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Technische Universität München

Workshop: Women in Random Geometric Systems

Forschung, Probability Theory | | Event

The workshop aims at bringing together female speakers in random geometric systems, as well as further members of the SPP 2265, in order to discuss recent advances in the field and offer networking opportunities for younger researchers. In this sense, we plan to have 13 long talks by renowned speakers and further seven talks by junior speakers. The workshop starts on Wednesday morning and ends on Friday at lunchtime.

The scientific program is organized by Alice Callegaro, Diana Conache, Nina Gantert, Silke Rolles.

There is no conference fee and there are no gender restrictions on the audience, everybody is welcome to attend. In particular, all the members of SPP 2265 are invited.

The workshop is funded by the DFG priority program SPP 2265 Random Geometric System. Members of the SPP who are interested in participating in the workshop should send an email to diana.conache(at)tum.de. In case they need financial support for the participation it should be mentioned in the email. There are some funds available for this.

Location and dates

The workshop takes place from March 5th (Wednesday) to March 7th (Friday) 2025, in room 5606.EG.011 (Hörsaal 3, Boltzmannstr. 3, 85748 Garching b. München) of the Department of Mathematics of the Technische Universität München.

Program

We will have several talks by renowned international speakers and a few short talks by PhD students.

Wednesday, March 5th

  • 09:00 -- 09:45 - Faggionato
  • 09:50 -- 10:35 - Magnanini
  • 10:40 -- 11:10 Coffee Break
  • 11:10 -- 11:30 - Short Talk Couillard
  • 11:35 -- 12:25 - Sturm
  • 12:30 -- 14:00 Lunch Break
  • 14:00 -- 14:45 - Pokalyuk
  • 14:50 -- 15:10 - Short Talk dai Pra
  • 15:15 -- 15:35 - Short Talk Mellis
  • 15:40 -- 16:10 Coffee Break

  • 16:10 -- 16:55 - Andreis

     

Thursday, March 6th

  • 09:00 -- 09:45 - Cotar
  • 09:50 -- 10:35 - Ruszel
  • 10:40 -- 11:10 Coffee Break
  • 11:10 -- 11:30 - Short Talk van Leeuwen
  • 11:35 -- 12:25 - Jacquier
  • 12:30 -- 14:00 Lunch Break
  • 14:00 -- 14:45 - Mu
  • 14:50 -- 15:10 - Short Talk Baran
  • 15:15 -- 15:35 - Short Talk Brockhaus
  • 15:40 -- 16:10 Coffee Break

  • 16:10 -- 16:55 - Bianchi

     

Friday, March 7th

  • 09:00 -- 09:45 - Deijfen
  • 09:50 -- 10:35 - Döring
  • 10:40 -- 11:10 Coffee Break
  • 11:10 -- 11:30 - Short Talk Kerriou
  • 11:35 -- 12:25- Winter

  • 12:30 -- 14:00 Lunch Break and End of the Conference

     

Confirmed Speakers

Confirmed Speakers - Short Talks

 

Abstracts

Confirmed Speakers

Luisa Andreis - Rare events in sparse random graphs, random trees and coagulation processes

In this talk, we would like to give an overview on recent and past results on large deviations for sparse random graphs and coagulation processes. In particular, recently there have been progresses in the study of inhomogeneous random graphs and random graphs with marks in this framework, with some interesting similarities between different models appear when studying their large deviation rate functions. On the other hand, an approach that exploits the description of interaction in spatial coagulation processes with random tree-like structures has proved useful in proving large deviations for such Markov processes. We will describe such approach and outline possible future directions. This talk is based on joint works with W. K¨onig, H. Langhammer, E. Magnanini (WIAS Berlin) and M. Kolodziejczyk (PoliMi).

 

Alessandra Bianchi - Mixing cutoff for random walks on directed inhomogeneous random graphs 

In this talk, we will consider a simple random walk on directed inhomogeneous random graphs and analyze its convergence to equilibrium under specific network conditions. The first part of the talk will focus on the Chung-Lu directed graph, a generalization of the Erdős–Rényi random digraph that incorporates edges independently and according to assigned Bernoulli laws. Assuming the average degree grows logarithmically with the graph size n (weakly sparse regime), we establish the occurrence of a cutoff phenomenon at the entropic time scale of order \log(n)/\log\log(n) and precisely characterize the cutoff profile within a specific window. The second part of the talk will address a variant of this graph model that includes community structure. Depending on the strength of inter-community interactions, we prove that the mixing behavior can display either a cutoff or a smooth exponential decay, with an intermediate behavior emerging in the critical regime. This is joint work with G. Passuello and M. Quattropani.

 

Mia Deijfen - Geometric random intersection graphs

An intersection graph is constructed by assigning each vertex a subset of some auxiliary set and then connecting two vertices if their subsets intersect. The model type has been popular in network modeling to describe networks arising from bipartite structures, for instance individuals who are connected if they share a social group, communication units connected via cell towers and scientists related through joint papers. We study a spatial version of the model type where both the vertex set and the auxiliary set are represented by Poisson processes on R^d, giving rise to a variation of the random connection model. Our results concern local quantities (e.g.  the degree distribution) and percolation properties of the resulting graph.

