@article{19548,
  abstract     = {We consider the BCS energy gap „.T / (essentially given by „.T /  .T; p/,
the BCS order parameter) at all temperatures 0  T  Tc up to the critical one, Tc, and show
that, in the limit of weak coupling, the ratio „.T /=Tc is given by a universal function of the relative temperature T =Tc. On the one hand, this recovers a recent result by Langmann and Triola
[Phys. Rev. B 108 (2023), no. 10, article no. 104503] on three-dimensional s-wave superconductors for temperatures bounded uniformly away from Tc. On the other hand, our result lifts these
restrictions, as we consider arbitrary spatial dimensions d 2 ¹1; 2; 3º, discuss superconductors
with non-zero angular momentum (primarily in two dimensions), and treat the perhaps physically most interesting (due to the occurrence of the superconducting phase transition) regime of
temperatures close to Tc.

​
 .},
  author       = {Henheik, Sven Joscha and Lauritsen, Asbjørn Bækgaard},
  issn         = {1664-0403},
  journal      = {Journal of Spectral Theory},
  number       = {1},
  pages        = {305–352},
  publisher    = {EMS Press},
  title        = {{Universal behavior of the BCS energy gap}},
  doi          = {10.4171/JST/540},
  volume       = {15},
  year         = {2025},
}

@article{13975,
  abstract     = {We consider the spectrum of random Laplacian matrices of the form Ln=An−Dn where An
 is a real symmetric random matrix and Dn is a diagonal matrix whose entries are equal to the corresponding row sums of An. If An is a Wigner matrix with entries in the domain of attraction of a Gaussian distribution, the empirical spectral measure of Ln is known to converge to the free convolution of a semicircle distribution and a standard real Gaussian distribution. We consider real symmetric random matrices An with independent entries (up to symmetry) whose row sums converge to a purely non-Gaussian infinitely divisible distribution, which fall into the class of Lévy–Khintchine random matrices first introduced by Jung [Trans Am Math Soc, 370, (2018)]. Our main result shows that the empirical spectral measure of Ln  converges almost surely to a deterministic limit. A key step in the proof is to use the purely non-Gaussian nature of the row sums to build a random operator to which Ln converges in an appropriate sense. This operator leads to a recursive distributional equation uniquely describing the Stieltjes transform of the limiting empirical spectral measure.},
  author       = {Campbell, Andrew J and O’Rourke, Sean},
  issn         = {1572-9230},
  journal      = {Journal of Theoretical Probability},
  pages        = {933--973},
  publisher    = {Springer Nature},
  title        = {{Spectrum of Lévy–Khintchine random laplacian matrices}},
  doi          = {10.1007/s10959-023-01275-4},
  volume       = {37},
  year         = {2024},
}

@article{14408,
  abstract     = {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 H20-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 z1,z2 with an improved error term in the entire mesoscopic regime |z1−z2|≫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.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1432-2064},
  journal      = {Probability Theory and Related Fields},
  pages        = {1131--1182},
  publisher    = {Springer Nature},
  title        = {{Mesoscopic central limit theorem for non-Hermitian random matrices}},
  doi          = {10.1007/s00440-023-01229-1},
  volume       = {188},
  year         = {2024},
}

@article{18762,
  abstract     = {Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded deterministic matrices $A_1,\dots,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\dots,f_k$ and the number of traceless matrices among $A_1,\dots,A_k$, thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of arbitrary length $k\geq2$. As an application, we consider the fluctuation of $\mathrm{Tr}(\mathrm{e}^{\mathrm{i} tW}A_1\mathrm{e}^{-\mathrm{i} tW}A_2)$ around its thermal value $\mathrm{Tr}(A_1)\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance.},
  author       = {Reker, Jana},
  issn         = {1083-6489},
  journal      = {Electronic Journal of Probability},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Multi-point functional central limit theorem for Wigner matrices}},
  doi          = {10.1214/24-EJP1247},
  volume       = {29},
  year         = {2024},
}

@article{15025,
  abstract     = {We consider quadratic forms of deterministic matrices A evaluated at the random eigenvectors of a large N×N GOE or GUE matrix, or equivalently evaluated at the columns of a Haar-orthogonal or Haar-unitary random matrix. We prove that, as long as the deterministic matrix has rank much smaller than √N, the distributions of the extrema of these quadratic forms are asymptotically the same as if the eigenvectors were independent Gaussians. This reduces the problem to Gaussian computations, which we carry out in several cases to illustrate our result, finding Gumbel or Weibull limiting distributions depending on the signature of A. Our result also naturally applies to the eigenvectors of any invariant ensemble.},
  author       = {Erdös, László and McKenna, Benjamin},
  issn         = {1050-5164},
  journal      = {Annals of Applied Probability},
  number       = {1B},
  pages        = {1623--1662},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Extremal statistics of quadratic forms of GOE/GUE eigenvectors}},
  doi          = {10.1214/23-AAP2000},
  volume       = {34},
  year         = {2024},
}

