Critical Behaviors and Finite-Size Scaling of Principal Fluctuation Modes in Complex Systems*

Complex systems consisting of N agents can be investigated from the aspect of principal fluctuation modes of agents. From the correlations between agents, an N x N correlation matrix C can be obtained. The principal fluctuation modes are defined by the eigenvectors of C. Near the critical point of a complex system, we anticipate that the principal fluctuation modes have the critical behaviors similar to that of the susceptibity. With the Ising model on a two-dimensional square lattice as an example, the critical behaviors of principal fluctuation modes have been studied. The eigenvalues of the first 9 principal fluctuation modes have been invesitigated. Our Monte Carlo data demonstrate that these eigenvalues of the system with size L and the reduced temperature t follow a finite-size scaling form lambda(n) (L, t) =L-gamma/nu f(n) (tL(1/nu)), where gamma is critical exponent of susceptibility and nu is the critical exponent of the correlation length. Using eigenvalues lambda(1), lambda(2) and lambda(6), we get the finite-size scaling form of the second moment correlation length xi (L, t) = L (xi) over tilde (tL(1/nu)) It is shown that the second moment correlation length in the two-dimensional square lattice is anisotropic.