Colloquium in probability

In this work, we consider a modification of the usual Branching Random Walk (BRW), where the position of each particle at the last generation 𝑛 is modified by an i.i.d. copy of a random variable 𝑌, which may differ from the driving increment distribution. This model was introduced by Bandyopadhyay and Ghosh (2021) and they termed it as Last Progeny Modified Branching Random Walk (LPM-BRW). Depending on the asymptotic properties of the tail of 𝑌, we describe the asymptotic behaviour of the extremal process of this model as 𝑛 → ∞.

Maker-Breaker is a two player game performed on a graph, in which Breaker tries to cut off a special vertex (e.g. origin or root) by erasing edges while Maker tries to prevent that by fixing them. In this talk we consider the game to be played on supercritical Galton-Watson trees and determine the corresponding winning probabilities given different information regimes.

I'll introduce a certain generalization of a martingale with the following property: at each time, the conditional expectation of a future value given the past, is a weighted average of all the values comprising the past. We'll assume only that more recent values are weighted no less than older values. We'll discuss motivations and constructions, and conditions under which martingale-like behaviors, such as maximal inequalities and convergence, are present in an appropriate form.

This talk is concerned with the following epidemic process: A set of nodes is partitioned into three states: susceptible, infectious, and recovered. We start with a single infectious node. Proceeding in rounds whose length is antiproportional to the population size, a fixed amount of nodes are drawn independently at random. If at least one of the selected nodes is infectious, every susceptible node in the sample becomes infected. Moreover, any infectious vertex recovers independently at a constant rate. If the expected amount of infections caused by single node is less than one, the epidemic dies out quickly and leaves almost the entire population untouched. If it is above one, either the infection dies out quickly or a large outbreak occurs, during which a non-vanishing fraction of the population is affected. Moreover, if enough nodes are infectious at the same time, the system’s behaviour is essentially deterministic.

In this talk I will present some new insights on so-called Poisson representable processes, a general class of {0,1}-valued processes recently introduced by Forsström, Gantert and Steif. Particularly, I will discuss a new characteristic of these in terms of certain mixing properties. As an application thereof, I will argue that the upper invariant measure of the contact process on Z^d is not Poisson representable, thereby answering a question raised in the above mentioned work. This relies on the upper invariant measure satisfying certain directional mixing properties, but not their spatial equivalent. Moreover, the general approach extends to other processes having similar properties, such as the plus phase of the Ising model on Z^2 in the phase transition regime.

Establishing regularity of transition probabilities is a standard focus for solutions to stochastic differential equations (SDEs). For diffusions in finite-dimensional spaces, the Hormander ``bracket generating'' condition for an SDE is a standard geometric assumption that ensures smoothness of the solution. The Hormander condition also often induces a natural geometry on the space which is tied to the analysis of the diffusion. The situation in infinite dimensions is more complicated and less understood. We'll consider a class of infinite dimensional spaces where we propose a different but equivalent analytic formulation of the Hormander condition. Under this assumption, we discuss the related geometry and establish some regularity properties of the associated diffusion.

In the study of Markov processes duality is an important tool used to prove various types of long-time behavior. There exist two approaches to Markov process duality: the algebraic one and the pathwise one. Using the well-known contact process as an example, this talk introduces the general idea of how to construct a pathwise duality for an interacting particle system. Afterwards, several different approaches how to construct pathwise dualities are presented. This is joint work with Jan M. Swart.

For a time dependent family of probability measure $(\mu_t)_{t\he 0}$ we consider a kinetic-type evolution equation $\partial \mu_t/\partial t + \mu_t = Q \mu_t$, where $Q$ is the smoothing transformation. During the talk we will present probabilistic representation of a solution of this equation in terms of continuous time branching random walks. Moreover, assuming that $\mu_0$ belongs to the domain of attraction of a stable law, we describe asymptotic behaviour of $\mu_t$. Literature: [1] Bogus, B., Marynych, SPA 2020 [2] B., Kolesko, Meiners, EJP 2021 [3] B., Dyszewski, Marynych, SPA 2023

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.