Abstracts
Tom Hutchcroft (Caltech) - Dimension dependence of critical phenomena in percolation
In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter p and study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability p_c(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.
In this course, I will give an overview of of what is known about critical percolation, focussing on the non-planar models and including a detailed treatment of recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.
No prior knowledge of percolation will be assumed.
Tuesday afternoon talks
Matthias Irlbeck (University of Groningen) - Poisson-Voronoi percolation in high dimensions
We consider a Poisson point process with constant intensity in Rd and independently color each cell of the resulting random Voronoi tessellation black with probability p. The critical probability p_c(d) is the value for p above which there exists almost surely an unbounded black component and almost surely does not for values below. In this talk, I aim to give an overview of the model and present our result that p_c(d)=(1+o(1)) e d-1 2-d , as d?8. We also obtain the corresponding result for site percolation on the Poisson-Gabriel graph, where p_c(d)=(1+o(1))2-d .
Frankie Higgs (University of Bath) - The random connection model is site percolation on an almost-transitive graph
The random connection model (RCM) built on a Poisson process of intensity λ can be constructed by site percolation with parameter 1/N on an RCM of intensity Nλ. We will see that as N tends to infinity, the RCM constructed in this way behaves more and more like a site percolation cluster on a vertex transitive graph, in a way we will make precise.
We call this method "asymptotic transitivity" and we will apply it to extend the recent exploration methods of Vanneuville to the RCM. In particular, for the subcritical RCM with any connection function, we will see that the probability the cluster of the origin has size at least n decays exponentially as n increases. To our knowledge this is the first proof of the exponential decay of the volume for any long-range percolation model.
Sima Mehri (University of Manchester) - On the Distribution of Sojourn Times in Tandem Queues
The presentation studies the (end-to-end) waiting and sojourn times in tandem queues with general arrivals and light-tailed service times. It is shown that the tails of the corresponding distributions are subject to polynomial-exponential upper bounds, whereby the degrees of the polynomials depend on both the number of bottleneck queues and the 'light-tailedness' of the service times. Closed-form bounds constructed for a two-queue tandem with exponential service times are shown to be numerically sharp, improve upon alternative large-deviations bounds by many orders of magnitude, and recover the exact results in the case of Poisson arrivals.
Leo Tyrpak (University of Oxford) - Responsive dormancy of a spatial population among a moving trap
We study a two-type branching random walk in a random environment modeling dormancy and activity. Individuals switch states in response to a moving trap, modeled by a symmetric random walk. Using the Feynman-Kac formula and the parabolic Anderson model, we quantify how responsive dormancy influences long-term survival and population growth. This is joint work with Helia Shafigh.
Thursday afternoon talks
Peter Mörters (University of Cologne) - Crossing probabilities in geometric inhomogeneous random graphs
In 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 independently forms 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 probability of existence of paths crossing annuli with increasing inner and outer radii in the quantitatively subcritical phase. Depending on the inner and outer radius of the annulus, 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. 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. Based on joint work with Emmanuel Jacob, Céline Kerriou and Amitai Linker.
We study the survival/extinction phase transition for contact processes with quenched disorder. The disorder is given by a locally finite random graph with vertices indexed by Z that is assumed to be invariant under index shifts and augments the nearest-neighbour lattice by additional long-range edges. We provide sufficient conditions that imply the existence of a subcritical phase and therefore the non-triviality of the phase transition. Our results particularly apply to instances of scale-free random geometric graphs with any integrable degree distribution. This contrasts the behaviour of the process on Galton-Watson trees for which the existence of an extinction phase is equivalent to light-tailedness of the offspring (i.e. degree) distribution. We further discuss potential extensions to higher dimension. The talk is based on joint work with Benedikt Jahnel and Christian Mönch.