We study the inhomogeneous random graph with preferential attachment kernel and degree distribution with power-law exponent τ∈(2,3) as a representative of the class of graphs of preferential attachment type with infinite variance degrees. Under bond percolation with a positive retention probability independent of the size n of the graph there is a unique macroscopic component with high probability. We therefore investigate percolation probabilities p_n↓0. We identify a moving critical window at p_c ∼ βn^(τ-3)/(2τ-2). Above this window, when p_n ≫ p_c, the maximal component has size of order n p^_(τ-1)/(3-τ)__n and it is unique. Below this window, when n^1/(1-τ)≪ p_n ≪ p_c, it is non-unique, star-shaped and has size of order n^1/(τ-1) p_n. In the critical window itself, the largest component scaled by √(n) converges in distribution to a positive random variable with a law given in terms of a subcritical Norros-Reittu graph. This behaviour is markedly different from that seen for other classes of scale-free graphs and is conjectured to persist throughout the broad class of growing graphs with infinite variance.

We identify the upper large deviation probability for the number of edges in scale-free geometric random graph models as the space volume goes to infinity. Our result covers the models of scale-free percolation, the Boolean model with heavy-tailed radius distribution, and the age-dependent random connection model. In all these cases the mechanism behind the large deviation is based on a condensation effect. Loosely speaking, the mechanism randomly selects a finite number of vertices and increases their power, so that they connect to a macroscopic number of vertices in the graph, while the other vertices retain a degree close to their expectation and thus make no more than the expected contribution to the large deviation event. We verify this intuition by means of limit theorems for the empirical distributions of degrees and edge-lengths under the conditioning. We observe that at large finite volumes, the edge-length distribution splits into a bulk and travelling wave part of asymptotically positive proportions.

We identify the size of the largest connected component in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The component is polynomial in the graph size with an explicitly given exponent, which is strictly larger than the exponent for the largest degree in the graph. This is in stark contrast to the behaviour of inhomogeneous random graphs with a kernel of rank one. Our proof uses local approximation by branching random walks going well beyond the weak local limit and novel results on subcritical killed branching random walks.

We show sharpness of the phase transition for a nearest-neighbour percolation model on ℤ^d, where vertices carry independent types and the percolation probability of edges depends on the type of the adjacent vertices. Our proof uses the OSSS inequality and adapts to our setup the method developed in Duminil-Copin et al. (2017) for the random cluster model. Additionally, we provide a more extensive study of the special case of combined Bernoulli bond and site percolation featuring a phase transition with two parameters.

We show existence of a non-trivial phase transition for the contact process, a simple model for infection without immunity, on a network which reacts dynamically to the infection trying to prevent an epidemic. This network initially has the distribution of an Erd\H{o}s-R\'enyi graph, but is made adaptive via updating in only the infected neighbourhoods, at constant rate. Under these graph dynamics, the presence of infection can help to prevent the spread and so many monotonicity-based techniques fail. Instead, we approximate the infection by an idealised model, the contact process on an evolving forest (CPEF). In this model updating events of a vertex in a tree correspond to the creation of a new tree with that vertex as the root. By seeing the newly created trees as offspring, the CPEF generates a Galton-Watson tree and we use that it survives if and only if its offspring number exceeds one. It turns out that the phase transition in the adaptive model occurs at a different infection rate when compared to the non-adaptive model.

We study a model of a branching process subject to selection, modeled by giving each family an individual fitness acting as a branching rate, and mutation, modeled by resampling the fitness of a proportion of offspring in each generation. For two large classes of fitness distributions of Gumbel type we determine the growth of the population, almost surely on survival. We then study the empirical fitness distribution in a simplified model, which is numerically indistinguishable from the original model, and show the emergence of a Gaussian travelling wave.

We investigate the contact process on four different types of scale-free inhomogeneous random graphs evolving according to a stationary dynamics, where each potential edge is updated with a rate depending on the strength of the adjacent vertices. Depending on the type of graph, the tail exponent of the degree distribution and the updating rate, we find parameter regimes of fast and slow extinction and in the latter case identify metastable exponents that undergo first order phase transitions.

We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex $v$ with probability proportional to a nonpositive power of the arrival time of $v$, continuing until the graph has $n$ vertices. This class includes uniformly grown random graphs and inhomogeneous random graphs of preferential-attachment type. Whenever the critical percolation threshold is positive, we show that the critical window has width of order $(\log n)^{-2}$ and a secondary phase transition at its finite upper boundary. Inside this window the largest component has size of order $\sqrt{n}/\log n$, and the susceptibility remains finite and independent of the position in the window. The proofs couple component explorations to branching random walks killed outside an interval of length $\log n$, allowing sharp control of the barely subcritical and critical regimes.

