Bose–Einstein-like condensation of deformed random matrix: a replica approach

Ikeda, Hisako

doi:10.1088/1742-5468/acb7d6

Cited by 4 publications

(2 citation statements)

References 48 publications

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“…We use a technique recently developed to find the smallest eigenvalue of random matrices [29]. One defines an artificial quadratic statistical mechanics model with configurations defined on the sphere, whose interaction tensor is given by the matrix of interest.…”

Section: Finding the Isolated Eigenvaluementioning

confidence: 99%

Arrangement of nearby minima and saddles in the mixed spherical energy landscapes

Kent-Dobias

2024

SciPost Phys.

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The mixed spherical models were recently found to violate long-held assumptions about mean-field glassy dynamics. In particular, the threshold energy, where most stationary points are marginal and that in the simpler pure models attracts long-time dynamics, seems to lose significance. Here, we compute the typical distribution of stationary points relative to each other in mixed models with a replica symmetric complexity. We examine the stability of nearby points, accounting for the presence of an isolated eigenvalue in their spectrum due to their proximity. Despite finding rich structure not present in the pure models, we find nothing that distinguishes the points that do attract the dynamics. Instead, we find new geometric significance of the old threshold energy, and invalidate pictures of the arrangement of most marginal inherent states into a continuous manifold.

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Section: Finding the Isolated Eigenvaluementioning

confidence: 99%

Arrangement of nearby minima and saddles in the mixed spherical energy landscapes

Kent-Dobias

2024

SciPost Phys.

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“…In particular, we perform a thorough study of the finite-N corrections to the average spectral density and compute the level compressibility in the intermediate phase, thereby providing a deeper understanding of the properties of the intermediate regime. Our analysis uses the replica method largely exploited in the analysis of spin glass models [45], but not only -for example, this tool has been recently applied to study the properties of the ground states in a deformed GOE ensemble [46]. Our replica study is developed for the case in which the entries of H are real numbers, so that H belongs to a deformed GOE ensemble (where the deformation is introduced by the addition of the diagonal random matrix A).…”

Section: Wigner-dyson Poissonmentioning

confidence: 99%

Replica approach to the generalized Rosenzweig-Porter model

et al. 2023

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The generalized Rosenzweig-Porter model with real (GOE) off-diagonal entries arguably constitutes the simplest random matrix ensemble displaying a phase with fractal eigenstates, which we characterize here by using replica methods. We first derive analytical expressions for the average spectral density in the limit in which the size NN of the matrix is large but finite. We then focus on the number of eigenvalues in a finite interval and compute its cumulant generating function as well as the level compressibility, i.e., the ratio of the first two cumulants: these are useful tools to describe the local level statistics. In particular, the level compressibility is shown to be described by a universal scaling function, which we compute explicitly, when the system is probed over scales of the order of the Thouless energy. Interestingly, the same scaling function is found to describe the level compressibility of the complex (GUE) Rosenzweig-Porter model in this regime. We confirm our results with numerical tests.

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Overlaps between eigenvectors of spiked, correlated random matrices: From matrix principal component analysis to random Gaussian landscapes

Pacco,

Ros

2023

Phys. Rev. E

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We consider pairs of Gaussian orthogonal ensemble matrices which are correlated with each others, and subject to additive and multiplicative rank-one perturbations. We focus on the regime of parameters in which the finite-rank perturbations generate outliers in the spectrum of the matrices. We investigate the statistical correlation (i.e., the typical overlap) between the eigenvectors associated to the outlier eigenvalues of each matrix in the pair, as well as the typical overlap between the outlier eigenvector of one matrix with the eigenvectors in the bulk of the spectrum of the other matrix. We discuss implications of these results for the signal recovery problem for spiked matrices, as well as for problems of high-dimensional random landscapes.

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Bose–Einstein-like condensation of deformed random matrix: a replica approach

Abstract: In this work, we investigate a symmetric deformed random matrix, which is obtained by perturbing the diagonal elements of the Wigner matrix. The eigenvector x m i n of the minimal eige… Show more

Cited by 4 publications

References 48 publications

Arrangement of nearby minima and saddles in the mixed spherical energy landscapes

Arrangement of nearby minima and saddles in the mixed spherical energy landscapes

Replica approach to the generalized Rosenzweig-Porter model

Overlaps between eigenvectors of spiked, correlated random matrices: From matrix principal component analysis to random Gaussian landscapes

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