Phase transitions and edge scaling of number variance in Gaussian random matrices

Abstract

We consider $N\times N$ Gaussian random matrices, whose average density of eigenvalues has the Wigner semi-circle form over $[-\sqrt{2},\sqrt{2}]$. For such matrices, using a Coulomb gas technique, we compute the large $N$ behavior of the probability ${\mathcal{P}}_{N,L}(N_L)$ that $N_L$ eigenvalues lie within the box $[-L,L]$. This probability scales as ${\mathcal{P}}_{N,L}(N_L={\kappa }_{L}N)\approx \exp \left(-\beta N^2{\psi }_L({\kappa }_L)\right)$, where $\beta$ is the Dyson index of the ensemble and ${\psi }_L({\kappa }_L)$ is a $\beta$-independent rate function that we compute exactly. We identify three regimes as $L$ is varied: (i) $N^{-1}\ll L<\sqrt{2}$ (bulk), (ii) $L\sim \sqrt{2}$ on a scale of $\mathcal{O}(N^{-2/3})$ (edge) and (iii) $L>\sqrt{2}$ (tail). We find a dramatic non-monotonic behavior of the number variance $V_N(L)$ as a function of $L$: after a logarithmic growth $\propto \ln (NL)$ in the bulk (when $L\sim \mathcal{O}(1/N)$), $V_N(L)$ decreases abruptly as $L$ approaches the edge of the semi-circle before it decays as a stretched exponential for $L>\sqrt{2}$. This "drop-off" of $V_N(L)$ at the edge is described by a scaling function ${\tilde{V}}_{\beta }$ which smoothly interpolates between the bulk (i) and the tail (iii). For $\beta =2$ we compute ${\tilde{V}}_2$ explicitly in terms of the Airy kernel. These analytical results, verified by numerical simulations, directly provide for $\beta =2$ the full statistics of particle-number fluctuations at zero temperature of 1d spinless fermions in a harmonic trap.

5 pag., 3 fig, published version

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• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
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• The entanglement entropy of one-dimensional gases
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Cited by in corpus (26)

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• Instability of many-body localized systems as a phase transition in a nonstandard thermodynamic limit
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• Free fermions at the edge of interacting systems
• Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions
• Statistics of fermions in a $d$-dimensional box near a hard wall
• Counting statistics for non-interacting fermions in a rotating trap
• Gap probability and full counting statistics in the one dimensional one-component plasma
• Joint statistics of quantum transport in chaotic cavities
• Dynamical phase transition in the occupation fraction statistics for non-crossing Brownian particles
• Dynamically emergent correlations in bosons via quantum resetting
• Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble
• Noninteracting particles in a harmonic trap with a stochastically driven center
• An exact formula for the variance of linear statistics in the one-dimensional jellium mode
• 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
• Nonequilibrium dynamics of noninteracting fermions in a trap
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle
• Importance Sampling for counting statistics in one-dimensional systems
• Full counting statistics of 1d short-range Riesz gases in confinement
• General truncated linear statistics for the top eigenvalues of random matrices
• Crossover in densities of confined particles with finite range of interaction
• Replica approach to the generalized Rosenzweig-Porter model
• The Tracy-Widom distribution at large Dyson index
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"