We prove a large deviation principle for the sum of n independent heavy-tailed random variables, which are subject to a moving cut-off boundary at location n. Conditional on the sum being large at scale n, we show that a finite number of summands take values near the cut-off boundary, while the remaining variables still obey the law of large numbers. This generalises the well-known single-big-jump principle for random variables without cut-off to a situation where just the minimal necessary number of jumps occur. As an application, we consider a random graph with vertex set given by the lattice points of a torus with sidelength 2N+1. Every vertex is the centre of a ball with random radius sampled from a heavy-tailed distribution. Oriented edges are drawn from the central vertex to all other vertices in this ball. When this graph is conditioned on having an exceptionally large number of edges we use our main result to show that, as N goes to infinity, the excess outdegrees condense in a fixed, finite number of randomly scattered vertices of macroscopic outdegree. By contrast, no condensation occurs for the indegrees of the vertices, which all remain microscopic in size.

For the inhomogeneous random graph with kernel of preferential attachment type and degree distribution with power-law exponent τ∈ (2,3) we study the decay of the size of the giant component when the edge density approaches zero. It turns out that the giant component is significantly smaller than for the inhomogeneous random graph with a kernel of rank one.

In a geometric inhomogeneous random graph vertices are given by the points of a Poisson process and are equipped with independent weights following a heavy tailed distribution. Any pair of distinct vertices is independently forming an edge with a probability decaying as a function of the product of the weights divided by the distance of the vertices. For this continuum percolation model we study the crossing probabilities of annuli, i.e. the probabilities that there exist paths starting inside a ball and ending outside a larger concentric ball with increasing inner and outer radii. Depending on the radii, the power-law exponent of the degree distribution and the decay of the probability of long edges, we identify regimes where the crossing probabilities by a path are equivalent to the crossing probabilities by one or by two edges. We also identify the escape probabilities from balls with strong centre, i.e. the asymptotics of the probability that there exists a path starting from a vertex with a given weight leaving a centred ball as radius and weight are going to infinity. As a corollary we get the subcritical one-arm exponents characterising the decay of the probability that a typical point is in a component not contained in a centred ball whose radius goes to infinity.

We study the Poisson Boolean model where the grains are random convex bodies with a rotation-invariant distribution. We say that a grain distribution is dense if the union of the grains covers the entire space and robust if the union of the grains has an unbounded connected component irrespective of the intensity of the underlying Poisson process. If the grains are balls of random radius, then density and robustness are equivalent, but in general this is not the case. We show that in any dimension $d\ge2$ there are grain distributions that are robust but not dense, and give general criteria for density, robustness and non-robustness of a grain distribution. We give examples which show that our criteria are sharp in many instances.

We investigate a disordered variant of Pitman's Chinese restaurant process where tables carry i.i.d. weights. Incoming customers choose to sit at an occupied table with a probability proportional to the product of its occupancy and its weight, or they sit at an unoccupied table with a probability proportional to a parameter $\theta>0$. This is a system out of equilibrium where the proportion of customers at any given table converges to zero almost surely. We show that for weight distributions in any of the three extreme value classes, Weibull, Gumbel or Fr\'echet, the proportion of customers sitting at the largest table converges to one in probability, but not almost surely, and the proportion of customers sitting at either of the largest two tables converges to one almost surely.

We study the size of the connected component of early typical vertices in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The principal tools in our analysis are, first, a coupling of the neighbourhood of a typical vertex in the graph to a killed branching random walk and, second, an asymptotic result for the number of particles absorbed at the killing barrier in this branching random walk.

We investigate two stochastic models of a growing population with discrete and non-overlapping generations, subject to selection and mutation. In our models each individual carries a fitness which determines its mean offspring number. Many of these offspring inherit their parent’s fitness, but some are mutants and obtain a fitness randomly sampled, as in Kingman’s house-of-cards model, from a distribution in the domain of attraction of the Fréchet distribution. We give a rigorous proof for the precise rate of superexponential growth of these stochastic processes and support the argument by a heuristic and numerical study of the mechanism underlying this growth. This study yields in particular that the empirical fitness distribution of one model in the long time limit displays periodic behaviour.