 

Hanna Döring - Crossings in Projected Random Geometric Graphs

We compare the crossing number of real world networks with the number of edge crossings in a random graph generated by projecting a higher-dimensional random geometric graph on a plane. Applying the Malliavin-Stein method we prove a central limit theorem together with a rate of convergence for the number of crossings in the projection of a random geometric graph in the thermodynamic regime. In the sparse regime, we show that these crossings converge to a Poisson point process. his talk is based on joint work with Lianne de Jonge.

 

Alessandra Faggionato - Large scale limit of the density fluctuation field of the SSEP on random graphs

We consider the simple symmetric exclusion process on a generic random weighted graph in R^d, built on a simple point process with positive weights (usually called conductances). Under the assumption of stationarity, ergodicity and a second moment condition, we investigate the scaling limit of the density fluctuation field. For d>2  we prove a universal  limit  towards a generalized Ornstein-Uhlenbeck process. For d=2 we discuss some relevant cases related to  Hölder regularity of solutions of discrete parabolic  equations. Joint work with A. Chiarini. 

 

Vanessa Jacquier - Discrete Nonlocal Isoperimetric Inequality and Analysis of the Long-Range Bi-Axial Ising Model

We consider a generalization of the classical perimeter, referred to as the nonlocal bi-axial discrete perimeter,  in which not only the external boundary of a polyomino contributes to the perimeter, but all its internal and external components. In particular, this nonlocal bi-axial perimeter is a fractional function that depends on the shape of the given geometrical figure and on a parameter λ>1, which accounts for the long-range effect.

We tackle the nonlocal discrete isoperimetric problem analyzing and characterizing the minimizers within the class of polyominoes with a fixed area. The solution of this isoperimetric problem provides a rigorous foundation for studying the metastable behavior of the long-range bi-axial Ising model.

 

Elena Magnanini - Convergence of subgraph densities in Exponential Random Graphs 

Exponential Random Graphs are a class of network models that can be seen as the generalization of the dense Erdős-Rényi random graph. They are defined, with a statistical mechanics approach, by introducing a Hamiltonian, a function that biases the occurrence of certain features, such as the number of edges or triangles. In this talk we will primarily, but not exclusively, focus on the so-called edge triangle model, where the Hamiltonian of the system only collects edge and triangle densities, properly tuned by real parameters. Using tools from statistical mechanics and large deviation theory, we establish limit theorems and concentration inequalities for subgraph densities (mainly focusing on edge and triangle density) in the replica-symmetric regime, where the limiting free energy of the model is known together with its phase diagram. Part of the results are concerned with a mean-field approximation, which allows for explicit computations and provides insights into the behavior of the original model in certain parameter region where rigorous results are hardly achievable. 
This talk is based on joint work with A. Bianchi, F. Collet, and G. Passuello.

 

Yingxin Mu - Indistinguishability of the Boolean model beyond Euclidean space

In this talk, we will consider the Boolean model in symmetric space, where each point of an isometry-invariant point process serves as the center of a ball with a random radius, partitioning the space into occupied and vacant regions. We focus on the indistinguishability of these infinite occupied or vacant components using isometry-invariant component properties, such as volume. We will show that for an insertion- (or deletion-) tolerant process, all infinite occupied (or vacant) components cannot be distinguished from each other by any isometry-invariant component property. This is joint work with A. Sapozhnikov.

 

Cornelia Pokalyuk - Survival and parasite spread in a spatial host-parasite model with host immunity

We introduce a generalized version of the frog model to describe the invasion of a parasite population in a spatially structured immobile host population with host immunity on the integer line. Parasites move according to simple symmetric random
walks and try to infect any host they meet. Hosts, however, own an immunity against the parasites that protects them from infection for a random number of attacks. Once a host gets infected, it and the infecting parasite die, and a random number of offspring parasites is generated. We show that the the positivity of the survival probability of parasites only depends on the mean offspring and mean height of immunity. Furthermore, we prove through the construction of a renewal structure that given survival of the parasite population parasites invade the host population at linear speed under relatively mild assumptions on the host immunity distribution.
This talk is based on a joint work with Sascha Franck, see also arXiv:2502.08475 for a preprint.
 

Wioletta Ruszel - Fermionic structures in random lattice models

The goal of the talk is to explain the (fermionic) discrete Gaussian free field  structure that appears in special random lattice models, namely the Abelian sandpile model, and the uniform spanning tree. In particular, we introduce the Abelian sandpile model, an example of a statistical mechanics model that displays self-organized criticality. We show its key properties, and present the burning bijection, a famous algorithm by Majumdar and Dhar which connects the sandpile to uniform spanning trees. To link both models to "fermions" we then introduce Grassmannian calculus. We give a brief presentation of its rules that are needed to work probabilistically, and we apply them to construct Grassmannian Gaussians. A nice application of these techniques is for scaling limits of local fields, like the height-one field of the sandpile or the degree field of the uniform spanning tree, for which we sketch a few proofs. This is based on the work in arXiv:2309.08349 and in collaboration with L. Chiarini (U Durham), A. Cipriani (UCL) and A. Rapoport (UU).

