Colloquium in probability

In the first part of the talk, we condition a Brownian motion on spending a total of at most $s > 0$ time units outside a bounded interval and discuss the behavior of the resulting process in the context of entropic repulsion. Moreover, we explicitly determine the exact asymptotic behavior of the probability that a Brownian motion on $[0,T]$ spends limited time outside a bounded interval, as $T \to \infty$. This is joint work with Frank Aurzada (Darmstadt) and Martin Kolb (Paderborn). In the second part, we condition a Brownian motion on having an atypically small $L_2$-norm on a long time interval and identify the resulting process as a well-known one. This is joint work with Frank Aurzada (Darmstadt) and Mikhail Lifshits (St. Petersburg).

Ginzburg-Landau fields are a class of models from statistical mechanics that describe the behavior of interfaces. The so-called Helffer-Sjöstrand representation relates them to a random walk in a time-dependent random environment. In the talk I will introduce these objects and survey some of the known results. I will then describe joint work with Wei Wu and Ofer Zeitouni on the asymptotics of the maximum of the Ginzburg-Landau fields in two dimensions.

In this talk, I will present a general framework which can be used to analyze the scaling limits of various stochastic ​spatial "population" models. Such models include ternary Branching Brownian motion subject to majority voting and several examples of interacting particle systems motivated by biology. The approach is based on moment duality and a PDE methodology introduced by Barles and Souganidis which can be used to study the asymptotic behaviour of rescaled reaction-diffusion equations. In the limit, the models exhibit phase separation which is governed by a global-in-time, generalized notion of mean-curvature flow. This talk is based on joint work in progress with Thomas Hughes (Bath).

This talk introduces face and cycle simplicial percolation as models for continuum percolation based on random simplicial complexes in Euclidean space. Face simplicial percolation is defined through infinite sequences of k-simplices sharing a (k-1)-dimensional face. In contrast, cycle simplicial percolation demands the existence of an infinite k-surface, thereby generalizing the lattice notion of plaquette percolation. We discuss the sharp phase transition for face simplicial percolation and derive several relationships between face simplicial percolation, cycle simplicial percolation, and classical vacant continuum percolation. We will also draw connections to a variety of topological models for percolation that have been proposed recently in the literature. This talk is based on joint work with Daniel Valesin

In this talk, we will discuss two dimensional random interlacements, both in discrete and continuous setups. We also discuss some (surprising) properties of their "noodles", which are (two-dimensional) simple random walks conditioned on never hitting the origin in the discrete case and Brownian motions conditioned on never hitting the unit disk in the continuous case. Of particular interest will be the properties of so-called vacant sets.

In recent years there has been significant progress in the study of extreme values of log-correlated Gaussian fields, thanks to the work of Bramson, Ding, Roy, Zeitouni and Biskup, Louidor. For instance, it has been shown that for the discrete Gaussian free field (DGFF) in d=2 and for log-correlated Gaussian fields the limiting law of the centred maximum is a randomly shifted Gumbel distribution. In this talk I will present analogous results for non-Gaussian fields such as the sine-Gordon field and the \Phi^4 field in d = 2. The main tool is a coupling at all scales between the field of interest and the DGFF which emerges from the Polchinski renormalisation group approach as well as the Boue-Dupuis variational formula. The talk is based on joint works with Roland Bauerschmidt and Trishen Gunaratnam, Nikolay Barashkov.

In the course of my habilitation, I researched random walks and some of their applications to statistical physics and random access algorithms. In this talk, I will first give a brief overview of the different papers which constitute the habilitation. I will then talk about the recent work "Off-diagonal long-range order for the free bosonic loop soup" in greater detail. In this work, we give a new (probabilistic) proof for condensation of the free Bose gas, irrespective of boundary condition. The result is based on the Feynman-Kac formula, combined with large deviation estimates and previous results on random partitions. Joint work with Wolfgang König and Alexander Zass.

In this talk we will consider random walks on supercritical Galton-Watson trees with random conductances. That is, given a Galton-Watson tree, we assign to each edge a positive random weight (conductance) and the random walk traverses an edge with a probability proportional to its conductance. On these trees, the random walk is transient and the distance of the walker to the root satisfies a law of large numbers with limit the speed of the walk. We show that the distance of the walker to the root satisfies a functional central limit theorem under the annealed law. In particular, we are interested how the variance changes when the conductances on a positive fraction of edges tend to zero.

The Abelian sandpile model on a graph G is a Markov chain whose state space is a subset of the set of functions with integer values defined on the vertices of G . The set of recurrent states of this Markov chain is called the sandpile group and the Abelian sandpile model can be then viewed as a random walk on a finite group. Then it is natural to ask about the stationary distribution and the speed of convergence to stationarity, and how do these quantities depend on the underlying graph . I will report on some recents results on Abelian sandpiles on fractal graphs, and state some open questions concerning the critical exponents for such processes. The talk is based on joint works with Nico Heizmann, Robin Kaiser and Yuwen Wang.

In 2001 Bolthausen, den Hollander and van den Berg obtained the asymptotics of the probability that the volume of a Wiener sausage at time t is smaller than expected by a fixed muliplicative constant. This asymptotics was given by a variational formula and they conjectured that the best strategy to achieve such a large deviation event is for the underlying Brownian motion to behave like a swiss cheese: stay most of the time inside a ball of subdiffusive size, visit most of the points but leave some random holes. They moreover conjectured that to do so the Brownian motion behaves like a Brownian motion in a drift field given by a function of the maximizer of the variational problem. In this talk I will talk about the corresponding problem for the random walk and will explain that conditioned to having a small range its properly defined empirical measure is indeed close to the maximizer of the above mentioned variational problem. This is joint work with Julien Poisat.

