Seminar Interacting Random Systems (Archive)

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Wednesday, 07.07.2021, 11:30 (Online Event)
Daoyi Wang, Leiden University:
The Parabolic Anderson Model on a Galton-Watson tree with unbounded degrees (online talk)
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Online Event

Abstract
A detailed analysis was given by den Hollander, König and dos Santos (2020), of the large-time asymptotics of the total mass of the solution to the parabolic Anderson model on a supercritical Galton-Watson random tree, with an i.i.d. random potential whose marginal distribution is double-exponential. Under the assumption that the degree distribution has bounded support, two terms in the asymptotic expansion were identified under the quenched law, i.e., conditional on the realisation of the random tree and the random potential. We extend the analysis to degree distributions with unbounded support. We identify the weakest condition on the tail of the degree distribution under which the arguments can be pushed through. This is joint work with my supervisor Frank den Hollander.

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Seminar Interacting Random Systems (Online Event)

Host
WIAS Berlin
Wednesday, 23.06.2021, 11:30 (Online Event)
Tejas Iyer, WIAS Berlin:
Degree and Edge Distributions in Inhomogeneous Random Recursive Trees (online talk)
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Online Event

Abstract
We introduce general models of inhomogeneous random recursive trees, where either one or several nodes arrive at a time, and are equipped with i.i.d random weights. At each time-step, an existing vertex is chosen with probability proportional to its fitness function, and the newly arriving node(s) connect to it. This may be a function of its weight, and possibly the weights of its neighbours. We study two main quantities: the empirical measures associated with the number of vertices with a given degree and weight and the empirical measure corresponding proportion of edges in the structure with endpoint having a given weight. We show that, under certain technical conditions, the limit of both quantities exists when normalised by the size of the network, as the number of nodes tends to infinity. However, when the trees take certain forms, we show that interesting, non-trivial behaviour can emerge when these conditions fail: in particular, the trees may exhibit emphcondensation where a positive proportion of edges accumulate around vertices with weight that maximises the reinforcement of their fitness, or, more drastically, may have a emphdegenerate limiting degree distribution where the entire proportion of edges accumulate around these vertices. This non-trivial behaviour may be of interest when considering the trees as toy models for the evolution of complex networks.

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Seminar Interacting Random Systems (Online Event)

Host
WIAS Berlin
Wednesday, 09.06.2021, 11:30 (Online Event)
Wen Sun, TU Berlin:
Pathwise large deviation for the pure jump k-nary interacting particle systems (online talk)
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Online Event

Abstract
A pathwise large deviation result is proved for the pure jump models of k-nary interacting particle system introduced by Kolokoltsov that generalize classical Boltzmann's collision model, Smoluchovski's coagulation model and many others. The upper bound is obtained by following the standard methods of using a process "perturbed" by a regular function. To show the lower bound, we propose a family of orthogonal martingale measures and prove a coupling for the general perturbations. The rate function is studied based on the idea of Léonard with a simplification by considering the conjugation of integral functionals on a subspace of L^infty.. General "gelling" solutions in the domain of the rate function are also discussed.

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Seminar Interacting Random Systems (Online Event)

Host
WIAS Berlin
Wednesday, 26.05.2021, 11:30 (Online Event)
András Tóbiás, TU Berlin:
Absence of percolation in graphs based on stationary point processes with degrees bounded by two (online talk)
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Online Event

Abstract
We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollobás that the bidirectional k-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for k=2. The subject of this talk is joint work with B. Jahnel.

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Seminar Interacting Random Systems (Online Event)

Host
WIAS Berlin
Wednesday, 12.05.2021, 11:30 (Online Event)
Tal Orenshtein, WIAS Berlin/TU Berlin:
Rough walks in random environment
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Online Event

Abstract
Random walks in random environment (RWRE) have been extensively studied in the last half-century. Functional central limit theorems (FCLT) hold in some prototypical classes such the reversible and the ballistic ones. The latter are treated using rather different techniques; Kipnis-Varadhan's theory for additive functionals of Markov processes is applicable in the reversible case whereas the main feature exploited in the ballistic class is a regeneration structure. Rough path theory is a deterministic theory which extends classical notions of integration to singular integrators in a continuous manner. It typically provides a framework for pathwise solutions of ordinary and partial stochastic differential equations driven by a singular noise. In the talk we shall discuss FCLT for additive functionals of Markov processes and regenerative processes lifted to the rough path space. The limiting rough path has two levels. The first one is the Brownian motion, whereas in the second we see a new feature: it is the iterated integral of the Brownian motion perturbed by a deterministic linear function called the area anomaly. The aforementioned classes of RWRE are covered as special cases. The results provide sharper information on the limiting path. In addition, the construction of new examples for SDE approximations is an immediate application. Based on collaborations (some are still in progress) with Johannes Bäumler, Noam Berger, Jean-Dominique Deuschel, Olga Lopusanschi, Nicolas Perkowski and Martin Slowik.

