CS17: Dependent percolation models: discrete and continuum

date: 7/17/2025, time: 14:00-15:30, room: ICS 25

Organizer: Bas Lodewijks (University of Augsburg)

Chair: Bas Lodewijks (University of Augsburg)

Oriented percolation in random spatial environment

Anh Duc Vu (WIAS Berlin)

We study an oriented north-east north-west percolation model on \(\mathbb{Z}^2\). Given an iid sequence \((\xi_x)_{x\in\mathbb{Z}}\subset\mathbb{N}\) (the random environment) and some \(p\in(0,1)\), vertices \((t,x)\) are open with probability \(p^{\xi_x}\). If \(\mathbb{E}[\xi_0^{1+\varepsilon}]<\infty\), then we show the existence of a percolation phase transition in \(p\). Our main result extends \([1]\) to the oriented bond-site percolation setting. This allows us to prove survival of contact processes featuring periodic recoveries, as well as survival of the contact process in random environment as introduced in \([2]\).

Bibliography

\([1]\) M.R. Hilário, M. Sá, R. Sanchis and A. Teixeira. Phase transition for percolation on a randomly stretched square lattice. The Annals of Applied Probability, vol. 33, no. 4, 2023, pp. 3145-3168.

\([2]\) M. Bramson, R. Durrett and R.H. Schonmann. The contact process in a random environment The Annals of Probability, vol. 19, no. 3, 1991, pp. 960-983.

Percolation in lattice k-neighbor graphs

András Tóbiás (Budapest University of Technology and Economics; Alfréd Rényi Institute of Mathematics)

We define a random graph obtained via connecting each point of \(\mathbb Z^d\) independently to a fixed number \(1 \leq k \leq 2d\) of its nearest neighbors via a directed edge. We call this graph the directed \(k\)-neighbor graph. Two natural associated undirected graphs are the undirected and the bidirectional \(k\)-neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed \(k\)-neighbor graph between them in at least one, respectively precisely two, directions. In these graphs we study the question of percolation, i.e., the existence of an infinite self-avoiding path. Using different kinds of proof techniques for different classes of cases, we show that for \(k=1\) even the undirected \(k\)-neighbor graph never percolates, but the directed one percolates whenever \(k \geq d+1\). Our main result is the following.

Theorem. The directed \(k\)-neighbor graph percolates when \(k \geq 3\) and \(d \geq 5\), or \(k=d=4\).

The proof of this result is based on a technique developed by Cox and Durrett \([2]\) (1983) for verifying the existence of an infinite oriented path in higher-dimensional oriented percolation.

We also show that the undirected 2-neighbor graph percolates for \(d=2\), the undirected 3-neighbor graph percolates for \(d=3\), and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the -nearest-neighbor graphs studied in continuum percolation, and our results support this interpretation.

Percolation in the directed graph for \(k=d=2\) has recently been proven by Coupier, Henry, Jahnel and Köppl \([1]\), which will be explained in another talk of our contributed session by Jonas Köppl.

Bibliography

\([1]\) David Coupier, Benoît Henry, Benedikt Jahnel and Jonas Köppl (2024). “The planar lattice two-neighbor graph percolates.” arXiv:2412.20781, 45 pp.

\([2]\) Theodore Cox and Richard Durrett (1983). “Oriented percolation in dimensions \(d \geq 4\): bounds and asymptotic formulas.” Math. Proc. Camb. Phil. Soc., vol. 93, pp. 151–162.

The variational principle for a marked Gibbs point process with infinite-range multibody interactions

Jonas Köppl (WIAS Berlin)

The Gibbs variational principle provides a fundamental connection between microscopic interactions and macroscopic thermodynamic behavior by characterizing equilibrium states as those minimizing the sum of energy and entropy densities—i.e., the free energy. In the setting of spatial stochastic systems, particularly marked Gibbs point processes, this principle plays a central role in understanding how geometry and randomness interact to produce large-scale structure.

In this talk, we present a rigorous formulation and proof of the Gibbs variational principle for a class of continuum point processes with unbounded marks and infinite-range multi-body, including the Asakura–Oosawa colloid–polymer system from soft matter physics. These models involve hard-core exclusion and a temperature-dependent area interaction that leads to genuinely multibody and long-range interactions.

We show that, despite the lack of locality and the presence of long-range dependencies, the variational principle holds and can be used to characterize all infinite-volume Gibbs measures via a free energy minimization problem. The essential control over the influence of boundary conditions can be established using the geometry of the model and the hard-core constraint.