Luisa Andreis (Milan) (back to the timetable)

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Marek Biskup (Los Angeles) (back to the timetable)

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Erwin Bolthausen (Zürick) (back to the timetable)

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Orphee Collin (Paris) (back to the timetable)

The random field Ising chain

We are going to present recent results concerning the Ising chain (i.e., the Ising model in dimension 1) with homogeneous spin-spin interaction, but subjected to a random external field, the latter being sampled from an i.i.d centered sequence. On the one hand we will give free energy estimates, obtained by expressing the free energy as the Lyapunov exponent of a random 2x2 matrix product and exploiting a heuristic by B. Derrida and H. J. Hilhorst. On the other hand we will discuss results at the level of configurations for large spin-spin interaction Gamma, which confirm a description by D. Fisher and collaborators appearing in the physics literature. This description is based on the notion of Gamma-extrema of the potential associated with the external field.

Alexander Drewitz (Cologne) (back to the timetable)

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.

Nina Gantert (Münich) (back to the timetable)

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Remco van der Hofstad (Eindhoven) (back to the timetable)

One-dimensional random polymers: Going ballistic with Wolfgang

Motivated by metastability in the zero-range process, we consider i.i.d. integer random variables and Weibull-like (stretched exponential) tails. We condition on large values of the sum and prove large deviation principles for the rescaled maximum and for the reversed order statistics - think of the canonical ensemble in a supersaturated gas and the sizes of the largest droplet, second-largest droplet, etc. We choose a sublinear power-law scale on which the the big-jump principle for heavy-tailed variables and a naive normal approximation for moderate deviations yield bounds of the same order. The rate function for the maximum is non-convex and solves a recursive equation similar to a Bellman equation.

Frank den Hollander (Leiden) (back to the timetable)

Parabolic Anderson model on random graphs

After presenting a bird-eye view of what is known for the PAM on lattices, I describe joint work with Wolfgang K\"onig, Renato dos Santos and Daoyi Wang identifying the scaling behaviour of the total mass of the PAM on Galton-Watson trees with a double-exponential potential. This work also identifies the scaling behaviour on sparse Configuration Graphs (whose local limit is a Galton-Watson tree), but only for times that are short in comparison to the size of the graph. I indicate what is needed to identify the scaling behaviour for all large times, and what can be done for other types of sparse random graphs. The latter is work in progress with Wolfgang and Renato.

Sabine Jansen (Münich) (back to the timetable)

Large deviations for the maximum and reversed order statistics of Weibull-like variables

Motivated by metastability in the zero-range process, we consider i.i.d. integer random variables and Weibull-like (stretched exponential) tails. We condition on large values of the sum and prove large deviation principles for the rescaled maximum and for the reversed order statistics - think of the canonical ensemble in a supersaturated gas and the sizes of the largest droplet, second-largest droplet, etc. We choose a sublinear power-law scale on which the the big-jump principle for heavy-tailed variables and a naive normal approximation for moderate deviations yield bounds of the same order. The rate function for the maximum is non-convex and solves a recursive equation similar to a Bellman equation.

Peter Mörters (Cologne) (back to the timetable)

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.

Nicolas Petrelis (Nantes) (back to the timetable)

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Michele Salvi (Rome) (back to the timetable)

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).

Nadia Sidorova (London) (back to the timetable)

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Florian Theil (Warwick) (back to the timetable)

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András Tóbiás (Budapest) (back to the timetable)

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.

Quirin Vogel (Münich) (back to the timetable)

Bose-Einstein condensation and long loops

Feynman's representation of the Bose gas in terms of interacting Brownian bridges provides a beautiful framework to study Bose–Einstein condensation. In this picture, condensation is conjectured to be reflected in the emergence of macroscopic permutation cycles. Whether the onset of Bose–Einstein condensation indeed coincides with the appearance of such infinite loops remains a central open problem. I will review key results in this area, present recent progress and new analytical tools, and outline some of the challenges that remain.

Simone Warzel (Münich) (back to the timetable)

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Alexander Zass (Berlin) (back to the timetable)

A dynamical model of interacting colloids

In this talk, we consider a dynamical version of the Asakura--Oosawa--Vrij model of interacting hard spheres of two different sizes. We study their infinite-dimensional random diffusion dynamics, modelled with collision local times; describe the reversible measures; and observe the emergence of an attractive short-range depletion interaction between the large spheres. We also study the Gibbs measures associated to this new interaction, exploring connections to percolation, optimal packing, and phase transitions.
This is joint work with Myriam Fradon.