In this work we focus on reactions on small-world networks (SWN's), disordered graphs of much recent interest. We study the target problem, since it allows an exact solution on regular lattices. On SWN's we find that the decay of the targets (for which we extend the formalism to disordered lattices) is again related to S(n), the mean number of distinct sites visited in n steps, although the S(n) vs n dependence changes here drastically in going from regular linear chains to their SWN.

We develop an efficient algorithm for evaluating divergent perturbation expansions of field theories in the bare coupling constant g B for which we possess a finite number L of expansion coefficients plus two more informations: The knowledge of the large-order behavior proportional to (−α) k k!k β g k B , with a known growth parameter α, and the knowledge of the approach to scaling being of the type c + c ′ /g ω B , with constants c, c ′ and a critical exponent of approach ω. The latter information leads to an increase in the speed of convergence and a high accuracy of the results. The algorithm is applied to the six-and seven-loop expansions for the critical exponents of O(N )-symmetric φ 4 -theories, and the result for the critical exponent α is compared with the recent satellite experiment.

We investigate the trapping of random walkers on small-world networks (SWN's), irregular graphs. We derive bounds for the survival probability Phi(SWN)(n) and display its analysis through cumulant expansions. Computer simulations are performed for large SWNs. We show that in the limit of infinite sizes, trapping on SWNs is equivalent to trapping on a certain class of random trees, which are grown during the random walk.

In this work we investigate the spectra of Laplacian matrices that determine many dynamic properties of scale-free networks below and at the percolation threshold. We use a replica formalism to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum for a general type of tree-like networks. Close to the percolation threshold we find characteristic scaling functions for the density of states ρ(λ) of scale-free networks. ρ(λ) shows characteristic power laws ρ(λ) ∼ λ α 1 or ρ(λ) ∼ λ α 2 for small λ, where α1 holds below and α2 at the percolation threshold. In the range where the spectra are accessible from a numerical diagonalization procedure the two methods lead to very similar results.

Treating the relaxation dynamics of an ensemble of random hyperbranched macromolecules in dilute solution represents a challenge even in the framework of Rouse-type approaches, which focus on generalized Gaussian structures (GGSs). The problem is that one has to average over a large class of realizations of molecular structures, and that each molecule undergoes its own dynamics. We show that a replica formalism allows to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum. Interestingly, for a specific probability distribution of the spring strengths of the GGSs, the integral equation takes a particularly simple form. Given that several dynamical observables, such as the mechanical moduli G'(omega) and G"(omega), as well as the averaged monomer displacement are relatively simple functions of the eigenvalues, we can use the obtained spectra to compute the corresponding averaged dynamical forms. Comparing the results obtained from this approach and from extensive diagonalizations of hyperbranched GGSs we find a very good agreement.