@article{64213,
  abstract     = {{A layered graph G^× is the Cartesian product of a graph G = (V, E) with the linear graph Z, e.g. Z^× is the 2D square lattice Z^2. For Bernoulli percolation with parameter p ∈ [0, 1] on G^× one intuitively would expect that P_p((o, 0) ↔ (v, n)) ≥ P_p((o, 0) ↔ (v, n + 1)) for all o, v ∈ V and n ≥ 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 ≥ 0 the above holds
for all n ≥ N o, v ∈ V and p ∈ [0, 1]. One might hope that this Markov chain approach could be useful for other problems concerning Bernoulli percolation on layered graphs}},
  author       = {{König, Philipp and Richthammer, Thomas}},
  issn         = {{0304-4149}},
  journal      = {{Stochastic Processes and their Applications}},
  publisher    = {{Elsevier BV}},
  title        = {{{Monotonicity properties for Bernoulli percolation on layered graphs— A Markov chain approach}}},
  doi          = {{10.1016/j.spa.2024.104549}},
  volume       = {{181}},
  year         = {{2024}},
}

@unpublished{64216,
  abstract     = {{Let X_n, n ≥ 0 be a Markov chain with finite state space M . If x, y ∈ M such that x is transient we have P_y (X_n = x) → 0 for n → ∞, and under mild aperiodicity conditions this convergence is monotone in that for some N we have ∀n ≥ N : P_y (X_n = x) ≥ Py (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 × Z. Utilizing a Markov chain describing infection patterns layer per layer, we thus show the following uniform result on the monotonicity of connection probabilities: ∀k ≥ 3 ∀n ≥ 500k^62^k ∀p ∈ (0, 1) ∀m ∈ C_k :
P_p((0, 0) ↔ (m, n)) ≥ P_p((0, 0) ↔ (m, n + 1)). In general these kind of monotonicity properties of connection probabilities are difficult to establish and there are only few pertaining results. }},
  author       = {{Richthammer, Thomas and König, Philipp}},
  title        = {{{Monotonicity of Markov chain transition probabilities via quasi-stationarity - an application to Bernoulli percolation on C_k × Z}}},
  year         = {{2022}},
}

@unpublished{64214,
  abstract     = {{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. }},
  author       = {{Richthammer, Thomas}},
  title        = {{{Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions)}}},
  year         = {{2022}},
}

@unpublished{64215,
  abstract     = {{Let G = (V, E) be a simple finite graph. The corresponding bunkbed graph G± consists of two copies G+ = (V +, E+), G− = (V −, E−) of G and additional edges connecting any two vertices v+ ∈ V+, v− ∈ V− that are the copies of a vertex v ∈ V . The bunkbed conjecture states that for independent bond percolation on G±, for all v, w ∈ V , it is more likely for
v−, w− to be connected than for v−, w+ to be connected. While recently a counterexample for the bunkbed conjecture was found, it should still hold for many interesting classes of graphs, and here we give a proof for complete bipartite graphs, complete graphs minus the edges of a complete subgraph, and symmetric complete k-partite graphs.}},
  author       = {{Richthammer, Thomas}},
  title        = {{{Bunkbed conjecture for complete bipartite graphs and related classes of graphs}}},
  year         = {{2022}},
}

@article{33481,
  abstract     = {{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 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 . 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.}},
  author       = {{Richthammer, Thomas and Fiedler, Michael}},
  journal      = {{Stochastic Processes and their Applications}},
  pages        = {{1--32}},
  publisher    = {{Elsevier}},
  title        = {{{A lower bound on the displacement of particles in 2D Gibbsian particle systems}}},
  doi          = {{https://doi.org/10.1016/j.spa.2020.10.003}},
  volume       = {{132}},
  year         = {{2021}},
}

@article{33344,
  abstract     = {{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 ×  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.}},
  author       = {{Richthammer, Thomas}},
  journal      = {{Communications in Mathematical Physics }},
  pages        = {{1077--1099}},
  title        = {{{Lower Bound on the Mean Square Displacement of Particles in the Hard Disk Model}}},
  doi          = {{https://link.springer.com/article/10.1007/s00220-016-2584-0}},
  volume       = {{345}},
  year         = {{2016}},
}

@article{33359,
  abstract     = {{We consider Gibbs distributions on permutations of a locally finite infinite set X⊂R, where a permutation σ of X is assigned (formal) energy ∑x∈XV(σ(x)−x). This is motivated by Feynman’s path representation of the quantum Bose gas; the choice X:=Z and V(x):=αx2 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. }},
  author       = {{Richthammer, Thomas and Biskup, Marek}},
  journal      = {{Communications in Mathematical Physics}},
  number       = {{2}},
  pages        = {{898--929}},
  publisher    = {{Springer Science+Business Media}},
  title        = {{{Gibbs measures on permutations over one-dimensional discrete point sets}}},
  doi          = {{https://doi.org/10.48550/arXiv.1310.0248}},
  volume       = {{25}},
  year         = {{2015}},
}