@article{10732, abstract = {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 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.}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {1096-0783}, journal = {Journal of Functional Analysis}, number = {8}, publisher = {Elsevier}, title = {{Thermalisation for Wigner matrices}}, doi = {10.1016/j.jfa.2022.109394}, volume = {282}, year = {2022}, } @article{11332, abstract = {We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size N converge to the Tracy–Widom laws at a rate O(N^{-1/3+\omega }), as N tends to infinity. For Wigner matrices this improves the previous rate O(N^{-2/9+\omega }) obtained by Bourgade (J Eur Math Soc, 2021) for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős et al. (Adv Math 229(3):1435–1515, 2012) to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein–Uhlenbeck process. Precise estimates on leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and recursive comparisons for correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian invariant ensembles.}, author = {Schnelli, Kevin and Xu, Yuanyuan}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, pages = {839--907}, publisher = {Springer Nature}, title = {{Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices}}, doi = {10.1007/s00220-022-04377-y}, volume = {393}, year = {2022}, } @article{11418, abstract = {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).}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {2168-894X}, journal = {Annals of Probability}, number = {3}, pages = {984--1012}, publisher = {Institute of Mathematical Statistics}, title = {{Normal fluctuation in quantum ergodicity for Wigner matrices}}, doi = {10.1214/21-AOP1552}, volume = {50}, year = {2022}, } @article{10623, abstract = {We investigate the BCS critical temperature Tc in the high-density limit and derive an asymptotic formula, which strongly depends on the behavior of the interaction potential V on the Fermi-surface. Our results include a rigorous confirmation for the behavior of Tc at high densities proposed by Langmann et al. (Phys Rev Lett 122:157001, 2019) and identify precise conditions under which superconducting domes arise in BCS theory.}, author = {Henheik, Sven Joscha}, issn = {1572-9656}, journal = {Mathematical Physics, Analysis and Geometry}, keywords = {geometry and topology, mathematical physics}, number = {1}, publisher = {Springer Nature}, title = {{The BCS critical temperature at high density}}, doi = {10.1007/s11040-021-09415-0}, volume = {25}, year = {2022}, } @article{12184, abstract = {We review recent results on adiabatic theory for ground states of extended gapped fermionic lattice systems under several different assumptions. More precisely, we present generalized super-adiabatic theorems for extended but finite and infinite systems, assuming either a uniform gap or a gap in the bulk above the unperturbed ground state. The goal of this Review is to provide an overview of these adiabatic theorems and briefly outline the main ideas and techniques required in their proofs.}, author = {Henheik, Sven Joscha and Wessel, Tom}, issn = {0022-2488}, journal = {Journal of Mathematical Physics}, number = {12}, publisher = {AIP Publishing}, title = {{On adiabatic theory for extended fermionic lattice systems}}, doi = {10.1063/5.0123441}, volume = {63}, year = {2022}, } @article{10642, abstract = {Based on a result by Yarotsky (J Stat Phys 118, 2005), we prove that localized but otherwise arbitrary perturbations of weakly interacting quantum spin systems with uniformly gapped on-site terms change the ground state of such a system only locally, even if they close the spectral gap. We call this a strong version of the local perturbations perturb locally (LPPL) principle which is known to hold for much more general gapped systems, but only for perturbations that do not close the spectral gap of the Hamiltonian. We also extend this strong LPPL-principle to Hamiltonians that have the appropriate structure of gapped on-site terms and weak interactions only locally in some region of space. While our results are technically corollaries to a theorem of Yarotsky, we expect that the paradigm of systems with a locally gapped ground state that is completely insensitive to the form of the Hamiltonian elsewhere extends to other situations and has important physical consequences.}, author = {Henheik, Sven Joscha and Teufel, Stefan and Wessel, Tom}, issn = {1573-0530}, journal = {Letters in Mathematical Physics}, keywords = {mathematical physics, statistical and nonlinear physics}, number = {1}, publisher = {Springer Nature}, title = {{Local stability of ground states in locally gapped and weakly interacting quantum spin systems}}, doi = {10.1007/s11005-021-01494-y}, volume = {112}, year = {2022}, } @article{11732, abstract = {We study the BCS energy gap Ξ in the high–density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential V on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3, 2022) on the critical temperature Tc at high densities, we prove the universality of the ratio of the energy gap and the critical temperature.