Finite-range Coulomb gas models. I. Some analytical results

Dyson has shown an equivalence between infinite-range Coulomb gas models and classical random matrix ensembles for the study of eigenvalue statistics. In this paper, we introduce finite-range Coulomb gas models as a generalization of the Dyson models with a finite range of eigenvalue interactions. As the range of interaction increases, there is a transition from Poisson statistics to classical random matrix statistics. These models yield distinct universality classes of random matrix ensembles. They also provide a theoretical framework to study banded random matrices, and dynamical systems the matrix representation of which can be written in the form of banded matrices.

Figure 1

Spacing distributions, $p_k\left(s\right)$ vs $s$, for $d=2$. Here, $k$ is the order of the distribution, with $k=0$ corresponding to the nearest neighbor. The different frames correspond to (a) $\beta =1$, (b) $\beta =2$, and (c) $\beta =4$. In each case, we show $p_k\left(s\right)$ up to $k=k_m\left(\beta \right)$. For $k\ge k_m$, the exact result is numerically indistinguishable from the MF result on this scale.

Figure 2

Comparison of spacing distributions for $d=2$ and $\beta =1,2,4$. The different frames show exact results for (a) $p_0\left(s\right)$, (b) $p_1\left(s\right)$, and (c) $p_2\left(s\right)$.

Figure 3

Nearest-neighbor spacing distribution, $p_0\left(s\right)$ vs $s$, for $d=0,1,2$ and (a) $\beta =1$, (b) $\beta =2$, and (c) $\beta =4$. The $d=0$ case corresponds to Poisson ensembles. For comparison, we also show the corresponding results for classical ensembles, which arise for $d=N-1$. These correspond to GOE, GUE, and GSE for $\beta =1,2,4$, respectively.

Figure 4

Comparison between exact and mean-field (MF) results for $p_{k_m}\left(s\right)$ in the $d=2$ case. The different frames show the cases (a) $\beta =1$, (b) $\beta =2$, and (c) $\beta =4$. For $k\ge k_m$, the exact and MF results are numerically indistinguishable on the scale of this plot.

Figure 5

Two-point correlation function $R_2\left(s\right)$ for (a) $\beta =1$, (b) $\beta =2$, and (c) $\beta =4$. In each frame, we plot $R_2\left(s\right)$ vs $s$ for $d=0,1,2$ and the classical-ensemble result ($d=N-1$). The $d=0$ case yields the Poisson result.

Figure 6

Analogous to Fig. 5, but for the two-point cluster function $Y_2\left(s\right)=1-R_2\left(s\right)$.