On memory effect in modified gravity theories

In this note, we discuss the gravitational memory effect in higher derivative and infinite derivative gravity theories and give the detailed relevant calculations whose results were given in our recent works. We show that the memory effect in higher derivative gravity takes the same form as in pure GR at large distances, whereas at small distances, the results are different. We also demonstrated that, in infinite derivative gravity, the memory is reduced via error function as compared to Einstein's gravity. For the lower bound on the mass scale of non-locality, the memory is essentially reproduces the usual GR result at distances above at very small distances.Comment: 14 pages, references added, published in Turkish Journal of Physic

PP-waves as Exact Solutions to Ghost-free Infinite Derivative Gravity

We construct exact pp-wave solutions of ghost-free infinite derivative gravity. These waves described in the Kerr-Schild form also solve the linearized field equations of the theory. We also find an exact gravitational shock wave with non-singular curvature invariants and with a finite limit in the ultraviolet regime of non-locality which is in contrast to the divergent limit in Einstein's theory.Comment: 13 pages, references added, version published in Phys. Rev.

Max-linear models on infinte graphs generated by Bernoulli bond percolation

We extend previous work of max-linear models on finite directed acyclic graphs to infinite graphs, and investigate their relations to classical percolation theory. We formulate results for the oriented square lattice graph $\mathbb{Z}^2$ and nearest neighbor bond percolation. Focus is on the dependence introduced by this graph into the max-linear model. As a natural application we consider communication networks, in particular, the distribution of extreme opinions in social networks.Comment: 18 page

The random walk on the random connection model

We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices $x$ and $y$ are connected with probability that asymptotically behaves like $|x-y|^{-\alpha}$ with $\alpha>d$, where $d$ denotes the dimension of the underlying Euclidean space. More precisely, focus is on the random connection model in which the vertex set is given by the realization of a homogeneous Poisson point process. We show that this random graph exhibits the same properties as classical discrete long-range percolation models studied in [3] with regard to recurrence and transience of the random walk. The recurrence results are valid for every intensity of the Poisson point process while the transience results hold for large enough intensity. Moreover, we address a question which is related to a conjecture in [16] for this graph.Comment: New version of the manuscript with some extension