}, author = {Henheik, Sven Joscha and Lauritsen, Asbjørn Bækgaard}, issn = {1572-9613}, journal = {Journal of Statistical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, publisher = {Springer Nature}, title = {{The BCS energy gap at high density}}, doi = {10.1007/s10955-022-02965-9}, volume = {189}, year = {2022}, } @article{11135, abstract = {We consider a correlated NxN Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one.}, author = {Reker, Jana}, issn = {2010-3271}, journal = {Random Matrices: Theory and Applications}, keywords = {Discrete Mathematics and Combinatorics, Statistics, Probability and Uncertainty, Statistics and Probability, Algebra and Number Theory}, number = {4}, publisher = {World Scientific Publishing}, title = {{On the operator norm of a Hermitian random matrix with correlated entries}}, doi = {10.1142/s2010326322500368}, volume = {11}, year = {2022}, } @article{15013, abstract = {We consider random n×n matrices X with independent and centered entries and a general variance profile. We show that the spectral radius of X converges with very high probability to the square root of the spectral radius of the variance matrix of X when n tends to infinity. We also establish the optimal rate of convergence, that is a new result even for general i.i.d. matrices beyond the explicitly solvable Gaussian cases. The main ingredient is the proof of the local inhomogeneous circular law [arXiv:1612.07776] at the spectral edge.}, author = {Alt, Johannes and Erdös, László and Krüger, Torben H}, issn = {2690-1005}, journal = {Probability and Mathematical Physics}, number = {2}, pages = {221--280}, publisher = {Mathematical Sciences Publishers}, title = {{Spectral radius of random matrices with independent entries}}, doi = {10.2140/pmp.2021.2.221}, volume = {2}, year = {2021}, } @article{15259, abstract = {We consider words Gi1⋯Gim involving i.i.d. complex Ginibre matrices and study tracial expressions of their eigenvalues and singular values. We show that the limit distribution of the squared singular values of every word of length m is a Fuss–Catalan distribution with parameter m+1. This generalizes previous results concerning powers of a complex Ginibre matrix and products of independent Ginibre matrices. In addition, we find other combinatorial parameters of the word that determine the second-order limits of the spectral statistics. For instance, the so-called coperiod of a word characterizes the fluctuations of the eigenvalues. We extend these results to words of general non-Hermitian matrices with i.i.d. entries under moment-matching assumptions, band matrices, and sparse matrices. These results rely on the moments method and genus expansion, relating Gaussian matrix integrals to the counting of compact orientable surfaces of a given genus. This allows us to derive a central limit theorem for the trace of any word of complex Ginibre matrices and their conjugate transposes, where all parameters are defined topologically.}, author = {Dubach, Guillaume and Peled, Yuval}, issn = {0091-1798}, journal = {The Annals of Probability}, keywords = {Statistics, Probability and Uncertainty, Statistics and Probability}, number = {4}, pages = {1886--1916}, publisher = {Institute of Mathematical Statistics}, title = {{On words of non-Hermitian random matrices}}, doi = {10.1214/20-aop1496}, volume = {49}, year = {2021}, } @article{10221, abstract = {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).}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, number = {2}, pages = {1005–1048}, publisher = {Springer Nature}, title = {{Eigenstate thermalization hypothesis for Wigner matrices}}, doi = {10.1007/s00220-021-04239-z}, volume = {388}, year = {2021}, } @article{10285, abstract = {We study the overlaps between right and left eigenvectors for random matrices of the spherical ensemble, as well as truncated unitary ensembles in the regime where half of the matrix at least is truncated. These two integrable models exhibit a form of duality, and the essential steps of our investigation can therefore be performed in parallel. In every case, conditionally on all eigenvalues, diagonal overlaps are shown to be distributed as a product of independent random variables with explicit distributions. This enables us to prove that the scaled diagonal overlaps, conditionally on one eigenvalue, converge in distribution to a heavy-tail limit, namely, the inverse of a γ2 distribution. We also provide formulae for the conditional expectation of diagonal and off-diagonal overlaps, either with respect to one eigenvalue, or with respect to the whole spectrum. These results, analogous to what is known for the complex Ginibre ensemble, can be obtained in these cases thanks to integration techniques inspired from a previous work by Forrester & Krishnapur.