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

Ricardo Marino; Satya N. Majumdar; Grégory Schehr; Pierpaolo Vivo

doi:10.1103/PhysRevLett.112.254101

paper

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

arXiv:1404.0575 · doi:10.1103/PhysRevLett.112.254101

Abstract

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

arXiv comment5 pag., 3 fig, published version

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

Abstract

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