@article{15378,
  abstract     = {We consider N×N non-Hermitian random matrices of the form X+A, where A is a general deterministic matrix and 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 N1+o(1) and (ii) that the expected condition number of any bulk eigenvalue is bounded by N1+o(1); both results are optimal up to the factor No(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.},
  author       = {Erdös, László and Ji, Hong Chang},
  issn         = {1097-0312},
  journal      = {Communications on Pure and Applied Mathematics},
  number       = {9},
  pages        = {3785--3840},
  publisher    = {Wiley},
  title        = {{Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices}},
  doi          = {10.1002/cpa.22201},
  volume       = {77},
  year         = {2024},
}

@article{17281,
  abstract     = {We extend the free convolution of Brown measures of R-diagonal elements introduced by Kösters and Tikhomirov [ 28] to fractional powers. We then show how this fractional free convolution arises naturally when studying the roots of random polynomials with independent coefficients under repeated differentiation. When the proportion of derivatives to the degree approaches one, we establish central limit theorem-type behavior and discuss stable distributions.},
  author       = {Campbell, Andrew J and O'Rourke, Sean and Renfrew, David T},
  issn         = {1687-0247},
  journal      = {International Mathematics Research Notices},
  number       = {13},
  pages        = {10189--10218},
  publisher    = {Oxford University Press},
  title        = {{The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation}},
  doi          = {10.1093/imrn/rnae062},
  volume       = {2024},
  year         = {2024},
}

@article{17375,
  abstract     = {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.},
  author       = {Cipolloni, Giorgio and Erdös, László and Xu, Yuanyuan},
  issn         = {0022-2488},
  journal      = {Journal of Mathematical Physics},
  number       = {6},
  publisher    = {AIP Publishing},
  title        = {{Precise asymptotics for the spectral radius of a large random matrix}},
  doi          = {10.1063/5.0209705},
  volume       = {65},
  year         = {2024},
}

@article{18554,
  abstract     = {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 obs ervables 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. Our results hold under very general conditions on the variance profile, even allowing many vanishing entries, demonstrating that Eigenstate Thermalization occurs robustly across a diverse class of random matrix ensembles, for which the underlying quantum system has a non-trivial spatial structure.},
  author       = {Erdös, László and Riabov, Volodymyr},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  number       = {12},
  publisher    = {Springer Nature},
  title        = {{Eigenstate Thermalization Hypothesis for Wigner-type matrices}},
  doi          = {10.1007/s00220-024-05143-y},
  volume       = {405},
  year         = {2024},
}

@article{18656,
  abstract     = {We consider the time evolution of the out-of-time-ordered correlator (OTOC) of two general observables 
 and 
 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. [10]) or the Hilbert-Schmidt norm (cf. [11]).},
  author       = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha},
  issn         = {1095-0753},
  journal      = {Advances in Theoretical and Mathematical Physics},
  number       = {6},
  pages        = {2025--2083},
  publisher    = {International Press},
  title        = {{Out-of-time-ordered correlators for Wigner matrices}},
  doi          = {10.4310/ATMP.241031013250},
  volume       = {28},
  year         = {2024},
}

@article{17049,
  abstract     = {We consider large non-Hermitian NxN 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.},
  author       = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha and Schröder, Dominik J},
  issn         = {1096-0783},
  journal      = {Journal of Functional Analysis},
  number       = {4},
  publisher    = {Elsevier},
  title        = {{Optimal lower bound on eigenvector overlaps for non-Hermitian random matrices}},
  doi          = {10.1016/j.jfa.2024.110495},
  volume       = {287},
  year         = {2024},
}

@unpublished{19545,
  abstract     = {We prove the Eigenstate Thermalisation Hypothesis for Wigner matrices
uniformly in the entire spectrum, in particular near the spectral edges, with a
bound on the fluctuation that is optimal for any observable. This complements
earlier works of Cipolloni et. al. (Comm. Math. Phys. 388, 2021; Forum Math.,
Sigma 10, 2022) and Benigni et. al. (Comm. Math. Phys. 391, 2022; arXiv:
2303.11142) that were restricted either to the bulk of the spectrum or to
special observables. As a main ingredient, we prove a new multi-resolvent local
law that optimally accounts for the edge scaling.},
  author       = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha},
  booktitle    = {arXiv},
  title        = {{Eigenstate thermalisation at the edge for Wigner matrices}},
  doi          = {10.48550/arXiv.2309.05488},
  year         = {2024},
}

