Interaction-correlated random matrices

Abstract

We introduce a family of random matrices where correlations between matrix elements are induced via interaction-derived Boltzmann factors. Varying these yields access to different ensembles. We find a universal scaling behavior of the finite-size statistics characterized by a heavy-tailed eigenvalue distribution whose extremes are governed by the Fréchet extreme value distribution for the case corresponding to a ferromagnetic Ising transition. The introduction of a finite density of nonlocal interactions restores standard random-matrix behavior. Suitably rescaled average extremes, playing a physical role as an order parameter, can thus discriminate aspects of the interaction structure; they also yield further nonuniversal information. In particular, the link between maximum eigenvalues and order parameters offers a potential route to resolving long-standing problems in statistical physics, such as deriving the exact magnetization scaling function in the two-dimensional Ising model at criticality.

Figures (4)

• Figure 1: Main: Distribution of the absolute eigenvalues of an ensemble of rescaled (see text) symmetric random matrices $\{\mathcal{M}\}$ as a realization of a 2d Ising model on a square lattice of linear size $L=2^8, 2^9, 2^{10}, 2^{11}$ and $2^{12}$, at the critical point $T=T_c$. The ensemble size is $10^5$ for the smallest and $10^4$ for the largest system size. The plots collapse for all system sizes. The distribution has a heavy tail $\sim |\lambda|^{-\text{f}-1}$ with tail exponent $11/3$ (dashed line) in the large size limit. The solid line shows our theoretical prediction for the asymptotic probability distribution compatible with a $t$-distribution with $\text{f}=8/3$ degrees of freedom (Eq. (\ref{['t-dist']})). Inset: Distribution of the eigenvalues for different system sizes compared with the corresponding $t$-distribution Eq. (\ref{['t-dist']}) (solid line). The red dashed line shows the semi-circle law of standard RMT for the PDF of bulk eigenvalues.
• Figure 2: Finite-size scaling collapse of the maximum eigenvalue $\lambda_{\text{max}}$ distributions for an ensemble of rescaled symmetric random matrices $\{\mathcal{M}\}$ at $T=T_c$. For each system size $L$, we have generated $10^6$ independent samples. The scaling exponent $\text{b}=0.375(15)$ is measured by examining the scaling relation $\langle \lambda_{\text{max}}\rangle\sim L^{\text{b}}$ shown in the inset. This exponent is further verified by examining the scaling relation of the standard deviation of $\lambda_{\text{max}}$ with system size (Inset). The solid line in the main panel shows our theoretical predictions based on Eqs. (\ref{['univ_Frechet']}) and (\ref{['Frechet']}) for the universal function describing the collapsed data as a Fréchet extreme value distribution with shape parameter $\text{k}=1/\text{f}=3/8$. The red dashed line shows the prediction by the standard RMT for comparison.
• Figure 3: Main: Distribution of the eigenvalues of an ensemble of $10^5$ rescaled symmetric random matrices as a realization of a 2d Ising model with additional $10$ nonlocal links per spin ($q=14$) on a square lattice of linear size $L=2^{12}$, at its critical point $T_c=q$ (symbols) compared with the Wigner's semicircle law (the red solid line). Inset: Distribution of the maximum eigenvalues (symbols) sitting at the right edge $\lambda\simeq\sqrt{2}$ compared with the GOE $\mathcal{TW}_{1}$ distribution (the solid line).
• Figure 4: Main: The average rescaled maximum eigenvalue $\tilde{\lambda}_{\text{max}}=L^{-1/2}\lambda_{\text{max}}$ as a function of the reduced temperature $T/T_c$ for the pure 2d Ising model (half-filled circles) and the 2d Ising model with nonlocal interaction links with different $q=14, 19$ and $34$ (triangle symbols) compared with their corresponding exact solution for order parameter shown by the solid lines. Upper-left inset: Our data agree well with the scaling relation $\langle\tilde{\lambda}_{\text{max}}\rangle\sim (T-T_c)^\beta$ with the known exact exponents $\beta=1/8$ and $1/2$ for the pure 2d Ising model and the mean-field Ising model, respectively. Lower-right inset: $\langle\tilde{\lambda}_{\text{max}}\rangle$ versus temperature for different values of $q=14, 19$ and $34$ from left to right, respectively, that vanishes at $T_c=q$.