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Seminar Interacting Random Systems (Online Event)

Host
WIAS Berlin
Wednesday, 28.04.2021, 11:30 (Online Event)
Alexander Hinsen, WIAS Berlin:
Percolation on a dynamical device-to-device communication system (online talk)
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Online Event

Abstract
We model a system of moving devices on a street system with the help of a general waypoint kernel. If there is an uninterrupted communication window between devices (their space-time trajectories are close to each other) they form a connection. We study regimes for percolation and absence of percolation of the cluster of connected devices, as it can be seen as an indication for the ability of the system to transmit messages over long distances.

Further Informations
Seminar Interacting Random Systems (Online Event)

Host
WIAS Berlin
Wednesday, 14.04.2021, 11:30 (Online Event)
Alexis Prevost, University of Cambridge, GB:
Cluster capacity functionals and isomorphism theorems for Gaussian free fields (online talk)
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Online Event

Abstract
Various isomorphisms theorem relate Gaussian free fields to random walk models, through their local times. Among them, the second Ray-Knight theorem can be extended in an isomorphism between the Gaussian free field and random interlacements on transient graphs. Following the work of Lupu and Sznitman, we will explain how the cable system method let us make this isomorphism more explicit, and how it relates to a certain cluster capacity functional for the Gaussian free field. Finally, we will present applications to percolation for the Gaussian free field on the cable system. Joint work with Alexander Drewitz (Cologne) and Pierre-François Rodriguez (London)

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Seminar Interacting Random Systems (Online Event)

Host
WIAS Berlin
Wednesday, 17.03.2021, 11:30 (Online Event)
Daniel Heydecker, University of Cambridge, GB:
Large deviations of Kac's elastic particle system (online talk)
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Online Event

Abstract
We consider Kac's stochastic model for a N-particle gas, with particles interacting by elastic collisions, and study the large deviations in the limit $Ntoinfty$ in the weak topology. There is a very natural candidate rate function, first proposed by Léonard, which amounts to the dynamic entropy of a flux measure plus an initial cost; with this rate function, we sketch the proof of an upper bound, and a lower bound restricted to a class of `sufficiently goods' paths. Perhaps surprisingly, we will see that the proposed rate function does not capture all of the possible large deviation behaviour. Lu and Wennberg showed that, even though the microscopic collisions preserve energy, there are solutions to the Boltzmann equation for which the energy is increasing, and these paths have 0 dynamic entropy. We will show that such paths can arise as large deviation limits with finite exponential cost, so cannot be excluded from large deviation analysis, but occur strictly more rarely than predicted by the rate function. At the level of the particle system, this occurs when a macroscopic proportion of the energy concentrates in $o(N)$ particles, which can happen with probability $e^-O(N)$.

Further Informations
Seminar Interacting Random Systems (Online Event)

Host
WIAS Berlin
Wednesday, 03.03.2021, 11:30 (Online Event)
Cecile Mailler, The University of Bath, GB:
The ants walk: finding geodesics in graphs using reinforcement learning
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Online Event

Abstract
How does a colony of ants find the shortest path between its nest and a source of food without any means of communication other than the pheromones each ant leave behind itself? In this joint work with Daniel Kious (Bath) and Bruno Schapira (Marseille), we introduce a new probabilistic model for this phenomenon. In this model, the nest and the source of food are two marked nodes in a finite graph. Ants perform successive random walks from the nest to the food, and ths distribution of the n-th walk depends on the trajectories of the (n-1) previous walks through some linear reinforcement mechanism. Using stochastic approximation methods, couplings with Pólya urns, and the electric conductances method for random walks on graphs, we prove that, in this model, the ants indeed eventually find the shortest path(s) between their nest and the source food.

Further Informations
Seminar Interacting Random Systems (Online Event)

Host
WIAS Berlin
Wednesday, 17.02.2021, 11:30 (Online Event)
Thorsten Lucke, Technische Universität Berlin:
Influence of mobility on telecommunication networks --- some thoughts (online talk)
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Online Event

Abstract
The Boolean model provides a nice framework for describing ad-hoc networks. It is well known that under certain assumptions on the distribution of the radii and the intensity of the Poisson process we can observe critical phenomena such as percolation or global connectivity. However, if we start in a subcritical regime, i.e., the intensity is below a critical threshold, can we manipulate the connectivity properties, if we allow the participants to move? This talk will provide a brief introduction to mobility models such as the random waypoint model.

