The study of random graphs has become very popular for real-life network modeling, such as social networks or financial networks. Inhomogeneous long-range percolation (or scale-free percolation) on the lattice Z^d, d ≥ 1, is a particular attractive example of a random graph model because it fulfills several stylized facts of real-life networks. For this model, various geometric properties, such as the percolation behavior, the degree distribution and graph distances, have been analyzed. In the present paper, we complement the picture of graph distances and we prove continuity of the percolation probability in the phase transition point. We also provide an illustration of the model connected to financial networks.

In this article we give a general criterion for some dependent Gaussian models to belong to maximal domain of attraction of Gumbel, following an application of the Stein–Chen method studied in Arratia et al. (Ann Probab 17(1):9–25, ). We also show the convergence of the associated point process. As an application, we show the conditions are satisfied by some of the well-known supercritical Gaussian interface models, namely, membrane model, massive and massless discrete Gaussian free field, fractional Gaussian free field.

In a recent work Levine et al. (Ann Henri Poincaré 17:1677–1711, 2016 . https://doi.org/10.1007/s00023-015-0433-x ) prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus.

Let {X _t , t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails, and let {Θ _t , t ≥ 1} be a sequence of positive random variables independent of the sequence {X _t , t ≥ 1}. We will discuss the tail probabilities and almost-sure convergence of X _(∞) = ∑ _t=1 ^∞Θ _t X _t ⁺ (where X ⁺ = max{0, X}) and max_1≤k<∞∑ _t=1 ^k Θ _t X _t , and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illustrate how the conditions on the slowly varying function involved in the tail probability of X ₁ help to control the tail behavior of the randomly weighted sums. Note that, the above results allow us to choose X ₁, X ₂,… as independent and identically distributed positive random variables. If X ₁ has a regularly varying tail of index -α, where α > 0, and if {Θ _t , t ≥ 1} is a positive sequence of random variables independent of {X _t }, then it is known – which can also be obtained from the sufficient conditions in this article – that, under some appropriate moment conditions on {Θ _t , t ≥ 1}, X _(∞) = ∑ _t=1 ^∞Θ _t X _t converges with probability 1 and has a regularly varying tail of index -α. Motivated by the converse problems in Jacobsen, Mikosch, Rosiński and Samorodnitsky (2009) we ask the question: if X _(∞) has a regularly varying tail then does X ₁ have a regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including the nonvanishing Mellin transform of ∑ _t=1 ^∞Θ _t along some vertical line in the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.