arXiv subjects

Thomas Richthammer

Publications and source records attributed to Thomas Richthammer.

13 recordsLinked to original sources

arXiv2025

Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions)

For Bernoulli percolation on a given graph $G = (V,E)$ we consider the cluster of some fixed vertex $o \in V$. We aim at comparing the number of vertices of this cluster in the set $V_+$ and in the set $V_-$, where $V_+,V_- \subset V$ have the same size. Intuitively, if $V_-$ is further away from $o$ than $V_+$, it should contain fewer vertices of the cluster. We prove such a result in terms of stochastic domination, provided that $o \in V_+$, and $V_+,V_-$ satisfy some strong symmetry conditions, and we give applications of this result in case $G$ is a bunkbed graph, a layered graph, the 2D square lattice or a hypercube graph. Our result only relies on general probabilistic techniques and a combinatorial result on group actions, and thus extends to fairly general random partitions, e.g. as induced by Bernoulli site percolation or the random cluster model.

math.PR↗

arXiv2025

Bunkbed conjecture for complete bipartite graphs and related classes of graphs

Let $G = (V,E)$ be a simple finite graph. The corresponding bunkbed graph $G^\pm$ consists of two copies $G^+ = (V^+,E^+),G^- = (V^-,E^-)$ of $G$ and additional edges connecting any two vertices $v_+ \in V_+,v_- \in V_-$ that are the copies of a vertex $v \in V$. The bunkbed conjecture states that for independent bond percolation on $G^\pm$, for all $v,w \in V$, it is more likely for $v_-,w_-$ to be connected than for $v_-,w_+$ to be connected. While this seems very plausible, so far surprisingly little is known rigorously. Recently the conjecture has been proved for complete graphs. Here we give a proof for complete bipartite graphs, complete graphs minus the edges of a complete subgraph, and symmetric complete $k$-partite graphs.

math.PR↗

arXiv2025

Monotonicity of Markov chain transition probabilities via quasi-stationarity -- an application to Bernoulli percolation on $C_k \times Z$

Let $X_n, n \ge 0$ be a Markov chain with finite state space $M$. If $x,y \in M$ such that $x$ is transient we have $P^y(X_n = x) \to 0$ for $n \to \infty$, and under mild aperiodicity conditions this convergence is monotone in that for some $N$ we have $\forall n \ge N: P^y(X_n = x)$ $\ge P^y(X_{n+1} = x)$. We use bounds on the rate of convergence of the Markov chain to its quasi-stationary distribution to obtain explicit bounds on $N$. We then apply this result to Bernoulli percolation with parameter $p$ on the cylinder graph $C_k \times Z$. Utilizing a Markov chain describing infection patterns layer per layer, we thus show the following uniform result on the monotonicity of connection probabilities: $\forall k \ge 3\, \forall n \ge 500k^6 2^k \,\forall p \in (0,1) \, \forall m \in C_k\!\!:$ $P_p((0,0) \leftrightarrow (m,n)) \ge P_p((0,0) \leftrightarrow (m,n+1))$. In general these kind of monotonicity properties of connection probabilities are difficult to establish and there are only few pertaining results.

math.PR↗

arXiv2022

Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach

A layered graph $G^\times$ is the Cartesian product of a graph $G = (V,E)$ with the linear graph $Z$, e.g. $Z^\times$ is the 2D square lattice $Z^2$. For Bernoulli percolation with parameter $p \in [0,1]$ on $G^\times$ one intuitively would expect that $P_p((o,0) \leftrightarrow (v,n)) \ge P_p((o,0) \leftrightarrow (v,n+1))$ for all $o,v \in V$ and $n \ge 0$. This is reminiscent of the better known bunkbed conjecture. Here we introduce an approach to the above monotonicity conjecture that makes use of a Markov chain building the percolation pattern layer by layer. In case of finite $G$ we thus can show that for some $N \ge 0$ the above holds for all $n \ge N$ $o,v \in V$ and $p \in [0,1]$. One might hope that this Markov chain approach could be useful for other problems concerning Bernoulli percolation on layered graphs.