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Seminar Interacting Random Systems (Online Event)

Host
WIAS Berlin
Wednesday, 03.02.2021, 11:30 (Online Event)
Elena Pulvirenti, Delft University of Technology, Niederlande:
The Widom-Rowlinson model: metastability, mesoscopic and microscopic fluctuations for the critical droplet (online talk)
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Online Event

Abstract
We introduce the equilibrium Widom-Rowlinson model on a two-dimensional finite torus in which the energy of a particle configuration is attractive and determined by the union of small discs centered at the positions of the particles. We then discuss the metastable behaviour of a dynamic version of the WR model. This means that the particle configuration is viewed as a continuous time Markov process where particles are randomly created and annihilated as if the outside of the torus were an infinite reservoir with a given chemical potential. In particular, we start with the empty torus and are interested in the first time when the torus is fully covered by discs in the regime at low temperature and when the chemical potential is supercritical. In order to achieve the transition from empty to full, the system needs to create a sufficiently large droplet, called critical droplet, which triggers the crossover. We compute the distribution of the crossover time and identify the size and the shape of the critical droplet. The analysis relies on a mesoscopic and microscopic description of the surface of the critical droplet. It turns out that the critical droplet is close to a disc of a certain deterministic radius, with a boundary that is random and consists of a large number of small discs that stick out by a small distance. We will show how the analysis of surface fluctuations in the WR model allows us to derive the leading order term of the condensation time and also the correction order term. This is a joint work with Frank den Hollander (Leiden), Sabine Jansen (Munich) and Roman Kotecky (Prague & Warwick).

Further Informations
Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 20.01.2021, 11:30 (Online Event)
Lorenzo Taggi, University of Rome “La Sapienza”, Italien:
Exponential decay of correlations for O(N) spin systems
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Online Event

Abstract
The Spin O(N) model is a classical statistical mechanics model whose configurations are collections of unit vectors, called spins, taking values on the surface of a N ?1 dimensional unit sphere, with each spin associated to the vertex of a graph. Some special cases of the spin O(N) model are the Ising model (N = 1), the XY model (N = 2), and the classical Heisenberg model (N = 3). Despite the fact that it is a very classical model, there remain important gaps in understanding, particularly in the case N > 2. This talk will provide an introduction on this class of models and present a new recent result about exponential decay of correlations for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N>1. The proof of this result employs a representation of the model as a system of coloured random paths and a sampling procedure, which allows us to bound from above the "typical lenght" of the open paths. The talk is based on a joint work with B. Lees.

Further Informations
Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 13.01.2021, 11:30 (Online Event)
Sabine Jansen, Universität München:
Virial inversion and density functionals (online talk)
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Online Event

Abstract
We present a novel inversion theorem for functionals that map measures to measures. Our main application is in statistical mechanics of inhomogeneous systems, in which the activity $z(x) = z_0 exp( - V_ext(x))$ is mapped to a density profile $rho(x)$ and one seeks to invert the density-activity relation. In probabilistic terms, the question is how to invert the relation between the intensity measure of an underlying Poisson point process and the intensity measure of a Gibbs point process, a question also of interest in spatial statistics. We apply the inversion theorem to the derivation of density functionals. The inversion theorem works in situations where the inverse function theorem in Banach spaces fails. Based on joint work with Tobias Kuna and Dimitrios Tsagkarogiannis (arXiv:1906.02322 [math-ph]).

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 09.12.2020, 11:30 (Online Event)
Robert Patterson:
Large deviations for population growth from near zero concentration
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Online Event

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 25.11.2020, 11:30 (Online Event)
Willem van Zuijlen:
Total mass asymptotics of the parabolic Anderson model
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Online Event

Abstract
We consider the parabolic Anderson model with a white noise potential in two dimensions. This model is also called the stochastic heat equation with a multiplicative noise. We study the large time asymptotics of the total mass of the solution. Due to the irregularity of the white noise, in two dimensions the equation is a priori not well-posed. Using paracontrolled calculus or regularity structures one can make sense of the equation by a renormalisation, which can be thought of as “subtracting infinity of the potential”. To obtain the asymptotics of the total mass we use the spectral decomposition, an alternative Feynman-Kac type representation and heat-kernel estimates which come from joint works with Khalil Chouk, Wolfgang König and Nicolas Perkowski.

