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Let T be a supercritical Galton–Watson tree with a bounded offspring distribution that has mean μ > 1 , conditioned to survive. Let φ T be a random embedding of T into Z d according to a simple random walk step distribution. Let T p be percolation on T with parameter p, and let p c = μ - 1 be the critical percolation parameter. We consider a random walk (X n) n ≥ 1 on T p and investigate the behavior of the embedded process φ T p (X n) as n → ∞ and simultaneously, T p becomes critical, that is, p = p n ↘ p c . We show that when we scale time by n / (p n - p c) 3 and space by (p n - p c) / n , the process (φ T p (X n)) n ≥ 1 converges to a d-dimensional Brownian motion. We argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments. [ABSTRACT FROM AUTHOR]

In [3, Theorem 1.2], we claim that the maximal cluster for critical percolation on the high-dimensional torus is non-concentrated. This proof contains an error. In this note, we replace this statement by a conditional statement instead. [ABSTRACT FROM AUTHOR]

One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade (Braunstein et al. in Phys Rev Lett 91(16):168701, 2003; Wu et al. in Phys Rev Lett 96(14):148702, 2006; Braunstein et al. Int J Bifurc Chaos 17(07):2215-2255, 2007; Chen et al. in Phys Rev Lett 96(6):068702, 2006) is as follows: for a wide array of random graph models with degree exponent $$\tau \in (3,4)$$ , distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $$n^{(\tau -3)/(\tau -1)}$$ . In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of Lévy processes 'without replacement' (Aldous and Limic Electron in J Probab 3(3):59, 1998), yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent $$\tau \in (3,4)$$ where edges are rescaled by $$n^{-(\tau -3)/(\tau -1)}$$ yielding the first rigorous proof of the above conjecture. The limits in this case are compact 'tree-like' random fractals with a dense collection of hubs (infinite degree vertices), a finite number of which are identified with leaves to form shortcuts. In a special case, we show that the Minkowski dimension of the limiting spaces equal $$(\tau -2)/(\tau -3)$$ a.s., in stark contrast to the Erdős-Rényi scaling limit whose Minkowski dimension is 2 a.s. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent $$\tau \in (3,4)$$ . Our proof makes crucial use of inhomogeneous continuum random trees (ICRT), which have previously arisen in the study of the entrance boundary of the additive coalescent. We show that tilted versions of the same objects using the associated mass measure, describe connectivity properties of the multiplicative coalescent. Since convergence of height processes of corresponding approximating $$\mathbf {p}$$ -trees is not known, we use general methodology in Athreya et al. (2014) and develop novel techniques relying on first showing convergence in the Gromov-weak topology and then extending this to Gromov-Hausdorff-Prokhorov convergence by proving a global lower mass-bound. [ABSTRACT FROM AUTHOR]

We consider critical spread-out oriented percolation above 4 + 1 dimensions. Our main result is that the extinction probability at time n (i.e., the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n + 1) decays like 1/ Bn 2 as $$n\to\infty$$ , where B is a finite positive constant. This in turn implies that the survival probability at time n (i.e., the probability that the origin is connected to the hyperplane at time n) decays like 1/ Bn as $$n\to\infty$$ . The latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster. The present paper is Part I in a series of two papers. In Part II, we derive a lace expansion for the survival probability, adapted so as to deal with point-to-plane connections. This lace expansion leads to a nonlinear recursion relation for the survival probability. In Part I, we use this recursion relation to deduce the asymptotics via induction. [ABSTRACT FROM AUTHOR]

The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by den Hollander and the authors. We give conditions under which the solution to the recursion relation behaves as a Gaussian, both in Fourier space and in terms of a local central limit theorem. These conditions are shown elsewhere to hold for sufficiently spread-out models of networks of self-avoiding walks in dimensions d > 4, and for sufficiently spread-out models of critical oriented percolation in dimensions d + 1 > 5, providing a unified approach and an essential ingredient for a detailed analysis of the branching behaviour of these models. [ABSTRACT FROM AUTHOR]

Abstract. We prove ballistic behaviour in dimension one for a model of weakly self-avoiding walks where loops of length m are penalized by a factor e[sup -beta/m[sup p]] with p is an element of [0, 1] and beta sufficiently large. Furthermore, we prove that the fluctuations around the linear drift satisfy a central limit theorem. The proof uses a variant of the lace expansion, together with an inductive analysis of the arising recursion relation. In particular, we derive the law of large numbers, first obtained by Greven and den Hollander, and the central limit theorem, first obtained by Konig, for the weakly self-avoiding walk (p = 0 and beta > 0). Their proofs use large deviation theory for the Markov description of the local times of one-dimensional simple random walk. It is the first time that the lace expansion is used to prove behaviour that is not diffusive. It has previously been used by van der Hofstad, den Hollander and Slade to prove diffusive behaviour in dimension d for p is greater than or equal to 0 such that p > 4-d/2 and beta > 0 sufficiently small. The lace expansion presented here compares the above weakly self-avoiding walk to strictly self-avoiding walk in dimension one, obtained when beta = Infinity, and shows that the difference in behaviour is small when beta is large. [ABSTRACT FROM AUTHOR]

Summary. We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding walk on Z[sup d] where loops of length m are penalised by a factor e[sup -beta/m[sup p]] (0 < beta much less than 1) when: (1) d > 4, p is greater than or equal to 0; (2) d less than or equal to 4, p > 4 - d/2. In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case d > 4, p = 0. In addition, we prove a local central limit theorem, with the exception of the case d > 4, p = 0. [ABSTRACT FROM AUTHOR]