We investigate random graphs on the points of a Poisson process in $d$-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random mark and given marks and positions of the points we form an edge between two points independently with a probability depending via a kernel on the two marks and the distance of the points. Different kernels allow the mark to play different roles, like weight, radius or birth time of a vertex. The kernels depend on a parameter~$\gamma$, which determines the power-law exponent of the degree distributions. A further independent parameter $\delta$ characterises the decay of the connection probabilities of vertices as their distance increases. We prove transience of the infinite cluster in the entire supercritical phase in regimes given by the parameters $\gamma$ and~$\delta$, and complement these results by recurrence results if $d=2$. Our results are particularly interesting for the soft Boolean graph model discussed in the preprint [arXiv:2108:11252] and the age-dependent random connection model recently introduced by Gracar et al.\ [Queueing Syst. 93.3-4 (2019)]}

The classical percolation problem is to find whether there is an infinite connected component in a random set created by removing edges from a $d$-dimensional lattice, independently at random. Since its introduction into the mathematical literature by Broadbent and Hammersley (1957) the subject of percolation has developed in many ways and is now one of the most exciting and active research areas in probability and statistical mechanics. In this workshop, we focused on current trends, including percolation on point sets with correlations, on spatial random graphs and networks with scale-free degree distribution or long-range edge distribution, percolation of random sets like level set of Gaussian fields or the vacant set of interlacements, conformally invariant percolation structures in the plane, and random walks or information diffusion on percolation clusters. The workshop brought together more than sixty experts and promising young researchers from probability, statistical mechanics and computer science working on all aspects of percolation and spatial networks for an unprecedented exchange of new ideas and methods, with outstanding high quality online talks.

We study geometric random graphs defined on the points of a Poisson process in $d$-dimensional space, which additionally carry independent random marks. Edges are established at random using the marks of the endpoints and the distance between points in a flexible way. Our framework includes the soft Boolean model (where marks play the role of radii of balls centred in the vertices), a version of spatial preferential attachment (where marks play the role of birth times), and a whole range of other graph models with scale-free degree distributions and edges spanning large distances. In this versatile framework we give sharp criteria for absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two points. Other than in the mean-field scale-free network models the boundary of the ultrasmall regime depends not only on the power-law exponent of the degree distribution but also on the spatial embedding of the graph, quantified by the rate of decay of the probability of an edge connecting typical points in terms of their spatial distance.

The Chinese restaurant process is a stochastic process closely related to the Dirichlet process that groups sequentially arriving objects into a variable number of classes, such that within each class objects are cyclically ordered. A popular description involves a restaurant, where customers arrive one by one and either sit down next to a randomly chosen customer at one of the existing tables or open a new table. The full state of the process after n steps is given by a permutation of the n objects and cannot be represented in sublinear space. In particular, if we only need specific information about a few objects or classes it would be preferable to obtain the answers without simulating the process completely. A recent line of research [15, 28, 5, 12] attempts to provide access to huge random objects without fully instantiating them. Such local access implementations provide answers to a sequence of queries about the random object, following the same distribution as if the object was fully generated. In this paper, we provide a local access implementation for a generalization of the Chinese restaurant process described above. Our implementation can be used to answer any sequence of adaptive queries about class affiliation of objects, number and sizes of classes at any time, position of elements within a class, or founding time of a class. The running time per query is polylogarithmic in the total size of the object, with high probability. Our approach relies on some ideas from the recent local access implementation for preferential attachment trees by Even et al. [12]. Such trees are related to the Chinese restaurant process in the sense that both involve a “rich-get-richer” phenomenon. A novel ingredient in our implementation is to embed the process in continuous time, in which the evolution of the different classes becomes stochastically independent [21]. This independence is used to keep the probabilistic structure manageable even if many queries have already been answered. As similar embeddings are available for a wide range of urn processes [2], we believe that our approach may be applicable more generally. Moreover, local access implementations for birth and death processes that we encounter along the way may be of independent interest.

Finding the most powerful node in a dynamic random network, the largest set in a partition-valued stochastic process, or the largest family in an evolving population at a given time, can be a very difficult problem. This is particularly the case when the underlying stochastic process has complex dependencies and the individual strength of an object has an impact that only plays out over time. We propose a novel technique to deal with such problems and show how it can be applied to a broad range of examples where it produces new insight and surprising results. The method relies on two steps: In the first step, which is highly problem dependent, the problem is embedded into continuous time so that the evolution of the sizes of objects after their individual birth times become approximately independent while we only need minimal control over the birth times themselves. Once such an embedding is achieved, the second step is to apply a Poisson limit theorem that allows a comparison of object sizes in a critical window and therefore allows a description of features of extremal objects. In this paper we prove such a versatile limit theorem, based on extreme value theory, and show how the technique can be used to study extremal behaviour in different types of preferential attachment networks with fitness, branching processes with selection and mutation, and random permutations with random cycle weights.