We identify the asymptotic probability of a configuration model CM n (d) to produce a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which the asymptotic number of simple connected graphs with a prescribed degree sequence follows.

c . In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation on H(d, n) for d = 2, 3, 4. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdős-Rényi random graph.MSC 2010. 60K35, 60K37, 82B43.1 Given a sequence of random variables (Xn) n∈N , we write Xn = Θ(f (n)) w.h.p. (with high probability) if there exist constants C ≥ c > 0 such that Pp(cf (n) ≤ Xn ≤ Cf (n)) → 1 as n → ∞.2 Subsequent results in [3,33,36,40] are much sharper and comprehensive than what is summarized here, and there is an extensive body of literature on the problem. 3 Given three sequences (an), (bn), (cn), we write that an = bn + O(cn) when there exists a constant K < ∞ such that |an − bn| ≤ Kcn for all n.

We study the connectivity of random subgraphs of the d-dimensional Hamming graph H(d, n), which is the Cartesian product of d complete graphs on n vertices. We sample the random subgraph with an i.i.d. Bernoulli bond percolation on H(d, n) with parameter p. We identify the window of the transition: when np−log n → −∞ the probability that the graph is connected goes to 0, while when np − log n → +∞ it converges to 1. We also investigate the connectivity probability inside the critical window, namely when np − log n → t ∈ R. We find that the threshold does not depend on d, unlike the phase transition of the giant connected component of the Hamming graph (see [1]). Within the critical window, the connectivity probability does depend on d. We determine how.

We introduce a model for inhomogeneous random graphs designed to have a lot of flexibility in the assignment of the degree sequence and the individual edge probabilities while remaining tractable. To achieve this we run a Poisson point process over the square [0, 1] 2 , with an intensity proportional to a kernel W (x, y) and identify every couple of vertices of the graph with a subset of the square, adding an edge between them if there is a point in such subset. This ensures unconditional independence among edges and makes many statements much easier to prove in this setting than in other similar models. Here we prove sharpness of the connectivity threshold under mild integrability conditions on W (x, y).

Tire particles (TPs) are one of the main emission sources of micro- and nano-plastics into the environment. Although most TPs are deposited in the soil or in the sediments of freshwater and although they have been demonstrated to accumulate in organisms, most research has focused on the toxicity of leachate, neglecting the potential effects of particles and their ecotoxicological impact on the environment. In addition, studies have focused on the impact on aquatic systems and there are many gaps in the biological and ecotoxicological information on the possible harmful effects of the particles on edaphic fauna, despite the soil ecosystem becoming a large plastic sink. The aim of the present study is to review the environmental contamination of TPs, paying particular attention to the composition and degradation of tires (I), transport and deposition in different environments, especially in soil (II), the toxicological effects on edaphic fauna (III), potential markers and detection in environmental samples for monitoring (IV), preliminary risk characterization, using Forlanini Urban Park, Milan (Italy), as an example of an urban park (V), and risk mitigation measures as possible future proposals for sustainability (VI).