Colloquium in probability

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.

TBA

We consider the random connection model with bounded edges which is generated by a Poisson point process with density $\lambda$ in $\mathbb{R}^d$. We prove that this model undergoes a sharp phase transition, i.e. we prove that in the subcritical phase the probability that the origin is connected to some point at distance $n$ decays exponentially in $n$, while in the supercritical phase the probability that the origin is connected to infinity is strictly positive and bounded from below by a term proportional to ($\lambda-\lambda_c)$, $\lambda_c$ being the critical density. This proof uses newly developed methods by Last, Peccati and Yogeshwaran in their recent work, in particular a continuous version of the OSSS inequality for Poisson functionals, relying on stopping sets and continuous-time decision trees. This approach simplifies an earlier result of Faggionato and Mimun, who proved sharp phase transition in the random connection model via the discrete OSSS inequality.

Based on a collection of works by Alessandra Faggionato, I will give a short introduction to Mott's Law and its rigorous derivation. It states that the conductivity in amorphous materials scales like $\exp( -c\beta^{1/4})$ for low temperatures, where $\beta$ denotes the inverse temperature. Using the Miller-Abrahams resistor network, A. Faggionato developed two approaches to rigorously prove such a limiting behavior. One via scaling limit of the conductivity of random resistor networks on simple point processes, and one via critical conductance of the Miller-Abrahams resistor network. A recent work of her presented at Paris CIRS connects these two approaches, showing that they both lead to the same sub-exponential decay of conductivity. Although, for a complete proof, a lower bound on LR-crossings in the supercritical regime for energy marks of both signs is still missing, and I will comment on this issue.

We consider a fully-connected loss network with dynamic alternative routing, each link of capacity K. Calls arrive to each link {i, j} at rate λ independently and depart at rate 1. If the link is full upon arrival, a third node k is chosen uniform and the call is routed via k: it uses a unit of capacity on both {i, k} and {k, j} if both have spare capacity; otherwise, the call is lost. This is a model for telephone networks, implemented by BT in the 1990s. We analyse the asymptotics of the mixing time of this process, depending on the traffic intensity α := λ/K. In particular, we determine a phase transition at an explicit threshold α*: there is fast mixing if α < α* or α > 1, but metastability if α* < α < 1. We also discuss a fixed for metastability—ie, an adjustment to the model which removes the slow-mixing phase. Again, this was implemented by BT in the UK telephone network.

Ziel des Vortrags ist es, mithilfe von Diffusionen auf Riemann Mannig- faltigkeiten eine Familie von austauschbaren Paaren im R^d für die Steinsche Meth- ode der austauschbaren Paare zu konstruieren und daraus eine Abschätzung zu gewinnen. Um die nötigen Schritte zu erklären, werden in dem Vortrag zuerst die Steinsche Methode vorgestellt und einige Grundlagen zu Mannigfaltigkeiten in Zusam- menhang mit Stochastik erklärt. Im Anschluss daran definiere ich eine geeignete Diffusion, erkläre ihre Eigenschaften, bilde sie auf R^d ab und wende die Steinsche Meth- ode der austauschbaren Paare an. Das Vorgehen entstammt dem Artikel ”Constructing exchangeable pairs by diffusion on manifold and its application” von Weitao Du, 2006.

Despite being very different in nature, martingales and rough paths have many similarities and their interplay is most fruitful. As a concrete example, I will introduce the recent notion of rough stochastic differential equations and explain its importance in filtering, pathwise control theory and option pricing under (possibly rough) stochastic volatility. (Joint work with numerous people, including Pavel Zorin-Kranich, Khoa Lê, Antoine Hocquet, Peter Bank, Christian Bayer and Luca Pelizzari.)

We consider a Poisson point process on \(\mathbb R ^d\) with intensity \(\lambda\) for \(d\ge2\). On each point, we independently center a ball whose radius is distributed according to some power-law distribution \(\mu\). When the distribution \(\mu\) has a finite \(d\)-moment, there exists a non-trivial phase transition in \(\lambda\) associated to the existence of an infinite connected component of balls. We aim here to prove subcritical sharpness that is that the subcritical regime behaves well in some sense. For distribution \(\mu\) with a finite \(5d−3\)-moment, Duminil-Copin--Raoufi--Tassion proved subcritical sharpness using randomized algorithm. We prove here using different methods that the subcritical regime is sharp for all but a countable number of power-law distributions. Joint work with Vincent Tassion.

The Weight-Dependent Random Connection model combines long-range percolation with scale-free network models. The talk focuses on the "weak decay regime" where connection probability tails are heavy enough to circumvent many geometrical difficulties that arise in short-range perclation models in low dimensions. I will summarise known sufficient conditions for existence and transience of an infinite component and discuss a new local existence theorem which improves upon a result of Berger (2002) and which implies the most general sufficient condition for transience hitherto known, as well as the continuity of the percolation function.

In this talk we describe a probabilistic methodology to derive the precise asymptotic for the probability of observing a maximal component containing more than n^{2/3} vertices in the (near-) critical Erdös-Rényi random graph. Our approach is mostly based on ballot-type estimates for one-dimensional, integer-valued random walks, and improves upon the martingale-based method introduced by Nachmias and Peres in 2009. We also briefly discuss how our method has been adapted to study the same type of problem for (near-) critical percolation on a random d-regular graph, as well as possible future developments.

The large deviation behavior of lacunary sums Michael Juhos Universität Passau We study the large deviation behavior of lacunary sums (Sn/n)n∈N with Sn := Pn k=1 f (akU ), n ∈ N, where U is uniformly distributed on [0, 1], (ak)k∈N is an Hadamard gap sequence, and f : R → R is a 1-periodic, (Lipschitz-)continuous mapping. In the case of large gaps, we show that the normalized partial sums satisfy a large deviation principle at speed n and with a good rate function which is the same as in the case of independent and identically distributed random variables Uk, k ∈ N, having uniform distribution on [0, 1]. When the lacunary sequence (ak)k∈N is a geometric progression, then we also obtain large deviation principles at speed n, but with a good rate function that is dif- ferent from the independent case, its form depending in a subtle way on the interplay between the function f and the arithmetic properties of the gap sequence. Our work generalizes some results recently obtained by Aistleitner, Gantert, Kabluchko, Prochno, and Ramanan [Large deviation principles for lacunary sums, 2023] who initiated this line of research for the case of lacunary trigonometric sums.

In this talk we consider continuous time random walks on $\mathbb{Z}^d$ among random conductances that permit jumps of arbitrary length, where the law of the conductances is assumed to be stationary and ergodic. Under a suitable moment condition we obtain a quenched local limit theorem and Hölder regularity estimates for solutions of the heat equation for the associated non-local discrete operator. Our results apply to random walks on long-range percolation graphs with connectivity exponents larger than 2d when all nearest-neighbour edges are present. This talk is based on a joint work with Martin Slowik (Mannheim).

The totally asymmetric simple exclusion process is a conservative particle system that has been studies though various mathematical lenses. Results for this particle system include hydrodynamic limits, invariant distributions, fluctuations and large deviations. It has connections to the celebrated KPZ class via a coupling with the corner growth model and last passage percolation; it is considered one of the exactly solvable models of the KPZ class. In this talk we will discuss a (non-exactly solvable) generalisation of TASEP in which the rates that govern the particle jumps depend on the location of the particle and the time that we are observing the process. The rates come from a background function that can be discontinuous in space and time. We will discuss the hydrodynamic limit of this version of TASEP (for particle current and density), which will be the solution to certain discontinuous PDEs.