Periodic Reporting for period 2 - RanMatRanGraCircEl (Random Matrices, Random Graphs and Circular Elements)
Reporting period: 2022-01-01 to 2022-12-31
The goal of this project is the extension of our knowledge about the structure and statistics of eigenvalues and eigenvectors of random matrices. Random matrix statistics are paradigms for collective behaviours arising when many strongly correlated random variables interact. Furthermore, random matrices frequently appear as models in applications, e.g. in natural sciences or engineering.
Therefore, random matrix theory has become a very active research area at the intersection of probability theory, analysis and mathematical physics. For the last decade, random matrix theory has seen tremendous progress with major very recent developments. A large portion of this research has been devoted to rigorously understand a universality phenomenon proposed by Wigner in the 1950’s. He conjectured that the fluctuations of eigenvalue statistics emerging on the microscopic scale, i.e. on the scale of the typical eigenvalue spacing, are universal in the sense that they coincide for large classes of random matrix models and depend only on their basic symmetry type. Such universality statement can be seen as the strongly correlated analogue of the central limit theorem for weak correlations. Since the breakthroughs by Erdős, Yau and collaborators in the 2010’s, and in certain cases by Tao and Vu, universality results have been established for larger and larger classes of random matrices and a number of eigenvalue statistics.
The present project aims at extending the range of validity of such universality statements to different random matrix models and observables as well as establishing and analysing the limitations of such universality phenomena. To that end, the adjacency matrices of random graphs as well as non-Hermitian random matrices are considered and the behaviour of their eigenvalues and eigenvectors is analysed. This is achieved through combining and further developing various tools from analysis, probability theory and mathematical physics.
The project obtained the following significant research results. The eigenvalue density of the elliptic random matrix ensemble, a class of non-Hermitian random matrices, was understood on all mesoscopic scales. This implied complete eigenvector delocalization, a strong indication for universality. Moreover, the eigenvalue and eigenvector behaviour of Erdős–Rényi graphs was characterised in extensive detail. To that end, it was shown that, at the spectral edges, the eigenvalues form asymptotically a Poisson point process if the edge probability is sufficiently small. In a larger region near the spectral edge, each eigenvector was proved to be localized around a vertex of large degree, thus, establishing a non-universal spectral region. Furthermore, the full spectral region of delocalized eigenvectors for Erdős–Rényi graphs was determined.
Therefore, random matrix theory has become a very active research area at the intersection of probability theory, analysis and mathematical physics. For the last decade, random matrix theory has seen tremendous progress with major very recent developments. A large portion of this research has been devoted to rigorously understand a universality phenomenon proposed by Wigner in the 1950’s. He conjectured that the fluctuations of eigenvalue statistics emerging on the microscopic scale, i.e. on the scale of the typical eigenvalue spacing, are universal in the sense that they coincide for large classes of random matrix models and depend only on their basic symmetry type. Such universality statement can be seen as the strongly correlated analogue of the central limit theorem for weak correlations. Since the breakthroughs by Erdős, Yau and collaborators in the 2010’s, and in certain cases by Tao and Vu, universality results have been established for larger and larger classes of random matrices and a number of eigenvalue statistics.
The present project aims at extending the range of validity of such universality statements to different random matrix models and observables as well as establishing and analysing the limitations of such universality phenomena. To that end, the adjacency matrices of random graphs as well as non-Hermitian random matrices are considered and the behaviour of their eigenvalues and eigenvectors is analysed. This is achieved through combining and further developing various tools from analysis, probability theory and mathematical physics.
The project obtained the following significant research results. The eigenvalue density of the elliptic random matrix ensemble, a class of non-Hermitian random matrices, was understood on all mesoscopic scales. This implied complete eigenvector delocalization, a strong indication for universality. Moreover, the eigenvalue and eigenvector behaviour of Erdős–Rényi graphs was characterised in extensive detail. To that end, it was shown that, at the spectral edges, the eigenvalues form asymptotically a Poisson point process if the edge probability is sufficiently small. In a larger region near the spectral edge, each eigenvector was proved to be localized around a vertex of large degree, thus, establishing a non-universal spectral region. Furthermore, the full spectral region of delocalized eigenvectors for Erdős–Rényi graphs was determined.
The project established eigenvector delocalization, a strong indication for universality, for a non-Hermitian random matrix ensemble and for a standard random graph model in certain spectral regions. In another spectral region of this random graph, universality was excluded by proving Poisson statistics for the eigenvalues and eigenvector localization.
More precisely, as a first main result of this project, Torben Krüger (FAU Erlangen-Nuremberg & Uni Copenhagen) and the MSC fellow showed [arXiv:2102.03335] that the eigenvalue density of the elliptic random matrix ensemble, an important class of non-Hermitian random matrices, converges to a deterministic limit on all mesoscopic scales, i.e. scales above the typical eigenvalue fluctuations. Moreover, they established complete delocalization of the corresponding eigenvectors. The mass of a completely delocalized vector is asymptotically almost uniformly distributed among all of its components.
Raphael Ducatez (Uni Lyon), Antti Knowles (Uni Geneva) and the MSC fellow studied the adjacency matrix of the Erdős–Rényi graph, a common model for a random graph. They proved [arXiv:2106.12519] that if the edge probability is small enough then the eigenvalues of the adjacency matrix at the edges of its spectrum form asymptotically a Poisson point process. They also established that each of the corresponding eigenvectors is localized in the vicinity of a vertex of large degree and the mass of the eigenvector decays exponentially around this vertex. Poisson statistics and eigenvector localization characterise non-universal behaviour.
