The preferential attachment network with fitness is a dynamic random graph model. New vertices are introduced consecutively and a new vertex is attached to an old vertex with probability proportional to the degree of the old one multiplied by a random fitness. We concentrate on the typical behaviour of the graph by calculating the fitness distribution of a vertex chosen proportional to its degree. For a particular variant of the model, this analysis was first carried out by Borgs, Chayes, Daskalakis and Roch. However, we present a new method, which is robust in the sense that it does not depend on the exact specification of the attachment law. In particular, we show that a peculiar phenomenon, referred to as Bose-Einstein condensation, can be observed in a wide variety of models. Finally, we also compute the joint degree and fitness distribution of a uniformly chosen vertex.

The continuous-space symbiotic branching model describes the evolution of two interacting populations that can reproduce locally only in the simultaneous presence of each other. If started with complementary Heaviside initial conditions, the interface where both populations coexist remains compact. Together with a diffusive scaling property, this suggests the presence of an interesting scaling limit. Indeed, in the present paper, we show weak convergence of the diffusively rescaled populations as measure-valued processes in the Skorokhod, respectively the Meyer-Zheng, topology (for suitable parameter ranges). The limit can be characterized as the unique solution to a martingale problem and satisfies a "separation of types" property. This provides an important step toward an understanding of the scaling limit for the interface. As a corollary, we obtain an estimate on the moments of the width of an approximate interface. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2016, Vol. 44, No. 2, 807-866. This reprint differs from the original in pagination and typographic detail. 1 4 J. BLATH, M. HAMMER AND M. ORTGIESE before Corollary 1.2. Recently, analogous results have been derived by Döring and Mytnik in the case ̺ ∈ (−1, 1) in [9, 10]. Returning to the continuous-space set-up, for ̺ = −1 (the stepping stone model) Tribe [27] proves a "functional limit theorem": For a pair of (continuous) functions (u, v), define R(u, v) := sup{x : u(x) > 0}, L(u, v) = inf{x : v(x) > 0}. (5)Note that for a solution (u t , v t ) t≥0 of the symbiotic branching model, the interface at time t is contained in the interval [L(u t , v t ), R(u t , v t )]. It is proved in [27] for ̺ = −1 and for continuous initial conditions u 0 = 1 − v 0 which satisfy −∞ < L(u 0 , v 0 ) ≤ R(u 0 , v 0 ) < ∞ that under Brownian rescaling, the motion of the position of the right endpoint of the interface t → 1 n R(u n 2 t , 1 − u n 2 t ), t ≥ 0, converges to a Brownian motion as n → ∞.The above results suggest the existence of an interesting diffusive scaling limit for the continuous-space symbiotic branching model (and its interface) for ̺ > −1. This is the starting point of our investigation. However, compared to the case ̺ = −1, the situation is more involved here: For example, the total mass of the solution is not necessarily bounded, and in particular, moments of the solution may diverge as t → ∞, depending on ̺. For instance, second moments diverge for ̺ ≥ 0. In order to state this result, which was obtained in [3]

The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model. MSc classification (2000): Primary 60K37 Secondary 82C441 Two recent papers, Dembo and Deuschel [DD07] and Aurzada and Doering [AD09], investigate weaker forms of ageing based on correlations. Both deal with a class of models which includes as a special case a parabolic Anderson model with time-variable potential and show absence of correlation-based ageing in this case. While this approach is probably the only way to deal rigorously with complicated models, it is not established that the effect picked up by these studies is actually really due to the existence or absence of ageing in our sense, or whether other moment effects are accountable.In the present work we show that the parabolic Anderson model exhibits ageing behaviour, at least if the underlying random potential is sufficiently heavy-tailed. As a lattice model with random disorder the parabolic Anderson model is a model of significant complexity, but its linearity and strong localization features make it considerably easier to study than, for example, the dynamics of most non-mean field spin glass models.Our work has led to three main results. The first one, Theorem 1.1, shows that the probability that during the time window [t, t + θt] the profiles of the solution of the parabolic Anderson problem remain within distance ε > 0 of each other converges to a constant I(θ), which is strictly between zero and one. This shows that ageing holds on a linear time scale. Our second main result, Theorem 1.3, is an almost sure ageing result. We define a function R(t) which characterizes the waiting time starting from time t until the profile changes again. We determine the precise almost sure upper envelope of R(t) in terms of an integral test. The third main result, Theorem 1.6, is a functional scaling limit theorem for the location of the peak, which determines the profile, and for the growth rate of the solution. We give the precise statements of the results in Section 1.2, and in Section 1.3 we provide a rough guide to the proofs. Statement of the main resultsThe parabolic Anderson model is given by the heat equation on the lattice Z d with a random po...

We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We describe the process, including a detailed shape theorem, in terms of a system of growing lilypads. As an application we show that the branching random walk is intermittent, in the sense that most particles are concentrated on one very small island with large potential. Moreover, we compare the branching random walk to the parabolic Anderson model and observe that although the two systems show similarities, the mechanisms that control the growth are fundamentally different.