Further Informations
Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 11.11.2020, 11:30 (WIAS-405-406)
Stein Andreas Bethuelsen:
Invariance principle for random walks on dynamically averaging random conductances
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406

Abstract
We prove an invariance principle for continuous-time random walks in a dynamically averaging environment on Z. In the beginning, the conductances may fluctuate substantially, but we assume that as time proceeds, the fluctuations decrease according to a typical diffusive scaling and eventually approach constant unit conductances. The proof relies on a coupling with the standard continuous time simple random walk. Based on joint work with Christian Hirsch (Groningen) and Christian Mönch (Mainz).

Further Informations
Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 21.10.2020, 11:30 (WIAS-405-406)
Michiel Renger:
Variational structures beyond gradient flows: a macroscopic-fluctuation-theory perspective joint work with Rob Patterson (WIAS) and Upanshu Sharma (FU Berlin)
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Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406

Abstract
Variational formulations in physics are often related to probability via large deviations. Although the idea goes back to Onsager, modern work shows that the large deviations of reversible Markov processes are related to gradient flows. The challenge of non-equilibrium thermodynamics is now to uncover meaningful variational structures for irreversible Markov processes. In that case one can not rule out the occurrence of non-equilibrium steady states, which are best studied by taken particle fluxes into account; this is the approach of “macroscopic fluctuation theory” (MFT). Most literature on MFT study quadratic rate functionals exclusively. These come with a natural concept of orthogonality that allows the decomposition of the flux rate functional into reversible and irreversible parts. We generalise beyond the quadratic case, which is for example relevant for chemical reactions, biological fitness models and epidemiology. As a byproduct we derive a connection to so-called “FIR” inequalities, and we find that the gap is quantified by a large-deviation rate functional of a rather curious process that should be considered `purely irreversible'. Disclaimer: this talk will hardly be about probability, but it will be about analysis of large-deviation rate functionals of Markov processes.

Further Informations
Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 07.10.2020, 11:30 (WIAS-405-406)
Tal Orenshtein:
Aging in the Edwards-Wilkinson and KPZ universality classes
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Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406

Abstract
Aging is an asymptotic property of non-equilibrium dynamical systems that captures non-trivial relaxation time temporal change; a canonical formulation is expressed in terms of the correlations of the system at two large times with a fixed relation. It was conjectured in Dembo-Deuschel '06 that one-dimensional KPZ models satisfy aging. In line with the progress on the one-time asymptotic behavior of KPZ in the past decade, this challenging problem gained attention in both the physics and the mathematical communities; there has been some experimental evidence for the phenomenon as well as related non-rigorous predictions and partial results. In the talk we shall see that for stationary systems one can use methods that rely solely on the variance asymptotics to achieve aging with an explicit aging function. We shall then derive aging for stationary models in the Edwards-Wilkinson universality class, which is easier to tackle. Moreover, we will demonstrate how to apply the methods to compute a formula for the space-time correlation scaling function in this case. In the remaining part of the talk we shall discuss aging for several stationary models in the KPZ class, including the KPZ fixed point, with the same aging function, matching the stationary KPZ equation prediction in Ferrari-Spohn '16. The talk is based on a recent work with Jean-Dominique Deuschel (TU Berlin) and Gregorio Moreno Flores (PUC Chile).

Further Informations
Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 23.09.2020, 11:30 (WIAS-405-406)
Benedikt Jahnel:
The free energy of a grid version of the Bose gas
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Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406

Abstract
A rigorous derivation of the famous Bose-Einstein condensation in terms of statistical mechanics is widely considered as one of the outstanding open problems in statistical physics.. Starting from a microscopic description of the system as a Gibbs measure of interacting particles in continuous finite volumes, the conjecture is that, in the thermodynamic limit and when the system is forced to have a large particle density, the particles become delocalized and macroscopic components begin to emerge, a condensation phenomenon. One way to make this observation is to consider a reformulation of the problem in terms of a variational formula for the free energy, a strategy which has been successful applied for the non-interacting case. In the talk, I will present a slightly simplified version of the model which allows us to derive the variational description in a way that includes the microscopic and macroscopic parts of the model using large-deviation theory. This is joint work with Wolfgang König.