}, author = {Dubach, Guillaume}, issn = {1083-6489}, journal = {Electronic Journal of Probability}, publisher = {Institute of Mathematical Statistics}, title = {{On eigenvector statistics in the spherical and truncated unitary ensembles}}, doi = {10.1214/21-EJP686}, volume = {26}, year = {2021}, } @article{8373, abstract = {It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted A#H-mean introduced by Kim, Lawson, and Lim.}, author = {Pitrik, József and Virosztek, Daniel}, issn = {0024-3795}, journal = {Linear Algebra and its Applications}, keywords = {Kubo-Ando mean, weighted multivariate mean, barycenter}, pages = {203--217}, publisher = {Elsevier}, title = {{A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means}}, doi = {10.1016/j.laa.2020.09.007}, volume = {609}, year = {2021}, } @article{9036, abstract = {In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.}, author = {Virosztek, Daniel}, issn = {0001-8708}, journal = {Advances in Mathematics}, keywords = {General Mathematics}, number = {3}, publisher = {Elsevier}, title = {{The metric property of the quantum Jensen-Shannon divergence}}, doi = {10.1016/j.aim.2021.107595}, volume = {380}, year = {2021}, } @unpublished{9230, abstract = {We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length (log T)θ, where θ is either fixed or tends to zero at a suitable rate. It is shown that the deterministic level of the maximum interpolates smoothly between the ones of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition ‘from 3/4 to 1/4’ in the second order. This provides a natural context where extreme value statistics of log-correlated variables with time-dependent variance and rate occur. A key ingredient of the proof is a precise upper tail tightness estimate for the maximum of the model on intervals of size one, that includes a Gaussian correction. This correction is expected to be present for the Riemann zeta function and pertains to the question of the correct order of the maximum of the zeta function in large intervals.}, author = {Arguin, Louis-Pierre and Dubach, Guillaume and Hartung, Lisa}, booktitle = {arXiv}, title = {{Maxima of a random model of the Riemann zeta function over intervals of varying length}}, doi = {10.48550/arXiv.2103.04817}, year = {2021}, } @unpublished{9281, abstract = {We comment on two formal proofs of Fermat's sum of two squares theorem, written using the Mathematical Components libraries of the Coq proof assistant. The first one follows Zagier's celebrated one-sentence proof; the second follows David Christopher's recent new proof relying on partition-theoretic arguments. Both formal proofs rely on a general property of involutions of finite sets, of independent interest. The proof technique consists for the most part of automating recurrent tasks (such as case distinctions and computations on natural numbers) via ad hoc tactics.}, author = {Dubach, Guillaume and Mühlböck, Fabian}, booktitle = {arXiv}, title = {{Formal verification of Zagier's one-sentence proof}}, doi = {10.48550/arXiv.2103.11389}, year = {2021}, } @article{9912, abstract = {In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via 𝑁≪𝑀 channels, the density 𝜌 of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio 𝜙:=𝑁/𝑀≤1; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit 𝜙→0, we recover the formula for the density 𝜌 that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker’s formula persists for any 𝜙<1 but in the borderline case 𝜙=1 an anomalous 𝜆−2/3 singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries.}, author = {Erdös, László and Krüger, Torben H and Nemish, Yuriy}, issn = {1424-0661}, journal = {Annales Henri Poincaré }, pages = {4205–4269}, publisher = {Springer Nature}, title = {{Scattering in quantum dots via noncommutative rational functions}}, doi = {10.1007/s00023-021-01085-6}, volume = {22}, year = {2021}, } @article{9412, abstract = {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.}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {1083-6489}, journal = {Electronic Journal of Probability}, publisher = {Institute of Mathematical Statistics}, title = {{Fluctuation around the circular law for random matrices with real entries}}, doi = {10.1214/21-EJP591}, volume = {26}, year = {2021}, } @article{8601, abstract = {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.}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {1432-2064}, journal = {Probability Theory and Related Fields}, publisher = {Springer Nature}, title = {{Edge universality for non-Hermitian random matrices}}, doi = {10.1007/s00440-020-01003-7}, year = {2021}, } @article{9550, abstract = {We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices. }, author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Equipartition principle for Wigner matrices}}, doi = {10.1017/fms.2021.38}, volume = {9}, year = {2021}, }