Alberto Berretti

New Monte Carlo method for the self-avoiding walk

We introduce a new Monte Carlo algorithm for the self-avoiding walk (SAW), and show that it is particularly efficient in the critical region (long chains). We also introduce new and more efficient statistical techniques. We employ these methods to extract numerical estimates for the critical parameters of the SAW on the square lattice. We findμ=2.63820 ± 0.00004 ± 0.00030γ=1.352 ± 0.006 ± 0.025νv=0.7590 ± 0.0062 ± 0.0042 where the first error bar represents systematic error due to corrections to scaling (subjective 95% confidence limits) and the second bar represents statistical error (classical 95% confidence limits). These results are based on SAWs of average length ≈ 166, using 340 hours CPU time on a CDC Cyber 170–730. We compare our results to previous work and indicate some directions for future research.

Natural Boundaries for Area-Preserving Twist Maps

We consider KAM invariant curves for generalizations of the standard map of the form (x′, y′)=(x+y′, y+ɛf(x)), wheref(x) is an odd trigonometric polynomial. We study numerically their analytic properties by a Padé approximant method applied to the function which conjugates the dynamics to a rotation θ↦θ+ω. In the complexɛ plane, natural boundaries of different shapes are found. In the complexθ plane the analyticity region appears to be a strip bounded by a natural boundary, whose width tends linearly to 0 asɛ tends to the critical value.

Scaling properties for the radius of convergence of a Lindstedt series: the standard map

Abstract By using a version of the tree expansion for the standard map, we prove that the radius of convergence of the corresponding Lindstedt series satisfies a scaling property as the (complex) rotation number tends to any rational (resonant) value, non-tangentially to the real axis. By suitably reseating the perturbative parameter , the function conjugating the dynamic on the (KAM) invariant curve with given rotation number to a linear rotation has a well defined limit, which can be explicitly computed.

Scaling properties for the radius of convergence of Lindstedt series: generalized standard maps

Abstract For a class of symplectic two-dimensional maps which generalize the standard map by allowing more general nonlinear terms, the radius of convergence of the Lindstedt series describing the homotopically non-trivial invariant curves is proved to satisfy a scaling law as the complexified rotation number tends to a rational value non-tangentially to the real axis, thus generalizing previous results of the authors. The function conjugating the dynamics to rotations by ω possesses a limit which is explicitly computed and related to hyperelliptic functions in the case of nonlinear terms which are trigonometric polynomials. The case of the standard map is shown to be non-generic.

Scaling of the critical function for the standard map: some numerical results

The behaviour of the critical function for the breakdown of the homotopically non-trivial invariant (KAM) curves for the standard map, as the rotation number tends to a rational number, is investigated using a version of Greenes residue criterion. The results are compared with the analogous ones for the radius of convergence of the Lindstedt series, in which case rigorous theorems have been proved. The conjectured interpolation of the critical function in terms of the Bryuno function is discussed.

Non-universal behaviour of scaling properties for generalized semistandard and standard maps

We consider two-dimensional maps generalizing the semistandard map by allowing more general analytic nonlinear terms having only Fourier components fν with positive label ν, and study the analyticity properties of the function conjugating the motion on analytic homotopically non-trivial invariant curves to rotations. Then we show that, if the perturbation parameter is suitably rescaled, when the rotation number tends to a rational value non-tangentially to the real axis from complex values, the limit of the conjugating function is a well defined analytic function. The rescaling depends not only on the limit value of the rotation number, but also on the map, and it is obtainable by the solution of a Diophantine problem: so no universality property is exhibited. We also show that the rescaling can be different from that of the corresponding generalized standard maps, i.e. of the maps also having the Fourier components f-ν = f*ν. The results allow us to give quantitative bounds, from above and from below, on the radius of convergence of the limit function for generalized standard maps in the case of nonlinear terms which are trigonometric polynomials, solving a problem left open in a previous work of ours.