Further Informations
Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 09.09.2020, 11:15 (WIAS-HVP-3.13)
Heide Langhammer:
Large deviations and the phase transition for the connected components of an inhomogeneous random graph
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Weierstraß-Institut, Hausvogteiplatz 11A, 10117 Berlin, 3. Etage, Raum: 3.13

Further Informations
Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 09.09.2020, 10:30 (WIAS-HVP-3.13)
Alexandra Quitmann:
Macroscopic cycles in the random loop model
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Weierstraß-Institut, Hausvogteiplatz 11A, 10117 Berlin, 3. Etage, Raum: 3.13

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 13.05.2020, 11:00 (Online Event)
Sam Baguley, Universität Mannheim:
Perpetual Integrals for stable SDEs
more ... Location
Online Event

Abstract
The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and completely understood for many several decades. For stochastic differential equations with jumps the picture is different, and even the most basic questions are only partially understood. We study the question of existence and uniqueness of weak solutions to $$dZ_t=sigma(Z_t-)dint X_t$$ driven by $$alpha$$-stable Lévy processes.

Further Informations
Diesen Vortrag können Sie mit Zoom verfolgen unter: https://zoom.us/j/86368292467.

Host
WIAS Berlin
Tuesday, 14.04.2020, 11:00 (Online Event)
Prof.Dr. Wolfgang König, WIAS Berlin:
Condensation in the interacting Bose gas in the semi-classical mean-field setting
more ... Location
Online Event

Further Informations
online seminar

Host
WIAS Berlin
Wednesday, 05.02.2020, 11:30 (WIAS-406)
Richard Pymar, Birkbeck University of London:
Mixing times of exclusion processes on regular graphs
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
Place k black particles and n-k white particles on the vertices of an n vertex graph, with one per vertex. Suppose each edge rings at rate 1 independently, and when an edge rings particles at the end-points switch positions. Oliveira conjectured that this "k-particle exclusion process" has mixing time of order at most that of k independent particles. Together with Jonathan Hermon we prove a bound for regular graphs which is in general within a log log n factor from this conjecture when k>n^c and which, in certain cases, verifies the conjecture. As a result we obtain new mixing time bounds for the exclusion process on expanders and the hypercube.

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 15.01.2020, 11:30 (WIAS-406)
Marcel Fenzl, Universität Zürich, Schweiz:
Asymptotic results for stabilizing geometric statistics
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
Even though global in nature, many geometric statistics like the germ-grain model or the $k$-nearest neighbour model can be realized as a sum of local contributions. This idea can be formalized by using the concept of stabilizing score functions. In this talk, we investigate geometric statistics arising from such stabilizing score functions for randomly chosen input points. More specifically, we analyse the asymptotic behaviour of such a statistic for general point processes with fast decaying correlation functions.

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 18.12.2019, 11:30 (WIAS-406)
Siragan Gailus, Hausdorff Institute for Mathematics/Boston University:
Homogenization of multiscale diffusion processes with small noise
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
Dynamical systems exhibiting multiple characteristic scales in space or time arise naturally as models in a great variety of applied fields. It is moreover common to incorporate random perturbations into these models in order to account for imperfect information or to capture random phenomena. We will discuss homogenization and fluctuations limit theorems for a model of diffusion type. Results on statistical inference will be introduced if time permits.

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 11.12.2019, 11:30 (WIAS-406)
Benjamin Lees, University of Bristol:
The phase transition for random loop models on trees
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
We show the existence of a sharp phase transition from non-existence to existence of infinite loops for a random loop model on d-regular trees, for all dimensions d ≥ 3. The loop model is built up by randomly placed 'crosses' and 'bars' whose relative intensity is controlled by a parameter u. We give a recursive scheme to obtain an expansion of the critical parameter in powers of 1/d, which in principle is explicit but whose combinatorial complexity grows very quickly. We were able to explicitly obtain the first 6 terms (the first two were previously found by Ueltschi and Bjornberg for the limit d → ∞), and observed that (as functions of u) they seem to have a very interesting structure. This is a joint work with Volker Betz and Johannes Ehlert.