For the adjacency matrix of the Erdős–Rényi graph, the same authors also fully characterised the spectral region, where the eigenvectors are completely delocalized [arXiv:2109.03227]. As mentioned above, eigenvector delocalization is a strong indication for universality in the corresponding spectral region.
In a further work, the same type of eigenvector localization as described above at the spectral edges was proved by Raphael Ducatez, Antti Knowles and the MSC fellow in a much larger region near the spectral edges for the Erdős–Rényi graph. The fundamental novelty was a lower bound on the minimal distance between two eigenvalues in this spectral region, which is obtained via an anti-concentration inequality for the sum of independent random variables due to Kesten.
More precisely, as a first main result of this project, Torben Krüger (FAU Erlangen-Nuremberg & Uni Copenhagen) and the MSC fellow showed [arXiv:2102.03335] that the eigenvalue density of the elliptic random matrix ensemble, an important class of non-Hermitian random matrices, converges to a deterministic limit on all mesoscopic scales, i.e. scales above the typical eigenvalue fluctuations. Moreover, they established complete delocalization of the corresponding eigenvectors. The mass of a completely delocalized vector is asymptotically almost uniformly distributed among all of its components.
Raphael Ducatez (Uni Lyon), Antti Knowles (Uni Geneva) and the MSC fellow studied the adjacency matrix of the Erdős–Rényi graph, a common model for a random graph. They proved [arXiv:2106.12519] that if the edge probability is small enough then the eigenvalues of the adjacency matrix at the edges of its spectrum form asymptotically a Poisson point process. They also established that each of the corresponding eigenvectors is localized in the vicinity of a vertex of large degree and the mass of the eigenvector decays exponentially around this vertex. Poisson statistics and eigenvector localization characterise non-universal behaviour.
For the adjacency matrix of the Erdős–Rényi graph, the same authors also fully characterised the spectral region, where the eigenvectors are completely delocalized [arXiv:2109.03227]. As mentioned above, eigenvector delocalization is a strong indication for universality in the corresponding spectral region.
In a further work, the same type of eigenvector localization as described above at the spectral edges was proved by Raphael Ducatez, Antti Knowles and the MSC fellow in a much larger region near the spectral edges for the Erdős–Rényi graph. The fundamental novelty was a lower bound on the minimal distance between two eigenvalues in this spectral region, which is obtained via an anti-concentration inequality for the sum of independent random variables due to Kesten.
The different parts of the project required novel techniques and insights as explained in the following.
The limiting eigenvalue density for the elliptic random matrix ensemble is not rotationally invariant. This posed a substantial difficulty in the fine stability analysis of the underlying self-consistent equation which was not present in previous works on the eigenvalue density of non-Hermitian random matrices on mesoscopic scales.
The analysis of the eigenvalues of Erdős–Rényi graphs at their spectral edges required a hierarchy of eigenvalue rigidity estimates. In order to identify the fluctuations of individual eigenvalues, the finest rigidity estimates had to be more precise than the typical size of the eigenvalue fluctuations, which was a major challenge. Then the asymptotic independence of the eigenvalue approximations was shown, which yielded Poisson statistics. These statistics implied a sufficient eigenvalue spacing to prove eigenvector localization.
The characterisation of the spectral region, where eigenvector delocalization occurs for Erdős–Rényi graphs, necessitated a stability analysis of the self-consistent equation describing the limiting eigenvalue density in the unstable edge regime in combination with distinguishing between typical and atypical vertices. This distinction was introduced in the authors' previous work on the bulk regime.
The key novelty for the eigenvector localization for Erdős–Rényi graphs in a large region near the spectral edge was a new lower bound on the eigenvalue spacing in this spectral region. In order to obtain this bound, the corresponding eigenvalues were characterised in terms of the resolvent of the adjacency matrix. Then, through a high-order resolvent expansion in a large ball around a vertex of high degree and after an appropriate conditioning, this characterisation could be expressed as a sum of independent random variables. Finally, the lower bound was shown by applying an anti-concentration inequality for such sums due to Kesten.
The limiting eigenvalue density for the elliptic random matrix ensemble is not rotationally invariant. This posed a substantial difficulty in the fine stability analysis of the underlying self-consistent equation which was not present in previous works on the eigenvalue density of non-Hermitian random matrices on mesoscopic scales.
The analysis of the eigenvalues of Erdős–Rényi graphs at their spectral edges required a hierarchy of eigenvalue rigidity estimates. In order to identify the fluctuations of individual eigenvalues, the finest rigidity estimates had to be more precise than the typical size of the eigenvalue fluctuations, which was a major challenge. Then the asymptotic independence of the eigenvalue approximations was shown, which yielded Poisson statistics. These statistics implied a sufficient eigenvalue spacing to prove eigenvector localization.
The characterisation of the spectral region, where eigenvector delocalization occurs for Erdős–Rényi graphs, necessitated a stability analysis of the self-consistent equation describing the limiting eigenvalue density in the unstable edge regime in combination with distinguishing between typical and atypical vertices. This distinction was introduced in the authors' previous work on the bulk regime.
The key novelty for the eigenvector localization for Erdős–Rényi graphs in a large region near the spectral edge was a new lower bound on the eigenvalue spacing in this spectral region. In order to obtain this bound, the corresponding eigenvalues were characterised in terms of the resolvent of the adjacency matrix. Then, through a high-order resolvent expansion in a large ball around a vertex of high degree and after an appropriate conditioning, this characterisation could be expressed as a sum of independent random variables. Finally, the lower bound was shown by applying an anti-concentration inequality for such sums due to Kesten.