---
_id: '64213'
abstract:
- lang: eng
text: "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\r\nwe 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\r\nfor 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"
article_number: '104549'
author:
- first_name: Philipp
full_name: König, Philipp
last_name: König
- first_name: Thomas
full_name: Richthammer, Thomas
id: '62054'
last_name: Richthammer
citation:
ama: König P, Richthammer T. Monotonicity properties for Bernoulli percolation on
layered graphs— A Markov chain approach. Stochastic Processes and their Applications.
2024;181. doi:10.1016/j.spa.2024.104549
apa: König, P., & Richthammer, T. (2024). Monotonicity properties for Bernoulli
percolation on layered graphs— A Markov chain approach. Stochastic Processes
and Their Applications, 181, Article 104549. https://doi.org/10.1016/j.spa.2024.104549
bibtex: '@article{König_Richthammer_2024, title={Monotonicity properties for Bernoulli
percolation on layered graphs— A Markov chain approach}, volume={181}, DOI={10.1016/j.spa.2024.104549},
number={104549}, journal={Stochastic Processes and their Applications}, publisher={Elsevier
BV}, author={König, Philipp and Richthammer, Thomas}, year={2024} }'
chicago: König, Philipp, and Thomas Richthammer. “Monotonicity Properties for Bernoulli
Percolation on Layered Graphs— A Markov Chain Approach.” Stochastic Processes
and Their Applications 181 (2024). https://doi.org/10.1016/j.spa.2024.104549.
ieee: 'P. König and T. Richthammer, “Monotonicity properties for Bernoulli percolation
on layered graphs— A Markov chain approach,” Stochastic Processes and their
Applications, vol. 181, Art. no. 104549, 2024, doi: 10.1016/j.spa.2024.104549.'
mla: König, Philipp, and Thomas Richthammer. “Monotonicity Properties for Bernoulli
Percolation on Layered Graphs— A Markov Chain Approach.” Stochastic Processes
and Their Applications, vol. 181, 104549, Elsevier BV, 2024, doi:10.1016/j.spa.2024.104549.
short: P. König, T. Richthammer, Stochastic Processes and Their Applications 181
(2024).
date_created: 2026-02-18T12:06:28Z
date_updated: 2026-02-18T12:32:13Z
doi: 10.1016/j.spa.2024.104549
intvolume: ' 181'
language:
- iso: eng
publication: Stochastic Processes and their Applications
publication_identifier:
issn:
- 0304-4149
publication_status: published
publisher: Elsevier BV
status: public
title: Monotonicity properties for Bernoulli percolation on layered graphs— A Markov
chain approach
type: journal_article
user_id: '62054'
volume: 181
year: '2024'
...
---
_id: '64216'
abstract:
- lang: eng
text: "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 :\r\nP_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:
- first_name: Thomas
full_name: Richthammer, Thomas
id: '62054'
last_name: Richthammer
- first_name: Philipp
full_name: König, Philipp
last_name: König
citation:
ama: Richthammer T, König P. Monotonicity of Markov chain transition probabilities
via quasi-stationarity - an application to Bernoulli percolation on C_k × Z. Published
online 2022.
apa: Richthammer, T., & König, P. (2022). Monotonicity of Markov chain transition
probabilities via quasi-stationarity - an application to Bernoulli percolation
on C_k × Z.
bibtex: '@article{Richthammer_König_2022, title={Monotonicity of Markov chain transition
probabilities via quasi-stationarity - an application to Bernoulli percolation
on C_k × Z}, author={Richthammer, Thomas and König, Philipp}, year={2022} }'
chicago: Richthammer, Thomas, and Philipp König. “Monotonicity of Markov Chain Transition
Probabilities via Quasi-Stationarity - an Application to Bernoulli Percolation
on C_k × Z,” 2022.
ieee: T. Richthammer and P. König, “Monotonicity of Markov chain transition probabilities
via quasi-stationarity - an application to Bernoulli percolation on C_k × Z.”
2022.
mla: Richthammer, Thomas, and Philipp König. Monotonicity of Markov Chain Transition
Probabilities via Quasi-Stationarity - an Application to Bernoulli Percolation
on C_k × Z. 2022.
short: T. Richthammer, P. König, (2022).
date_created: 2026-02-18T12:27:28Z
date_updated: 2026-02-18T12:27:38Z
language:
- iso: eng
status: public
title: Monotonicity of Markov chain transition probabilities via quasi-stationarity
- an application to Bernoulli percolation on C_k × Z
type: preprint
user_id: '62054'
year: '2022'
...
