Distribution of the Ratio of Consecutive Level Spacings in Random Matrix Ensembles

We derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices. This ratio distribution was recently introduced to study spectral properties of many-body problems, as, contrary to the standard level spacing distributions, it does not depend on the local density of states. Our Wigner-like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit. Quantitative improvements are found through a polynomial expansion. Examples from a quantum many-body lattice model and from zeros of the Riemann zeta function are presented.

Figure 1

Distribution of the ratio of consecutive level spacings $P(r)$ for Poisson and RMT ensembles: full lines are the surmise Eq. (7), points are numerical results obtained by diagonalizing matrices of size $N=1000$ with Gaussian distributed entries, averaged over ${10}^5$ histograms. Inset: the distribution $P(\tilde{r})$.

Figure 2

Difference $\delta P(r)=P_{\mathrm{num}}(r)-P_W(r)$ between the numerics and the surmise (7), with the same data as in Fig. 1. The fit function is given by Eq. (8). Green diamonds are results of exact calculations obtained from (9) for GUE.

Figure 3

(a) $P(r)-P_{\infty }(r)$ for GUE and various matrix sizes. Inset: constant $C_N$ from the fit (8) as a function of matrix size $N$ (solid line is a fit $1/N$). (b) Density distributions for the overlapping ratio $r_n^{(2)}=(e_{n+2}-e_n)/(e_{n+1}-e_{n-1})$ for Poisson variables and for the three classical RMT ensembles (same color code as in Fig. 1). In the Poisson case it is given by $P(r^{(2)})=r^{(2)}(r^{(2)}+2)/(1+r^{(2)}{)}^2$ for $0\le r\le 1$, and obtained from (4) for $r\ge 1$.

Figure 4

Histogram of the ratio of consecutive level spacings $P(r)$. Black: Quantum Ising model in fields $\lambda =\alpha =0.5$ in sector of eigenvectors of $\widehat{T}$ with eigenvalue ${\omega }_3$ [${\omega }_j=\exp (2i\pi j/L)$, $j=0,1,\dots ,L-1$], for $L=18$ spins (dimension of $\mathrm{\text{eigenspace}}=14541$). Violet: The same for zeros of Riemann zeta function up the critical line (${10}^4$ levels starting from the ${10}^{22}$th zero, taken from Ref. 18). Full lines correspond to the Wigner-like surmise Eq. (7) with, respectively, $\beta =1$ and $\beta =2$. Inset: Difference between the numerics and these surmises.