Further Informations
Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 20.11.2019, 11:30 (WIAS-HVP-3.13)
D. R. Michiel Renger:
Dynamical Phase Transitions on Finite Graphs
more ... Location
Weierstraß-Institut, Hausvogteiplatz 11A, 10117 Berlin, 3. Etage, Raum: 3.13

Abstract
We consider systems where particles jump through edges of a finite graph with nonlinear intensities, and study time-averaged particle fluxes as both the number of particles and the end time go to infinity. The corresponding large-deviation rate functional involve a minimisation over trajectories with a given time-averaged flux. The minimisation can often be restricted to constant paths, resulting in a very simple expression for the rate functional. However, for specific models and specific regimes of the average flux, such expression overshoot the large-deviation rate functional, which typically happens if small oscillations in the trajectories are more profitable. In that case we say that a dynamical phase transition occurs. The literature on dynamical phase transition is almost exclusively restricted to models where the graph becomes continuous in the limit, yielding quadratic rate functionals. In our work we focus on graphs that remain discrete in the limit, which leads to entropic rate functionals. We present conditions that rule out dynamical phase transition, and for zero-range processes on a discrete ring, we give sufficient conditions under which a dynamical phase transition can be constructed. (Joint work with Davide Gabrielli, l'Aquila)

Further Informations
Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 06.11.2019, 11:30 (WIAS-HVP-3.13)
Tal Orenshtein, WIAS:
Random walks in random environment as rough paths
more ... Location
Weierstraß-Institut, Hausvogteiplatz 11A, 10117 Berlin, 3. Etage, Raum: 3.13

Abstract
Random walk in random environment (RWRE) is a model to describe propagation of heat or diffusion of matter through a highly irregular medium. The latter is expressed locally in the model in terms of a random environment according to which the process evolves randomly in time. In a few fundamental classes the phenomenon of homogenization of the media takes place. One way this can be expressed is in the fact that on large scales, the RWRE looks like a Brownian motion with a deterministic covariance matrix given in terms the (law of the) environment. Rough path theory enables the construction of solutions to SDEs so that the solution map is continuous with respect to the noise. One important application guarantees that if the approximation converges to the noise in the rough path topology, the SDEs driven by the noise approximations converge, in an appropriate sense, to a well-defined SDE different than the original one, so that the correction term is explicit in terms of the noise approximation. In this talk we shall present our current program, in which one lifts RWRE in various classes to the rough path space and shows a convergence to an enhanced Brownian motion in the rough path topology. Interestingly, the limiting second level of the lifted RWRE may have a linear correction, called area anomaly, which we identify. Except for the immediate application to approximations of SDEs (and potentially of SPDEs), this adds some new information on the RWRE limiting path. Time permitted, we shall elaborate on the tools to tackle these problems. Based on joint works with Olga Lopusanschi, with Jean-Dominique Deushcel and Nicolas Perkowski and with Johaness Bäumler, Noam Berger, and Martin Slowik.

Further Informations
Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 30.10.2019, 11:30 (WIAS-406)
Dr. Niccolo Torri, Université Paris Nanterre, Frankreich:
Directed polymer in a heavy-tail random environment
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
A classical problem in the field of disordered systems is to understand the behaviour of a directed polymer in interaction with a random environment. Mathematically, the directed polymer is modeled by a directed random walk and the random environment is a sequence of random variables which can interact with the random walk perturbing its behaviour, giving rise to super-diffusive transversal fluctuations and localisation phenomena. Understanding the typical trajectories of the the walk is a very challenging issue and several results have been obtained when the random environment has exponential moments. In this talk we consider the case of a heavy-tail random environment, that is, the environment?s distribution function decades polynomially with exponent $alpha >0$. We mainly focus on the case in which the random environment has no second moment ($alphain (0,2)$), finding the super-diffusive transversal fluctuations of the polymer as function of the parameter $alpha$. Joint work with Quentin Berger (Sorbonne Université).

Further Informations
Seminar Interacting Random Suystems

Host
WIAS Berlin
Wednesday, 16.10.2019, 11:30 (WIAS-406)
Dr. Christian Mönch, Johannes-Gutenberg-Universität:
Universality of persistence exponents for self-similar processes with stationary increments
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
n 1999, G. Molchan showed that for a centered fractional Brownian motion X on the real line Prob[X(t) <1, 1 <t<T]=TH-1+o(1), where H is the index of self-similarity of X. Furthermore, he showed that the same tail exponent occurs for sevaral other path functionals of X and conjectured that it also governs the tail of Prob[L(t)<1, 1<t<T], where L is the local time at 0 of X, but was only able to prove a lower bound. In this talk, I present an entirely novel approach to persistence problems for self-similar processes with stationary increments based on Palm theory. The technique is not limited to Gaussian processes, allows us to resolve Molchan's conjecture for ANY H-self-similar process with stationary increments that admits a sufficiently smooth local time and provides a better error estimate even for the known lower bound of Prob[L(t)<1, 1<t<T] in the fractional Brownian case. Furhermore, I will discuss some work in progress concerning fractional variants of Brownian excursion-type processes.

