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The problem of random walk is considered in one dimension in the simultaneouspresence of a quenched random force field and long-range connections theprobability of which decays with the distance algebraically as p_l ~ \betal^{-s}. The dynamics are studied mainly by a numerical strong disorderrenormalization group method. According to the results, for s>2 the long-rangeconnections are irrelevant and the mean-square displacement increases as<x^2(t)> ~ (ln t)^{2/\psi} with the barrier exponent \psi=1/2, which is knownin one-dimensional random environments. For s<2, instead, the quenched disorderis found to be irrelevant and the dynamical exponent is z=1 like in ahomogeneous environment. At the critical point, s=2, the interplay betweenquenched disorder and long-range connections results in activated scaling,however, with a non-trivial barrier exponent \psi(\beta), which decayscontinuously with \beta, but is independent of the form of the quencheddisorder. Upper and lower bounds on \psi(\beta) are established and numericalestimates are given for various values of \beta. Beside random walks, accuratenumerical estimates of the graph dimension and the resistance exponent aregiven for various values of \beta at s=2.

Competition between quenched disorder and long-range connections: A numerical study of diffusion