Scaling laws for the largest Lyapunov exponent in long-range systems: A random matrix approach

We investigate the laws that rule the behavior of the largest Lyapunov exponent (LLE) in many particle systems with long-range interactions. We consider as a representative system the so-called Hamiltonian $\alpha -\mathrm{XY}$ model where the adjustable parameter $\alpha$ controls the range of the interactions of N ferromagnetic spins in a lattice of dimension d. In previous work the dependence of the LLE with the system size N, for sufficiently high energies, was established through numerical simulations. In the thermodynamic limit, the LLE becomes constant for $\alpha >d$ whereas it decays as an inverse power law of N for $\alpha <d.\mathrm{}$ A recent theoretical calculation based on a geometrization of the dynamics is consistent with these numerical results. Here we show that the scaling behavior can also be explained by a random matrix approach, in which the tangent mappings that define the Lyapunov exponents are modeled by random simplectic matrices drawn from a suitable ensemble.