math.PR↗

arXiv2019

A lower bound on the displacement of particles in 2D Gibbsian particle systems

While 2D Gibbsian particle systems might exhibit orientational order resulting in a lattice-like structure, these particle systems do not exhibit positional order if the interaction between particles satisfies some weak assumptions. Here we investigate to which extent particles within a box of size $2n \times 2n$ may fluctuate from their ideal lattice position. We show that particles near the center of the box typically show a displacement at least of order $\sqrt{log n}$. Thus we extend recent results on the hard disk model to particle systems with fairly arbitrary particle spins and interaction. Our result applies to models such as rather general continuum Potts type models, e.g. with Widom-Rowlinson or Lenard-Jones-type interaction.

math-ph↗

arXiv2016

Lower bound on the mean square displacement of particles in the hard disk model

The hard disk model is a 2D Gibbsian process of particles interacting via pure hard core repulsion. At high particle density the model is believed to show orientational order, however, it is known not to exhibit positional order. Here we investigate to what extent particle positions may fluctuate. We consider a finite volume version of the model in a box of dimensions $2n \times 2n$ with arbitrary boundary configuration,and we show that the mean square displacement of particles near the center of the box is bounded from below by $c \log n$. The result generalizes to a large class of models with fairly arbitrary interaction.

math-ph↗

arXiv2015

Gibbs measures on permutations over one-dimensional discrete point sets

We consider Gibbs distributions on permutations of a locally finite infinite set $X\subset\mathbb{R}$, where a permutation $σ$ of $X$ is assigned (formal) energy $\sum_{x\in X}V(σ(x)-x)$. This is motivated by Feynman's path representation of the quantum Bose gas; the choice $X:=\mathbb{Z}$ and $V(x):=αx^2$ is of principal interest. Under suitable regularity conditions on the set $X$ and the potential $V$, we establish existence and a full classification of the infinite-volume Gibbs measures for this problem, including a result on the number of infinite cycles of typical permutations. Unlike earlier results, our conclusions are not limited to small densities and/or high temperatures.

math.PR↗

arXiv2010

The two-type continuum Richardson model: Non-dependence of the survival of both types on the initial configuration

We consider the model of Deijfen et al. for competing growth of two infection types in R^d, based on the Richardson model on Z^d. Stochastic ball-shaped infection outbursts transmit the infection type of the center to all points of the ball that are not yet infected. Relevant parameters of the model are the initial infection configuration, the (type-dependent) growth rates and the radius distribution of the infection outbursts. The main question is that of coexistence: Which values of the parameters allow the unbounded growth of both types with positive probability? Deijfen et al. conjectured that the initial configuration basically is irrelevant for this question, and gave a proof for this under strong assumptions on the radius distribution, which e.g. do not include the case of a deterministic radius. Here we give a proof that doesn't rely on these assumptions. One of the tools to be used is a slight generalization of the model with immune regions and delayed initial infection configurations.

math.PR↗

arXiv2009

Proof of Aldous' spectral gap conjecture

Aldous' spectral gap conjecture asserts that on any graph the random walk process and the random transposition (or interchange) process have the same spectral gap. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates. The proof of the latter inequality uses suitable coset decompositions of the associated matrices on permutations.

math.PR↗

arXiv2008

Percolation of arbitrary words in one dimension

We consider a type of long-range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and provide partial results in others.

math.PR↗

arXiv2007

Translation-invariance of two-dimensional Gibbsian systems of particles with internal degrees of freedom

One of the main objectives of equilibrium state statistical physics is to analyze which symmetries of an interacting particle system in equilibrium are broken or conserved. Here we present a general result on the conservation of translational symmetry for two-dimensional Gibbsian particle systems. The result applies to particles with internal degrees of freedom and fairly arbitrary interaction, including the interesting cases of discontinuous, singular, and hard core interaction. In particular we thus show the conservation of translational symmetry for the continuum Widom Rowlinson model and a class of continuum Potts type models.

math.PR↗

arXiv2007

Translation-invariance of two-dimensional Gibbsian point processes

The conservation of translation as a symmetry in two-dimensional systems with interaction is a classical subject of statistical mechanics. Here we establish such a result for Gibbsian particle systems with two-body interaction, where the interesting cases of singular, hard-core and discontinuous interaction are included. We start with the special case of pure hard core repulsion in order to show how to treat hard cores in general.

math.PR↗

arXiv2004

Two-dimensional Gibbsian point processes with continuous spin-symmetries

We consider two-dimensional marked point processes which are Gibbsian with a two-body-potential U. U is supposed to have an internal continuous symmetry. We show that under suitable continuity conditions the considered processes are invariant under the given symmetry. We will achieve this by using Ruelle`s superstability estimates and percolation arguments.

math.PR↗