalization of the widely studied random features model to deeper architectures. First, we prove Gaussian universality of the test error in a ridge regression setting where the learner and target networks share the same intermediate layers, and provide a sharp asymptotic formula for it. Establishing this result requires proving a deterministic equivalent for traces of the deep random features sample covariance matrices which can be of independent interest. Second, we conjecture the asymptotic Gaussian universality of the test error in the more general setting of arbitrary convex losses and generic learner/target architectures. We provide extensive numerical evidence for this conjecture. In light of our results, we investigate the interplay between architecture design and implicit regularization.
On the Condition Number of the Shifted Real Ginibre Ensemble
Cipolloni, Giorgio; Erdős, László; Schröder, Dominik (2022)
SIAM Journal on Matrix Analysis and Applications
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].
Edge universality for non-Hermitian random matrices
Cipolloni, Giorgio; Erdős, László; Schröder, Dominik (2021)
Probability Theory and Related Fields
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.
Functional central limit theorems for Wigner matrices
Cipolloni, Giorgio; Erdős, László; Schröder, Dominik (2023)
The Annals of Applied Probability
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).

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Dominik Schröder

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