 

Anja Sturm - On min-max games on trees and beyond

We study a random game in which two players in turn play a fixed number of moves. For each move, there are two possible choices. To each possible outcome of the game we assign a winner in an i.i.d. fashion with a fixed parameter p. In the case where all different game histories lead to different outcomes, a classical result due to Pearl (1980) says that in the limit when the number of moves is large, there is a sharp threshold in the parameter p that separates the regimes in which either player has with high probability a winning strategy. 

We are interested in a modification of this game where the outcome is determined by the exact sequence of moves played by the first player (as in a game tree) and by the number of times the second player has played each of the two possible moves. We show that also in this case, there is a sharp threshold in the parameter p that separates the regimes in which either player has with high probability a winning strategy. Since in the modified game, different game histories can lead to the same outcome, the graph associated with the game is no longer a tree which means independence is lost. As a result, the analysis becomes more complicated and open problems remain.

This is joint work with Jan Swart (UTIA Prague) and Natalia Cardona Tobon (Universidad Nacional de Colombia).

 

Short Talks

Zsuzsa Baran - Intersections of branching random walks on $\mathbb{Z}^8$

We consider the intersections between the ranges of two independent branching random walks on $\mathbb{Z}^8$, and discuss two results that are analogous to the results by Lawler from the '90s about two independent simple random walks on $\mathbb{Z}^4$. We also show a weak law of large numbers for the branching capacity of the range of a branching random walk.

 

Marta dai Pra - The genealogy of a logistic branching process with selection

In this talk we present a model for growth in a multi-species population. We consider two types evolving as a logistic branching process with mutation, where one of the type as a selective advantage. We first study the frequency of the disadvantageous type and show that, once the population approaches the carrying capacity, such frequency converges to a Wright—Fisher diffusion process. We then study the dynamics backward in time: we fix a time horizon at which the population is at carrying capacity and we study the ancestral relations of a sample of individuals . We prove that, provided that the advantageous and disadvantageous branching measures are ordered, this ancestral line process converges to the moment dual of the limiting diffusion. This talk is based on ongoing joint work with Julian Kern.

 

Sophia-Marie Mellis - Coalescents with migration in the moderate regime

Multi-type models have recently experienced renewed interest in the stochastic modeling of evolution. This is partially due to their mathematical analysis often being more challenging than their single-type counterparts; an example of this is the site-frequency spectrum of a colony-based population with moderate migration. 

In this talk, we model the genealogy of such a population via a multi-type coalescent starting with $N(K)$ colored singletons with $d \geq 2$ possible colors (colonies). The process is described by a continuous-time Markov chain with values on the colored partitions of the colored integers in $\{1, \ldots, N(K)\}$; blocks of the same color coalesce at rate $1$, while they are also allowed to change color at a rate proportional to $K$ (migration). 

Given this setting, we study the asymptotic behavior, as $K\to\infty$ at small times, of the vector of empirical measures, whose $i$-th component keeps track of the blocks of color $i$ at time $t$ and of the initial coloring of the integers composing each of these blocks. We show that, in the proper time-space scaling, it converges to a multi-type branching process (case $N(K) \sim K$) or a multi-type Feller diffusion (case $N(K) \gg K$). Using this result, we derive an applicable representation of the site-frequency spectrum.

This is joint work with Fernando Cordero and Emmanuel Schertzer.

 

Céline Kerriou - Temporal connectivity of Random Geometric Graphs

A temporal random geometric graph is a random geometric graph in which all edges are endowed with a uniformly random time-stamp. We determine a threshold for the existence of monotone increasing paths between all pairs of vertices in temporal random geometric graphs. Our results hold for a family of soft random geometric graphs as well as the hard random geometric graph. This talk is based on joint work with A. Brandenberger, S. Donderwinkel, G. Lugosi and R. Mitchell.
 

Tess van Leeuwen - Complex abstract Wiener spaces

In this talk we investigate Gaussian random fields on an arbitrary complex Banach space. We see that a standard Gaussian field on a Hilbert space cannot exist, but that an abstract Wiener space is a viable alternative. This concept turns out to be equivalent to that of a centred Gaussian field, and in some cases it may be constructed by way of a straightforward Hilbert-Schmidt operator. We may relate the complex abstract Wiener space to the real version if and only if the complex Banach space in question admits a bounded real structure. In this case, the complex Gaussian field is a complex linear combination of two independent, identical real Gaussian fields. Finally, this theory is applied to the construction of the fractional Gaussian fields. This talk is based on joint work with W. M. Ruszel.

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