---
_id: '64214'
abstract:
- lang: eng
text: '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:
- first_name: Thomas
full_name: Richthammer, Thomas
id: '62054'
last_name: Richthammer
citation:
ama: Richthammer T. Comparing the number of infected vertices in two symmetric sets
for Bernoulli percolation (and other random partitions). Published online 2022.
apa: Richthammer, T. (2022). Comparing the number of infected vertices in two
symmetric sets for Bernoulli percolation (and other random partitions).
bibtex: '@article{Richthammer_2022, title={Comparing the number of infected vertices
in two symmetric sets for Bernoulli percolation (and other random partitions)},
author={Richthammer, Thomas}, year={2022} }'
chicago: Richthammer, Thomas. “Comparing the Number of Infected Vertices in Two
Symmetric Sets for Bernoulli Percolation (and Other Random Partitions),” 2022.
ieee: T. Richthammer, “Comparing the number of infected vertices in two symmetric
sets for Bernoulli percolation (and other random partitions).” 2022.
mla: Richthammer, Thomas. Comparing the Number of Infected Vertices in Two Symmetric
Sets for Bernoulli Percolation (and Other Random Partitions). 2022.
short: T. Richthammer, (2022).
date_created: 2026-02-18T12:13:09Z
date_updated: 2026-02-18T12:13:21Z
language:
- iso: eng
status: public
title: Comparing the number of infected vertices in two symmetric sets for Bernoulli
percolation (and other random partitions)
type: preprint
user_id: '62054'
year: '2022'
...
---
_id: '64215'
abstract:
- lang: eng
text: "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\r\nv−, 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:
- first_name: Thomas
full_name: Richthammer, Thomas
id: '62054'
last_name: Richthammer
citation:
ama: Richthammer T. Bunkbed conjecture for complete bipartite graphs and related
classes of graphs. Published online 2022.
apa: Richthammer, T. (2022). Bunkbed conjecture for complete bipartite graphs
and related classes of graphs.
bibtex: '@article{Richthammer_2022, title={Bunkbed conjecture for complete bipartite
graphs and related classes of graphs}, author={Richthammer, Thomas}, year={2022}
}'
chicago: Richthammer, Thomas. “Bunkbed Conjecture for Complete Bipartite Graphs
and Related Classes of Graphs,” 2022.
ieee: T. Richthammer, “Bunkbed conjecture for complete bipartite graphs and related
classes of graphs.” 2022.
mla: Richthammer, Thomas. Bunkbed Conjecture for Complete Bipartite Graphs and
Related Classes of Graphs. 2022.
short: T. Richthammer, (2022).
date_created: 2026-02-18T12:17:42Z
date_updated: 2026-02-18T12:23:17Z
language:
- iso: eng
status: public
title: Bunkbed conjecture for complete bipartite graphs and related classes of graphs
type: preprint
user_id: '62054'
year: '2022'
...
---
_id: '33481'
abstract:
- lang: eng
text: 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:
- first_name: Thomas
full_name: Richthammer, Thomas
id: '62054'
last_name: Richthammer
- first_name: Michael
full_name: Fiedler, Michael
last_name: Fiedler
citation:
ama: Richthammer T, Fiedler M. A lower bound on the displacement of particles in
2D Gibbsian particle systems. Stochastic Processes and their Applications.
2021;132:1-32. doi:https://doi.org/10.1016/j.spa.2020.10.003
apa: Richthammer, T., & Fiedler, M. (2021). A lower bound on the displacement
of particles in 2D Gibbsian particle systems. Stochastic Processes and Their
Applications, 132, 1–32. https://doi.org/10.1016/j.spa.2020.10.003
bibtex: '@article{Richthammer_Fiedler_2021, title={A lower bound on the displacement
of particles in 2D Gibbsian particle systems}, volume={132}, DOI={https://doi.org/10.1016/j.spa.2020.10.003},
journal={Stochastic Processes and their Applications}, publisher={Elsevier}, author={Richthammer,
Thomas and Fiedler, Michael}, year={2021}, pages={1–32} }'
chicago: 'Richthammer, Thomas, and Michael Fiedler. “A Lower Bound on the Displacement
of Particles in 2D Gibbsian Particle Systems.” Stochastic Processes and Their
Applications 132 (2021): 1–32. https://doi.org/10.1016/j.spa.2020.10.003.'
ieee: 'T. Richthammer and M. Fiedler, “A lower bound on the displacement of particles
in 2D Gibbsian particle systems,” Stochastic Processes and their Applications,
vol. 132, pp. 1–32, 2021, doi: https://doi.org/10.1016/j.spa.2020.10.003.'
mla: Richthammer, Thomas, and Michael Fiedler. “A Lower Bound on the Displacement
of Particles in 2D Gibbsian Particle Systems.” Stochastic Processes and Their
Applications, vol. 132, Elsevier, 2021, pp. 1–32, doi:https://doi.org/10.1016/j.spa.2020.10.003.
short: T. Richthammer, M. Fiedler, Stochastic Processes and Their Applications 132
(2021) 1–32.
date_created: 2022-09-26T06:53:59Z
date_updated: 2022-09-26T06:54:06Z
department:
- _id: '96'
doi: https://doi.org/10.1016/j.spa.2020.10.003
intvolume: ' 132'
language:
- iso: eng
page: 1-32
publication: Stochastic Processes and their Applications
publication_status: published
publisher: Elsevier
status: public
title: A lower bound on the displacement of particles in 2D Gibbsian particle systems
type: journal_article
user_id: '85821'
volume: 132
year: '2021'
...
