| Time | Wednesday | Thursday | Friday |
|---|---|---|---|
| 08:45 - 09:15 | Registration | ||
| 09:15 - 10:00 | Frank den Hollander | Talk | Talk |
| 10:00 - 10:45 | Coffee break | Coffee break | Talk |
| 10:45 - 11:30 | Talk | Talk | Coffee break |
| 11:30 - 12:15 | Talk | Talk | Talk |
| 12:15 - 13:00 | Lunch break | Lunch break | Talk |
| 13:00 - 14:00 | Lunch break | Lunch break | |
| 14:00 - 14:45 | Talk | Talk | |
| 14:45 - 15:30 | Talk | Talk | |
| 15:30 - 16:15 | Coffee break | Coffee break | |
| 16:15 - 17:00 | Talk | Talk | |
| 17:00 - 17:45 | Talk | Talk | |
| 18:30 | Get together | Conference Dinner |
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Abstract:
Branching Processes and the Fisher–KPP Equation in Random Environments
Branching Brownian motion, branching random walks, and the Fisher–KPP equation have been central objects of study in probability theory and mathematical physics over the past decades. Through the Feynman–Kac and McKean representations, the behavior of extremal particles in the branching models is intimately linked to the position of the traveling front in the Fisher–KPP equation.
In this talk, I will present recent progress on extensions of these classical models to spatially random environments, incorporating random branching rates and random nonlinearities. Such inhomogeneities give rise to a significantly richer and more delicate phenomenology than in the homogeneous case.
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One-dimensional random polymers: Going ballistic with Wolfgang
Abstract:
Self-Reinforced Preferential Attachment
We consider a preferential attachment random graph with self-reinforcement. Each time a new vertex comes in, it attaches itself to an old vertex with a probability that is proportional to the sum of the degrees of that old vertex at all prior times. The resulting growing graph is a random tree whose vertices have degrees that grow polynomially fast in time. We compute the growth exponent, show that it is strictly larger than the growth exponent in the absence of self-reinforcement, and develop insight into how the self-reinforcement affects the growth. Proofs are based on a stochastic approximation scheme.
Joint work with Yogesh Dahiya (IISER, Mohali).
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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 the component of a typical point is not contained in a centred ball whose radius goes to infinity. Based on joint work with Emmanuel Jacob, Céline Kerriou and Amitai Linker.
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Random Spanning Trees in Random Environment
A spanning tree of a graph G is a connected subset of G without cycles. The Uniform Spanning Tree (UST) is obtained by choosing one of the possible spanning trees of G at random. The Minimum Spanning Tree (MST) is realised instead by putting random weights on the edges of G and then selecting the spanning tree with the smallest weight. These two models exhibit markedly different behaviours: for example, their diameter on the complete graph with n nodes transitions from n^1/2 for the UST to n^1/3 for the MST. What lies in between?
We introduce a model of Random Spanning Trees in Random Environment (RSTRE) designed to interpolate between UST and MST. In particular, when the environment disorder is sufficiently low, the RSTRE on the complete graph has a diameter of n^1/2 as the UST. Conversely, when the disorder is high, the diameter behaves like n^1/3 as for the MST. We conjecture a smooth transition between these two values for intermediate levels of disorder.
This talk is based on joint work with Rongfeng Sun and Luca Makowiec (NUS Singapore).
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Excursions to the world of bounded-degree percolation
Among models of continuum percolation, apart from the classical distance-based ones (such as variants of the Gilbert graph/Boolean model), bounded-degree models and in particular k-nearest neighbour graphs represent another important class. Such models were first studied by Häggström and Meester in 1996. In my talk, I will explain how I encountered this class of models during my PhD and what Wolfgang's role was in guiding me towards them. I will mention my joint results with Benedikt Jahnel on the absence of percolation in variants of the bidirectional 2-neighbour graph based on deletion-tolerant point processes, and then I will turn to lattice analogues of the k-nearest neighbour graphs. Here, the directed k-neighbour graph is the directed graph where each vertex of \Z^d sends an outgoing edge towards k uniformly chosen nearest neighbours, and we also study undirected variants of this model. I will summarize joint results with Benedikt, Jonas Köppl and Bas Lodewijks on the existence/absence of an infinite connected component in these models (which have recently been improved by Benedikt, Jonas, David Coupier and Benoît Henry). Finally, I will mention some current ongoing work by the four of us together with Johannes Bäumler and Lily Reeves on obtaining further positive percolation results in these models based on local comparison arguments with independent lattice percolation models.
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