@unpublished{19551,
  abstract     = {We introduce a notion of a \emph{local gap} for interacting many-body quantum lattice systems and prove the validity of response theory and Kubo's formula for localized perturbations in such settings.
On a high level, our result shows that the usual spectral gap condition, concerning the system as a whole, is not a necessary condition for understanding local properties of the system.
More precisely, we say that an equilibrium state ρ0 of a Hamiltonian H0 is locally gapped in Λgap⊂Λ, whenever the Liouvillian −i[H0,⋅] is almost invertible on local observables supported in Λgap when tested in ρ0.
To put this into context, we provide other alternative notions of a local gap and discuss their relations.
The validity of response theory is based on the construction of \emph{non-equilibrium almost stationary states} (NEASSs).
By controlling locality properties of the NEASS construction, we show that response theory holds to any order, whenever the perturbation \(\epsilon V\) acts in a region which is further than |logϵ| away from the non-gapped region Λ∖Λgap.},
  author       = {Henheik, Sven Joscha and Wessel, Tom},
  booktitle    = {arXiv},
  title        = {{Response theory for locally gapped systems}},
  doi          = {10.48550/arXiv.2410.10809},
  year         = {2024},
}

@unpublished{19550,
  abstract     = {We introduce a multi-band BCS free energy functional and prove that for a
multi-band superconductor the effect of inter-band coupling can only increase
the critical temperature, irrespective of its attractive or repulsive nature
and its strength. Further, for weak coupling and weaker inter-band coupling, we
prove that the dependence of the increase in critical temperature on the
inter-band coupling is (1) linear, if there are two or more equally strongly
superconducting bands, or (2) quadratic, if there is only one dominating band.},
  author       = {Henheik, Sven Joscha and Langmann, Edwin and Lauritsen, Asbjørn Bækgaard},
  booktitle    = {arXiv},
  title        = {{Multi-band superconductors have enhanced critical temperatures}},
  doi          = {10.48550/arXiv.2409.17297},
  year         = {2024},
}

@unpublished{19547,
  abstract     = {For correlated real symmetric or complex Hermitian random matrices, we prove
that the local eigenvalue statistics at any cusp singularity are universal.
Since the density of states typically exhibits only square root edge or cubic
root cusp singularities, our result completes the proof of the
Wigner-Dyson-Mehta universality conjecture in all spectral regimes for a very
general class of random matrices. Previously only the bulk and the edge
universality were established in this generality [arXiv:1804.07744], while cusp
universality was proven only for Wigner-type matrices with independent entries
[arXiv:1809.03971, arXiv:1811.04055]. As our main technical input, we prove an
optimal local law at the cusp using the Zigzag strategy, a recursive tandem of
the characteristic flow method and a Green function comparison argument.
Moreover, our proof of the optimal local law holds uniformly in the spectrum,
thus also re-establishing universality of the local eigenvalue statistics in
the previously studied bulk [arXiv:1705.10661] and edge [arXiv:1804.07744]
regimes.},
  author       = {Erdös, László and Henheik, Sven Joscha and Riabov, Volodymyr},
  booktitle    = {arXiv},
  title        = {{Cusp universality for correlated random matrices}},
  doi          = {10.48550/arXiv.2410.06813},
  year         = {2024},
}

@article{14542,
  abstract     = {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.},
  author       = {Henheik, Sven Joscha and Lauritsen, Asbjørn Bækgaard and Roos, Barbara},
  issn         = {1793-6659},
  journal      = {Reviews in Mathematical Physics},
  number       = {9},
  publisher    = {World Scientific Publishing},
  title        = {{Universality in low-dimensional BCS theory}},
  doi          = {10.1142/s0129055x2360005x},
  volume       = {36},
  year         = {2024},
}

