Luisa Andreis (Torino) (back to the timetable)

Some recent progress in coagulation processes

Since Smoluchowski introduced his well-known coagulation equation in 1917, there has been an active line of research focused on understanding the properties of the solutions to this equation and related models for coagulation. In particular, in 2000, Norris introduced a generalised version of the model, which he named the cluster coagulation model. This model was intended to extend the framework established by Smoluchowski, allowing particles to have additional properties beyond their mass, such as shape or spatial location.
In this talk we focus on some recent progress in the study of the particle system that converges to such limiting (spatial) coagulation equation, often called the Marcus-Lushnikov process. In particular, we present an approach based on Poisson Point Processes to study large deviations of the trajectory of such a Markov process in the large volume limit. We explain how this also provides insight into gelation phenomena and phase transitions for related random graphs, highlighting new directions in which such an approach seems promising. This talk is based on a series of joint works with T. Iyer, W. König, M. Kolodjekzyk, H. Langhammer, E. Magnanini and R.I.A. Patterson.

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

Eigenvalue fluctuations for the crushed-ice problem

The crushed-ice problem is concerned with the spectral and transport properties of the Dirichlet Laplacian in bounded domains perforated by many randomly-placed tiny holes. Through the work of Kac, Rauch and Taylor in 1970s and many others later it is known that, for the number of holes diverging as their size tends to zero so that the aggregate capacity density of the holes converges, the eigenvalues of the Laplacian tend to those of a suitable (deterministic) Schrödinger operator. Using a martingale representation I will show that, for the eigenvalues that are simple in the limit, a multivariate CLT holds under centering by expected values. This extends an earlier work of Figari, Orlandi and Teta and, independently, Ozawa from 1980s who proved a CLT in spatial dimension 3, albeit using functional-analytic methods that do not apply in higher dimensions. Based on joint work with Ryoki Fukushima.

Erwin Bolthausen (Zürich) (back to the timetable)

Large deviations for the capacity of the Wiener sausage

Consider the Wiener sausage with radius 1 of a time length t in the d-dimensional Euclidean space, and its Newtonian capacity. We show that for d>4, this capacity satisfies a downward large deviation principle with rate t^(d-4)/(d-2), and we identify the rate function in terms of a variational formula. (Joint work with M. van den Berg, and F. den Hollander).

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 (Munich) (back to the timetable)

TBA

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

One-dimensional random polymers: Going ballistic with Wolfgang

One-dimensional random polymers played a central role in the early career of Wolfgang, and mine. Wolfgang wrote pioneering works in his PhD project with Erwin Bolthausen, and in his PostDoc period with Frank den Hollander, while I was doing my PhD with Frank. In the latter period, we started collaborating, and wrote several papers on laws of large numbers for one-dimensional random polymers, as well as central limit theorems, for the end-to-end displacement of the polymer. The proofs use large deviations and deep functional analysis. In this talk, I review the highlights of this early work by, and with, Wolfgang.

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 (Munich) (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 Pétrélis (Nantes) (back to the timetable)

Non local random deposition models for earthquakes

In this presentation, we consider a class of deposition models that originate from the study of geophysical problems, such as earthquakes. The basic model is as follows: We consider a sequence (h_n)_{n \in \N} of real-valued functions defined on [0, D]^d (with d = 1 or d = 2). The first function h_0 is uniformly zero. Then, recursively, we compute h_{n+1} by adding to $h_n$ a symmetric function that is centered at a random point Y_{n+1} and is non-zero on a width determined by a heavy-tailed random variable Z_{n+1}.
The distribution of those points (Y_n)_{n\in \N} determines the type of physical system we aim to study:
a) the rand. model if (Y_n)_{n\in \N} are sampled independently and uniformly over [0, D]^d (for instance to model the filtration of radiations by aerosols)
b) the min. model if the center of the $(n+1)$-th transformation corresponds to the point where the minimum of h_n is attained (for instance to model earthquakes close to the boundary of two tectonic plates).
For both models we will study, as n diverges, the convergence in distribution of (h_n)_{n\in \N} and of the fluctuations of (h_n)_{n\in \N}.
This is joint work with P. Carmona and F. Pétrélis.

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)

Localisation and delocalisation in the parabolic Anderson model

The parabolic Anderson problem is the Cauchy problem for the heat equation on the integer lattice with random potential. It describes the mean-field behaviour of a continuous-time branching random walk. It is well-known that, unlike the standard heat equation, the solution of the parabolic Anderson model exhibits strong localisation. In particular, for a wide class of iid potentials it is localised at just one point. However, in a partially symmetric parabolic Anderson model, the one-point localisation breaks down for heavy-tailed potentials and remains unchanged for light-tailed potentials, exhibiting a range of phase transitions. Based on joint work with S. Muirhead and R. Pymar.

Florian Theil (Warwick) (back to the timetable)

Finite size corrections for kinetic equations

The justification of kinetic equations like the Boltzmann equation as a scaling limit of particle dynamics for large times is a longstanding problem; the main challenge is to control the size of the relevant density functions. We propose a regularisation strategy where the number of interactions per particles is bounded and derive uniform-in-time bounds for the discrepancy between the solution of the kinetic equation and the particle dynamics. This result is a consequence of a careful study of cancellations associated with recollisions.

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 (Munich) (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 (Munich) (back to the timetable)

Dynamical Phase Diagram of the REM

Dynamical expressions of glassy behavior are a central topic in the non-equilibrium theory of spin glass models. Much of the mathematical work concerns one of the versions of Glauber or trap dynamics in the simplest spin glass, the random energy model (REM) and its relatives. By contrast, in this talk, I will discuss the atypical behavior of the REM's energy for rare trajectories of the infinite-temperature limit of the Glauber dynamics, namely the simple random walk. I will present a closed formula for the quenched cumulant generating function of the time integral of the REM energy, discuss the underlying phase transitions for trajectories of any time extent, and identify phases distinguished by the activity and value of the time integral. This is achieved by relating the dynamical behavior to the spectral properties of Hamiltonians associated with the Quantum Random Energy Model. (Based on joint work with C. Manai.)

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.