Multi-dimensional chaos II: String scattering amplitudes, curve repulsion, and RMT

Abstract

Multi-dimensional chaos refers to processes described by erratic functions of several dynamical variables. In this letter we analyze the string scattering amplitudes of highly-excited states and ground states. We show that the amplitudes, which depend on a scattering angle and a polarization angle, are characterized by two sets of non-intersecting curves associated with the vanishing of the derivatives with respect to the angles. We introduce the notion of the "area eigenvalue" $A_n$ associated with the $n$-th curve. We compute the spacings $\delta_{n}= A_{n+1}-A_n$ and their ratios $r_{n}=\frac{\delta_{n+1}}{\delta_n}$. We show that the distributions of the spacing ratios take the form of the RMT Gaussian $\beta$-ensembles. The curves associated with the scattering angle tend to converge to the Gaussian Orthogonal Ensemble value of $\beta=1$ and those related to the polarization angle to the Gaussian Unitary Ensemble $\beta=2$. We also compute the ``areas form factor" associated with the areas and discover the regions of decline, ramp and plateau which characterize chaotic processes. The slope of the ramp seems to agree with the $\beta$ values extracted from the distribution of the spacing ratios.