We consider some variations of the edge-reinforced random walk. The focus will be on multiple (but finitely many) walkers which influence the edge weights together. Methods which have been used previously for studying reinforced walks break down and we therefore look at very basic models. First, we consider 2 walkers with linear reinforcement on a line graph comprising three nodes. We show that the edge weights evolve similarly to the setting with a single walker which corresponds to a Pólya urn. We then look at an arbitrary number of walkers on Z with very general reinforcement. We show that in this case, the behaviour is also the same as for a single walker. If there is enough time, we will also have a look at unfinished work on reinforced walks with a bias and on evolving graphs.

TBA

The Zero Range Process is an important example of particle movements in physics. It models particles jumping on a finite set, which surprisingly results in independent occupation numbers in the limit. We will give an overview of Large Deviations in the Zero Range Process and present some important results that arise in the chosen setting of heavy tailed occupation numbers. This gives rise to some related theory like the Catastrophe Principle and the Large Deviations Principle which we will also give a brief introduction to.

Wigner’s jellium is a theoretical model that describes a gas composed of electrons.In this concept, the overall charge is neutralised by n particles, each with a negativeunit charge, floating in a medium of uniformly distributed positive charges. The interactionsbetween the particles are dictated by the Coulomb potential. In this thesis, the Maxwell-Boltzmann distribution is used to describe the statistical behaviour of the quantum jelliummodel in a one-dimensional environment. We state a process-level large deviation principlefor the empirical field and prove it using similar techniques as done by Hirsch, Jansen and Jung (2022).

Consider a biased random walk in positive random conductances on Z^d in dimension 5 and above. In the sub-ballistic regime, Fribergh and Kious (2018) proved the convergence, under the annealed law, of the properly rescaled random walk towards a Fractional Kinetics. I will explain that a quenched equivalent of this theorem is true and a strategy to simplify the question. This is joint work with A. Fribergh and T. Lions

In statistical physics, the zero-freeness property of the grand canonical partition function guarantees the analyticity of the pressure as we approach the infinite volume limit, as shown by Lee and Yang in 1952. Moreover, computer scientists have leveraged the zero-freeness property of the grand canonical partition function to approximate it using various algorithms, such as Barvinok's algorithm. We introduce a novel approach, rooted in computer science, known as the recursion method. This method gives a zero-free region of the partition function. Specifically, we investigate the application of this method to the hard-core lattice gas model, following the work by Peters and Regts in 2019. Additionally, we briefly discuss how Michelen and Perkins (2023) adapted this method for studying gas particles in a continuum space, which interact via a repulsive potential.

The theory of Abstract Wiener Spaces is the basis for many fundamental results of Gaussian measure theory: Large Deviations, Cameron-Martin theorems, Malliavin Calculus, Support theorems, etc. Analogues of these classical theorems exist also in the context of Gaussian Rough Paths and Regularity Structures. It is our goal to investigate the role of an “enhanced” Cameron-Martin subspace in this setting. In particular, we present two approaches to a generalization based on Large Deviation theory and apply them to examples of Rough Path theory and Regularity Structures.

We consider an electric network on the $d$-dimensional integer lattice with an edge between every two points $x$ and $y$. The conductance of the edge $\{x,y\}$ equals $\|x-y\|^{-s}$, for some $s>d$. We show that the random walk on this network is recurrent if and only if $d \in \{1,2\}$ and $s\geq 2d$. We also discuss how this result relates to the return properties of random walks on percolation clusters, particularly on the two-dimensional weight-dependent random connection model.

A spanning tree of a finite connected graph G is a connected subgraph of G that touches every vertex and contains no cycles. In this talk we will consider uniformly drawn spanning trees of ``high-dimensional’‘ graphs, and show that, under appropriate rescaling, they converge in distribution as metric-measure spaces to Aldous’ Brownian CRT. This extends an earlier result of Peres and Revelle (2004) who previously showed a form of finite-dimensional convergence. Based on joint works with Asaf Nachmias and Matan Shalev.

The Double Bubble problem is a generalization of the isoperimetric problem asking the following: given two volumes, what are the two shapes admitting these volumes with the smallest perimeter, where the perimeter of the joint boundary is counted once. We study the DB problem over the l_1 norm and show that one can approximate the solutions very well in the discrete lattice. I will also discuss solutions over other norms and their connection to the Euclidean solutions.

For an inverse temperature β>0, we define the β-circular Riesz gas on Rd as any microscopic thermodynamic limit of Gibbs particle systems on the torus interacting via the Riesz potential g(x)=∥x∥^(−s). We focus on the non integrable case d−10, the existence of a β-circular Riesz gas which is not number-rigid. Recall that a point process is said number rigid if the number of points in a bounded Borel set Δ is a function of the point configuration outside Δ. It is the first time that the non number-rigidity is proved for a Gibbs point process interacting via a non integrable potential. We follow a statistical physics approach based on the canonical DLR equations. It is inspired by Dereudre-Hardy-Leblé and Maïda (2021) where the authors prove the number-rigidity of the Sineβ process.