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This paper considers inhomogeneous Erdős–Rényi random graphs 𝔾 N on N vertices in the non-sparse non-dense regime. The edge between the pair of vertices { i , j } is retained with probability 𝜀 N f (i N , j N) , 1 ≤ i ≠ j ≤ N , independently of other edges, where f : [ 0 , 1 ] × [ 0 , 1 ] → [ 0 , ∞) is a continuous function such that f (x , y) = f (y , x) for all x , y ∈ [ 0 , 1 ]. We study the empirical distribution of both the adjacency matrix A N and the Laplacian matrix Δ N associated with 𝔾 N , in the limit as N → ∞ when lim N → ∞ 𝜀 N = 0 and lim N → ∞ N 𝜀 N = ∞. In particular, we show that the empirical spectral distributions of A N and Δ N , after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where f (x , y) = r (x) r (y) with r : [ 0 , 1 ] → [ 0 , ∞) a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, we apply our results to constrained random graphs, Chung–Lu random graphs and social networks. [ABSTRACT FROM AUTHOR]

Abstract: We derive a lace expansion for the survival probability for critical spread-out oriented percolation above dimensions, i.e., the probability that the origin is connected to the hyperplane at time n, at the critical threshold . Our lace expansion leads to a non-linear recursion relation for , with coefficients that we bound via diagrammatic estimates. This lace expansion is for point-to-plane connections and differs substantially from previous lace expansions for point-to-point connections. In particular, to be able to deduce the asymptotics of for large n, we need to derive the recursion relation up to quadratic order. The present paper is Part II in a series of two papers. In Part I, we use the recursion relation and the diagrammatic estimates to prove that , and also deduce consequences of this asymptotics for the geometry of large critical clusters and for the incipient infinite cluster. [Copyright &y& Elsevier]

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]

Abstract: In this paper we consider an arbitrary irreducible random walk on , , with i.i.d. increments, together with an arbitrary i.i.d. random scenery. Walk and scenery are assumed to be independent. Random walk in random scenery (RWRS) is the random process where time is indexed by , and at each unit of time both the step taken by the walk and the scenery value at the site that is visited are registered. Bad configurations for RWRS are the discontinuity points of the conditional probability distribution for the configuration at the origin of time given the configuration at all other times. We show that the set of bad configurations is non-empty. We give a complete description of this set and compute its probability under the random scenery measure. Depending on the type of random walk, this probability may be zero or positive. For simple symmetric random walk we get three different types of behavior depending on whether , or . Our classification is actually valid for a class of subshifts having a certain determinative property, which we call specifiable, of which RWRS is an example. We also consider bad configurations w.r.t. a finite time interval (replacing the origin) and obtain an almost complete generalization of our results. Remarkably, this extension turns out to be somewhat delicate. [Copyright &y& Elsevier]

It is generally believed that, in the thermodynamic limit, the microcanonical description as a function of energy coincides with the canonical description as a function of temperature. However, various examples of systems for which the microcanonical and canonical ensembles are not equivalent have been identified. A complete theory of this intriguing phenomenon is still missing. Here we show that ensemble nonequivalence can manifest itself also in random graphs with topological constraints. We find that, while graphs with a given number of links are ensemble equivalent, graphs with a given degree sequence are not. This result holds irrespective of whether the energy is nonadditive (as in unipartite graphs) or additive (as in bipartite graphs). In contrast with previous expectations, our results show that (1) physically, nonequivalence can be induced by an extensive number of local constraints, and not necessarily by longrange interactions or nonadditivity, (2) mathematically, nonequivalence is determined by a different large-deviation behavior of microcanonical and canonical probabilities for a single microstate, and not necessarily for almost all microstates. The latter criterion, which is entirely local, is not restricted to networks and holds in gener [ABSTRACT FROM AUTHOR]