@phdthesis{17164,
  abstract     = {This thesis is structured into two parts. In the first part, we consider the random
variable X := Tr(f1(W)A1 . . . fk(W)Ak) where W is an N × N Hermitian Wigner matrix, k ∈ N, and we choose (possibly N-dependent) regular functions f1, . . . , fk as well as
bounded deterministic matrices A1, . . . , Ak. In this context, we prove a functional central
limit theorem on macroscopic and mesoscopic scales, showing that the fluctuations of X
around its expectation are Gaussian and that the limiting covariance structure is given
by a deterministic recursion. We further give explicit error bounds in terms of the scaling
of f1, . . . , fk and the number of traceless matrices among A1, . . . , Ak, thus extending
the results of Cipolloni, Erdős and Schröder [40] to products of arbitrary length k ≥ 2.
Analyzing the underlying combinatorics leads to a non-recursive formula for the variance
of X as well as the covariance of X and Y := Tr(fk+1(W)Ak+1 . . . fk+ℓ(W)Ak+ℓ) of similar
build. When restricted to polynomials, these formulas reproduce recent results of Male,
Mingo, Peché, and Speicher [107], showing that the underlying combinatorics of noncrossing partitions and annular non-crossing permutations continue to stay valid beyond
the setting of second-order free probability theory. As an application, we consider the
fluctuation of Tr(eitW A1e
−itW A2)/N around its thermal value Tr(A1) Tr(A2)/N2 when t
is large and give an explicit formula for the variance.
The second part of the thesis collects three smaller projects focusing on different random
matrix models. In the first project, we show that a class of weakly perturbed Hamiltonians
of the form Hλ = H0 + λW, where W is a Wigner matrix, exhibits prethermalization.
That is, the time evolution generated by Hλ relaxes to its ultimate thermal state via an
intermediate prethermal state with a lifetime of order λ
−2
. As the main result, 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 global law
for the deformed Wigner matrix Hλ.
The second project focuses on correlated random matrices, more precisely on a correlated N × N Hermitian random matrix with a polynomially decaying metric correlation
structure. A trivial a priori bound shows that the operator norm of this model is stochastically dominated by √
N. However, by calculating the trace of the moments of the matrix
and using the summable decay of the cumulants, the norm estimate can be improved to a
bound of order one.
In the third project, we consider a multiplicative perturbation of the form UA(t) where U
is a unitary random matrix and A = diag(t, 1, ..., 1). This so-called UA model was
first introduced by Fyodorov [73] for its applications in scattering theory. We give a
general description of the eigenvalue trajectories obtained by varying the parameter t and
introduce a flow of deterministic domains that separates the outlier resulting from the
rank-one perturbation from the typical eigenvalues for all sub-critical timescales. The
results are obtained under generic assumptions on U that hold for various unitary random
matrices, including the circular unitary ensemble (CUE) in the original formulation of
the model.},
  author       = {Reker, Jana},
  issn         = {2663-337X},
  keywords     = {Random Matrices, Spectrum, Central Limit Theorem, Resolvent, Free Probability},
  pages        = {206},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Central limit theorems for random matrices: From resolvents to free probability}},
  doi          = {10.15479/at:ista:17164},
  year         = {2024},
}

@article{17154,
  abstract     = {We compute the deterministic approximation for mixed fluctuation moments of products of deterministic matrices and general Sobolev functions of Wigner matrices. Restricting to polynomials, our formulas reproduce recent results of Male et al. (Random Matrices Theory Appl. 11(2):2250015, 2022), showing that the underlying combinatorics of non-crossing partitions and annular non-crossing permutations continue to stay valid beyond the setting of second-order free probability theory. The formulas obtained further characterize the variance in the functional central limit theorem given in the recent companion paper (Reker in Preprint, arXiv:2204.03419, 2023). and thus allow identifying the fluctuation around the thermal value in certain thermalization problems.},
  author       = {Reker, Jana},
  issn         = {1572-9656},
  journal      = {Mathematical Physics, Analysis and Geometry},
  number       = {3},
  publisher    = {Springer Nature},
  title        = {{Fluctuation moments for regular functions of Wigner Matrices}},
  doi          = {10.1007/s11040-024-09483-y},
  volume       = {27},
  year         = {2024},
}

@article{17047,
  abstract     = {We provide a dynamical study of a model of multiplicative perturbation of a unitary matrix introduced by Fyodorov. In particular, we identify a flow of deterministic domains that bound the spectrum with high probability, separating the outlier from the typical eigenvalues at all sub-critical timescales. These results are obtained under generic assumptions on U that hold for a variety of unitary random matrix models.},
  author       = {Dubach, Guillaume and Reker, Jana},
  issn         = {2010-3271},
  journal      = {Random Matrices: Theory and Applications},
  number       = {2},
  publisher    = {World Scientific Publishing},
  title        = {{Dynamics of a rank-one multiplicative perturbation of a unitary matrix}},
  doi          = {10.1142/s2010326324500072},
  volume       = {13},
  year         = {2024},
}

@article{11741,
  abstract     = {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.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1432-2064},
  journal      = {Probability Theory and Related Fields},
  pages        = {1183–1218},
  publisher    = {Springer Nature},
  title        = {{Quenched universality for deformed Wigner matrices}},
  doi          = {10.1007/s00440-022-01156-7},
  volume       = {185},
  year         = {2023},
}