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We investigate the laws that rule the behavior of the largest Lyapunovexponent (LLE) in many particle systems with long range interactions. Weconsider as a representative system the so-called Hamiltonian alpha-XY modelwhere the adjustable parameter alpha controls the range of the interactions ofN ferromagnetic spins in a lattice of dimension d. In previous work thedependence of the LLE with the system size N, for sufficiently high energies,was established through numerical simulations. In the thermodynamic limit, theLLE becomes constant for alpha greater than d whereas it decays as an inversepower law of N for alpha smaller than d. A recent theoretical calculation basedon Pettini's geometrization of the dynamics is consistent with these numericalresults (M.-C. Firpo and S. Ruffo, cond-mat/0108158). Here we show that thescaling behavior can also be explained by a random matrix approach, in whichthe tangent mappings that define the Lyapunov exponents are modeled by randomsimplectic matrices drawn from a suitable ensemble.Comment: 5 pages, no figure

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