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Seminar Interacting Random Systems

Host
WIAS Berlin
DFG Research Center MATHEON
Wednesday, 09.10.2019, 11:30 (WIAS-406)
Dr. Lorenzo Taggi, WIAS:
Phase transition in random lattice permutations
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
We consider the model of random lattice permutations, whose realisations are permutations of the vertices of Z^d and can be viewed as a system of oriented self-avoiding loops interacting via mutual exclusion, with parameter beta (”inverse temperature”) which rewards the number ”jumps”. This model attracts interest from different perspectives: it is related to the interacting quantum Bose gas through a representation which is due to Feynman, it can be viewed as representation of the dimer model (when the inverse temperature is infinite) and it is a variant of loop O(N) models. A central question for all these models is whether a regime of ”macroscopic loops” and uniformly positive "two point correlations" occurs when the inverse temperature is large enough and d > 2. This talk presents the results and the proof techniques of a recent paper where a positive answer to this question is provided.

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 04.09.2019, 11:30 (WIAS-406)
Alexander Hinsen, WIAS:
Phase transition for the white knight model
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Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 27.03.2019, 10:30 (WIAS-406)
Dr. Giovanni Luca Torrisi, Istituto per le Applicazioni del Calcolo “Mauro Picone” (C.N.R), Italien:
The Clark-Ocone formula for point processes
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Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
Clark-Ocone formulas are powerful results in stochastic analysis with a variety of applications. In the talk we provide the Clark-Ocone formula for square-integrable functionals of point processes with stochastic intensity. Then we present two applications of the formula: the Poincare' inequality and a deviation bound for those functionals. Our results generalize the corresponding ones on the Poisson space

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 13.02.2019, 11:30 (WIAS-406)
Peter Neijjar, Institute of Science and Technology Austria (IST Austria), Österreich:
Product limit laws at shocks in (T)ASEP
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Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
We consider the asymmetric simple exclusion process (ASEP) on Z with an initial data such that in the large time particle density r a discontinuity at the origin is created, where the value of r jumps from zero to one, but r (resp. 1-r) is strictly positive to the left (resp. right) of the origin. We consider the position of a particle macroscopically located at the discontinuity, and show that its limit law has a cutoff at the origin. Inside the discontinuity region, we show that a discrete product limit law arises, which bounds from above the limiting fluctuations of the particle in the general ASEP, and equals them in the totally ASEP. Sending the size of the shock region to infinity, we recover the product of GUE distributions previously observed at shocks in the totally ASEP.

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 23.01.2019, 11:30 (WIAS-406)
Pierre Houdebert, Aix-Marseille Université, Frankreich:
Sharp phase transition for the Widom-Rowlinson model
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Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
The Widom-Rowlinson model is formally defined as two homogeneous Poisson point processes forbidding the points of different type to be too close. For this Gibbs model the question of uniqueness/ non-uniqueness depending on the two intensities is relevant. This model is famous because it was the first continuum Gibbs model for which phase transition was proven, in the symmetric case of equal intensities large enough. But nothing was known in the non-symmetric case, where it is conjectured that uniqueness would hold. In a recent work with D. Dereudre (Lille), we partially solved this conjecture, proving that for large enough activities the phase transition is only possible in the symmetric case of equal intensities. The proof uses percolation and stochastic domination arguments.

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Seminar Interacting Random Systems

Host
WIAS Berlin
Friday, 11.01.2019, 15:00 (WIAS-406)
Dr. Darcy Camargo, The Weizmann Institute of Science, Israel:
The meteor process stationary distribution
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Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
The Potlatch and Smoothing processes have a long story of studies and application, but even today not much is known about their stationary distributions when they exist. In this talk we will present the simplest version of such models and expose recent developments and techniques used to get them, especially the coupling by WIMPs and Matching.

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 19.12.2018, 11:30 (WIAS-406)
Carina Betken, Universität Osnabrück:
Dynamic random graphs: Sedentary random waypoint and preferential attachment models
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Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 12.12.2018, 11:30 (WIAS-406)
Prof. Dr. Kwabena Doku-Amponsah, University of Ghana School of Physical and Mathematical Sciences, Ghana, Republik:
Joint large deviation result for empirical measures of the random geometric graphs
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Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
We prove joint large deviation principle for the empirical pair measure and empirical locality measure of the near intermediate coloured random geometric graph models on n points picked uniformly in the space [0; 1]d; for d 2 N: From this result we obtain large deviation principles for the number of edges per vertex, the degree distribution and the proportion of isolated vertices for the near intermediate random geometric graph models.