---
_id: '33344'
abstract:
- lang: eng
text: 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:
- first_name: Thomas
full_name: Richthammer, Thomas
id: '62054'
last_name: Richthammer
citation:
ama: Richthammer T. Lower Bound on the Mean Square Displacement of Particles in
the Hard Disk Model. Communications in Mathematical Physics . 2016;345:1077-1099.
doi:https://link.springer.com/article/10.1007/s00220-016-2584-0
apa: Richthammer, T. (2016). Lower Bound on the Mean Square Displacement of Particles
in the Hard Disk Model. Communications in Mathematical Physics , 345,
1077–1099. https://link.springer.com/article/10.1007/s00220-016-2584-0
bibtex: '@article{Richthammer_2016, title={Lower Bound on the Mean Square Displacement
of Particles in the Hard Disk Model}, volume={345}, DOI={https://link.springer.com/article/10.1007/s00220-016-2584-0},
journal={Communications in Mathematical Physics }, author={Richthammer, Thomas},
year={2016}, pages={1077–1099} }'
chicago: 'Richthammer, Thomas. “Lower Bound on the Mean Square Displacement of Particles
in the Hard Disk Model.” Communications in Mathematical Physics 345 (2016):
1077–99. https://link.springer.com/article/10.1007/s00220-016-2584-0.'
ieee: 'T. Richthammer, “Lower Bound on the Mean Square Displacement of Particles
in the Hard Disk Model,” Communications in Mathematical Physics , vol.
345, pp. 1077–1099, 2016, doi: https://link.springer.com/article/10.1007/s00220-016-2584-0.'
mla: Richthammer, Thomas. “Lower Bound on the Mean Square Displacement of Particles
in the Hard Disk Model.” Communications in Mathematical Physics , vol.
345, 2016, pp. 1077–99, doi:https://link.springer.com/article/10.1007/s00220-016-2584-0.
short: T. Richthammer, Communications in Mathematical Physics 345 (2016) 1077–1099.
date_created: 2022-09-13T08:01:29Z
date_updated: 2022-09-13T08:01:36Z
department:
- _id: '96'
doi: https://link.springer.com/article/10.1007/s00220-016-2584-0
intvolume: ' 345'
language:
- iso: eng
page: 1077-1099
publication: 'Communications in Mathematical Physics '
publication_status: published
status: public
title: Lower Bound on the Mean Square Displacement of Particles in the Hard Disk Model
type: journal_article
user_id: '85821'
volume: 345
year: '2016'
...
---
_id: '33359'
abstract:
- lang: eng
text: '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:
- first_name: Thomas
full_name: Richthammer, Thomas
id: '62054'
last_name: Richthammer
- first_name: Marek
full_name: Biskup, Marek
last_name: Biskup
citation:
ama: Richthammer T, Biskup M. Gibbs measures on permutations over one-dimensional
discrete point sets. Communications in Mathematical Physics. 2015;25(2):898-929.
doi:https://doi.org/10.48550/arXiv.1310.0248
apa: Richthammer, T., & Biskup, M. (2015). Gibbs measures on permutations over
one-dimensional discrete point sets. Communications in Mathematical Physics,
25(2), 898–929. https://doi.org/10.48550/arXiv.1310.0248
bibtex: '@article{Richthammer_Biskup_2015, title={Gibbs measures on permutations
over one-dimensional discrete point sets}, volume={25}, DOI={https://doi.org/10.48550/arXiv.1310.0248},
number={2}, journal={Communications in Mathematical Physics}, publisher={Springer
Science+Business Media}, author={Richthammer, Thomas and Biskup, Marek}, year={2015},
pages={898–929} }'
chicago: 'Richthammer, Thomas, and Marek Biskup. “Gibbs Measures on Permutations
over One-Dimensional Discrete Point Sets.” Communications in Mathematical Physics
25, no. 2 (2015): 898–929. https://doi.org/10.48550/arXiv.1310.0248.'
ieee: 'T. Richthammer and M. Biskup, “Gibbs measures on permutations over one-dimensional
discrete point sets,” Communications in Mathematical Physics, vol. 25,
no. 2, pp. 898–929, 2015, doi: https://doi.org/10.48550/arXiv.1310.0248.'
mla: Richthammer, Thomas, and Marek Biskup. “Gibbs Measures on Permutations over
One-Dimensional Discrete Point Sets.” Communications in Mathematical Physics,
vol. 25, no. 2, Springer Science+Business Media, 2015, pp. 898–929, doi:https://doi.org/10.48550/arXiv.1310.0248.
short: T. Richthammer, M. Biskup, Communications in Mathematical Physics 25 (2015)
898–929.
date_created: 2022-09-14T04:57:58Z
date_updated: 2022-09-14T04:58:02Z
department:
- _id: '96'
doi: https://doi.org/10.48550/arXiv.1310.0248
intvolume: ' 25'
issue: '2'
language:
- iso: eng
page: 898-929
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer Science+Business Media
status: public
title: Gibbs measures on permutations over one-dimensional discrete point sets
type: journal_article
user_id: '85821'
volume: 25
year: '2015'
...