Johannes
Bäumler
UCLA
Interacting Random Systems  📅
Institute
Head
Alexander Zass
Usual time
Wednesdays, 11:30 (CET)
Usual venue
Weierstrass Lecture Room (WIAS-405-406)
Number of talks
6
July 2025
Jonas
Köppl
WIAS Berlin
Two edges suffice: the planar lattice two-neighbor graph percolates
Abstract.
The $k$-neighbor graph is a directed percolation model on the hypercubic lattice $\mathbb{Z}^d$ in which each vertex independently picks exactly $k$ of its $2d$ nearest neighbors at random, and we open directed edges towards those. We prove that the $2$-neighbor graph percolates on $\mathbb{Z}^2$, i.e., that the origin is connected to infinity with positive probability. The proof rests on duality, an exploration algorithm, a comparison to i.i.d. bond percolation under constraints as well as enhancement arguments. As a byproduct, we show that i.i.d. bond percolation with forbidden local patterns has a strictly larger percolation threshold than $1/2$. Additionally, our main result provides further evidence that, in low dimensions, less variability is beneficial for percolation.
June 2025
D.R. Michiel
Renger
TU MĂĽnchen
Collisions in the exclusion process II
Abstract.
(Ongoing work together with Mario Ayala (TUM)) The simple symmetric exclusion process (SSEP) and a system of independent random walkers both converge to the same diffusion equation. To understand the difference, we count the number of times a SSEP particle would have jumped if the particles were independent but does not jump because the target site is occupied. This number can also be interpreted as the number of times two SSEP particles 'collide'. We study the hydrodynamic limit of various related variables in different scaling regimes (by playing with the lattice spacing and particle numbers).
Michiel
Renger
TU MĂĽnchen
Elena
Pulvirenti
TU Delft
Metastability for the Curie–Weiss–Potts model with unbounded random interactions
Abstract.
I will first introduce the model, i.e. a disordered version of the mean-field q-spin Potts model, where the interaction coefficients between spins are general independent random variables. These random variables are chosen to have fixed mean (for simplicity taken to be 1), well defined log-moment generating function and finite variance. I will then present quantitative estimates of metastability in the regime of large number of particles at fixed temperature, when the system evolves according to a Glauber dynamics. This means that the spin configuration is viewed as a Markov chain where spins flip according to Metropolis rates at a fixed inverse temperature. Our main result identifies conditions ensuring that, with high probability, the system behaves like the corresponding system where the random couplings are replaced by their averages. More precisely, we prove that the metastability of the former system is implied with high probability by the metastability of the latter. Moreover, we consider relevant metastable hitting times of the two systems and find the asymptotic tail behaviour and the moments of their ratio. Our proofs use the potential-theoretic approach to metastability in combination with concentration inequalities. Based on a joint work in collaboration with Johan Dubbeldam, Vicente Lenz and Martin Slowik.
May 2025
Adrian
Röllin
University of Leeds
Centered Subgraph Counts in Dense Random Graphs