Number statistics for $\beta$-ensembles of random matrices: applications to trapped fermions at zero temperature

Abstract

Let ${\mathcal{P}}_{\beta }^{(V)}(N_{\mathcal{I}})$ be the probability that a $N\times N$ $\beta$-ensemble of random matrices with confining potential $V(x)$ has $N_{\mathcal{I}}$ eigenvalues inside an interval $\mathcal{I}=[a,b]$ of the real line. We introduce a general formalism, based on the Coulomb gas technique and the resolvent method, to compute analytically ${\mathcal{P}}_{\beta }^{(V)}(N_{\mathcal{I}})$ for large $N$. We show that this probability scales for large $N$ as ${\mathcal{P}}_{\beta }^{(V)}(N_{\mathcal{I}})\approx \exp \left(-\beta N^2{\psi }^{(V)}(N_{\mathcal{I}}/N)\right)$, where $\beta$ is the Dyson index of the ensemble. The rate function ${\psi }^{(V)}(k_{\mathcal{I}})$, independent of $\beta$, is computed in terms of single integrals that can be easily evaluated numerically. The general formalism is then applied to the classical $\beta$-Gaussian ($\mathcal{I}=[-L,L]$), $\beta$-Wishart ($\mathcal{I}=[1,L]$) and $\beta$-Cauchy ($\mathcal{I}=[-L,L]$) ensembles. Expanding the rate function around its minimum, we find that generically the number variance $\mathrm{Var}(N_{\mathcal{I}})$ exhibits a non-monotonic behavior as a function of the size of the interval, with a maximum that can be precisely characterized. These analytical results, corroborated by numerical simulations, provide the full counting statistics of many systems where random matrix models apply. In particular, we present results for the full counting statistics of zero temperature one-dimensional spinless fermions in a harmonic trap.

34 pages, 19 figures

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Cited by in corpus (34)

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• Counting statistics for non-interacting fermions in a $d$-dimensional potential
• Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble
• Linear statistics and pushed Coulomb gas at the edge of beta random matrices: four paths to large deviations
• Full counting statistics for interacting trapped fermions
• Truncated linear statistics associated with the top eigenvalues of random matrices
• Exact large deviation function of spin current for the one dimensional XX spin chain with domain wall initial condition
• Periodic Airy process and equilibrium dynamics of edge fermions in a trap
• Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature
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• Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble
• Fluctuations of observables for free fermions in a harmonic trap at finite temperature
• Dynamical Signatures of Chaos to Integrability Crossover in $2\times 2$ Generalized Random Matrix Ensembles
• Free fermions and $\alpha$-determinantal processes
• Correlations vs connectivity in R-charge
• Current fluctuations in the Dyson Gas
• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• Dynamically emergent correlations in Brownian particles subject to simultaneous non-Poissonian resetting protocols
• Hole probability for noninteracting fermions in a $d$-dimensional trap
• Theory for the conditioned spectral density of non-invariant random matrices
• Full counting statistics of 1d short-range Riesz gases in confinement
• Replica-symmetric approach to the typical eigenvalue fluctuations of Gaussian random matrices
• General truncated linear statistics for the top eigenvalues of random matrices
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle
• Importance Sampling for counting statistics in one-dimensional systems
• Analytic approach for the number statistics of non-Hermitian random matrices
• Linear statistics at the microscopic scale for the 2D Coulomb gas
• Replica approach to the generalized Rosenzweig-Porter model
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"