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 05.12.2018, 11:30 (WIAS-HVP-3.13)
Prof. Dr. Serguei Popov, University of Campinas --- UNICAMP, Brasilien:
Two-dimensional random interlacements, conditional SRW, and the cryptocurrencies
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Weierstraß-Institut, Hausvogteiplatz 11A, 10117 Berlin, 3. Etage, Raum: 3.13

Abstract
Serguei Popov will briefly describe his recent research topics mentioned in the title.

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Seminar Interacting Random Systems

Host
WIAS Berlin
Wednesday, 21.11.2018, 10:45 (WIAS-HVP-3.13)
Wen Sun, TU Berlin:
On the asymptotic distribution of nucleation times of polymerization processes
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Weierstraß-Institut, Hausvogteiplatz 11A, 10117 Berlin, 3. Etage, Raum: 3.13

Abstract
Experiments have shown that there is a sharp phase transition in polymerization: it takes long time to have small amount of stable polymers and once some amount of stable polymers appear, very quickly all particles are polymerized. Moreover, the lag time usually have a very high variance. We propose a growth-fragmentation model with a critical mass to explain these phenomenon. Particles having a mass less than this critical mass are unstable -- they are fragmented much more quicker than the larger particles. A scaling approach is used, by taking the initial total mass N as a scaling parameter and assuming that the ratio of the unstable fragmentation rates to stable fragmentation rates are of order Phi(N), which is a non-decreasing function of N. We study the time evolution of this infinite dimension process under a certain class of fragmentation distributions. We show that 1) with a proper scaling parameter, the time (T) spent for the stable polymers that generating from small particles is asymptotically exponential distributed; 2) the time for the growth of stable particles has a much smaller order than T. The exponential distribution explains the high variance and the different time scales explain the sharp phase transition. These results are proved via stochastic calculus, estimations for occupation measures on different time scales, some coupling techniques and branching processes. It is a joint work with Philippe Robert.

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Forschungsgruppenseminar Interacting Random System

Host
WIAS Berlin
Wednesday, 07.11.2018, 11:30 (WIAS-406)
András Tóbiás, Technische Universität Berlin:
Signal to interference ratio percolation for Cox point processes
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Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
We study signal-to-interference plus noise ratio (SINR) percolation for Cox point processes, i.e., Poisson point processes with a random intensity measure. SINR percolation was first studied by O. Dousse et al. in the case of a two-dimensional Poisson point process. It is a version of continuum percolation where the connection between two points depends on the locations of all points of the point process. Continuum percolation for Cox point processes was recently studied by C. Hirsch, B. Jahnel and E. Cali. We study the SINR graph model for a stationary Cox point process in two or higher dimensions. We show that under suitable moment or boundedness conditions on the path-loss function and the intensity measure, this graph has an infinite connected component if the spatial density of points is large enough and the interferences are sufficiently reduced (without vanishing). This holds in all dimensions larger than 1 if the intensity measure is asymptotically essentially connected, and in two dimensions also if the intensity measure is only stabilizing but the connection radius is large. A prominent example of the intensity measure is the two-dimensional Poisson--Voronoi tessellation. We show that its total edge length in a given square has some exponential moments. We conclude that its SINR graph has an infinite cluster if the path-loss function is bounded and has a power-law decay of exponent at least 3.

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Forschungsgruppenseminar Interacting Random System

Host
WIAS Berlin
Thursday, 25.10.2018, 11:30 (WIAS-406)
Andrea Agazzi, Duke University, USA:
Large deviations and recurrence for chemical reaction networks
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Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Weierstraß-Hörsaal (Raum: 406)

Abstract
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. The extension of such results to the estimation of transitions times between metastable states further requires the positive recurrence of such processes. This property depends critically on dynamics at of the jump Process when the concentration of some of the components are small. Precise statements on the connection between this property and the structure of the underlying network are an active area of research.

Further Informations
Seminar Interacting Random Systems

Host
WIAS Berlin
Due to the COVID 19 pandemic, events are currently taking place under restricted circumstances. We offer the seminar online, besides the opportunity to attend it in person in rooms 405/406. As these rooms have a limited capacity of 12 persons, due to hygiene recommendations, we kindly ask those who want to participate in person to register (put your name in the list) in the following google document. If 12 have already registered, please follow the streamed talk. The organisers will send the zoom link shortly before the time of the talk. Please contact the organisers if you are not included in the email list and you want to follow the talk or be kept up to date with the seminar.

Zoom link

Organisers: Alexandra Quitmann and Alexander Zass