A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting $k$-body fermionic Embedded Gaussian Ensembles

Abstract

In spite of its simplicity, the central limit theorem captures one of the most outstanding phenomena in mathematical physics, that of universality. While this classical result is well understood it is still not very clear what happens to this universal behaviour when the random variables become correlated. A fruitful mathematically laboratory to investigate the rising of new universal properties is offered by the set of eigenvalues of random matrices. In this regard a lot of work has been done using the standard random matrix ensembles and focusing on the distribution of extreme eigenvalues. In this case, the distribution of the largest -- or smallest -- eigenvalue departs from the Fisher-Tippett-Gnedenko theorem yielding the celebrated Tracy-Widom distribution. One may wonder, yet again, how robust is this new universal behaviour captured by the Tracy-Widom distribution when the correlation among eigenvalues changes. Few answers have been provided to this poignant question and our intention in the present work is to contribute to this interesting unexplored territory. Thus, we study numerically the probability distribution for the normalized largest eigenvalue of the interacting $k$-body fermionic orthogonal and unitary Embedded Gaussian Ensembles in the diluted limit. We find a smooth transition from a slightly asymmetric Gaussian-like distribution, for small $k/m$, to the Tracy-Widom distribution as $k/m\to 1$, where $k$ is the rank of the interaction and $m$ is the number of fermions. Correlations at the edge of the spectrum are stronger for small values of $k/m$, and are independent of the number of particles considered. Our results indicate that subtle correlations towards the edge of the spectrum distinguish the statistical properties of the spectrum of interacting many-body systems in the dilute limit, from those expected for the standard random matrix ensembles.

arXiv comment10 pages, 4 figures

References in corpus (41)

• Random-Matrix Theory of Quantum Transport
• Random Matrix Theories in Quantum Physics: Common Concepts
• Level-Spacing Distributions and the Airy Kernel
• On Orthogonal and Symplectic Matrix Ensembles
• Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles
• Universality Classes for Extreme Value Statistics
• Level-Spacing Distributions and the Airy Kernel
• Application of Random Matrix Theory to Biological Networks
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Extreme value statistics of correlated random variables: a pedagogical review
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• The Index Distribution of Gaussian Random Matrices
• Review of the k-Body Embedded Ensembles of Gaussian Random Matrices
• Spectral Properties of the k-Body Embedded Gaussian Ensembles of Random Matrices
• Exact extremal statistics in the classical $1d$ Coulomb gas
• Large Deviations of the Smallest Eigenvalue of the Wishart-Laguerre Ensemble
• Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase
• Extremes of $2d$ Coulomb gas: universal intermediate deviation regime
• Large deviation function for the number of eigenvalues of sparse random graphs inside an interval
• Large deviations of the maximum of independent and identically distributed random variables
• Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble
• Near-extreme eigenvalues and the first gap of Hermitian random matrices
• The Two-Body Random Ensemble in Nuclei
• Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions
• Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles
• Disordered mesoscopic systems with interactions: induced two-body ensembles and the Hartree-Fock approach
• Gap probability and full counting statistics in the one dimensional one-component plasma
• Symmetry Properties of the k-Body Embedded Unitary Ensemble of Random Matrices
• Spectral statistics of the k-body random-interaction model
• Large deviation theory for diluted Wishart random matrices
• Spectral Properties of the Jacobi Ensembles via the Coulomb Gas approach
• Particle Diagrams and Statistics of Many-Body Random Potentials
• Quenched many-body quantum dynamics with $k$-body interactions using $q$-Hermite polynomials
• Universal behavior of the full particle statistics of one-dimensional Coulomb gases with an arbitrary external potential
• Phase transitions in the condition number distribution of Gaussian random matrices
• Chaos in fermionic many-body systems and the metal-insulator transition
• Theory for the conditioned spectral density of non-invariant random matrices
• Analytic approach for the number statistics of non-Hermitian random matrices
• Truncated linear statistics in the one dimensional one-component plasma

Cited by in corpus (5)

• Two species $k$-body embedded Gaussian unitary ensembles: $q$-normal form of the eigenvalue density
• Bivariate moments of the two-point correlation function for embedded Gaussian unitary ensemble with $k$-body interactions
• Distribution of lowest eigenvalue in $k$-body bosonic random matrix ensembles
• On the extremal spectral properties of random graphs
• Surmise for random matrices' level spacing distributions beyond nearest-neighbors