# On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field

@article{Bogachev2019OnTU, title={On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field}, author={Leonid V. Bogachev and Utkir A. Rozikov}, journal={Journal of Statistical Mechanics: Theory and Experiment}, year={2019} }

The paper concerns the $q$-state Potts model (i.e., with spin values in $\{1,\dots,q\}$) on a Cayley tree $\mathbb{T}^k$ of degree $k\geq 2$ (i.e., with $k+1$ edges emanating from each vertex) in an external (possibly random) field. We construct the so-called splitting Gibbs measures (SGM) using generalized boundary conditions on a sequence of expanding balls, subject to a suitable compatibility criterion. Hence, the problem of existence/uniqueness of SGM is reduced to solvability of the…

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## References

SHOWING 1-10 OF 83 REFERENCES

Gibbs Measures For SOS Models On a Cayley Tree

- Mathematics, Physics
- 2004

We consider a nearest-neighbor SOS model, spin values $0,1,..., m$, $m\geq 2$, on a Cayley tree of order $k$ . We mainly assume that $m=2$ and study translation-invariant (TI) and `splitting' (S)…

Fuzzy transformations and extremality of Gibbs measures for the potts model on a Cayley tree

- Mathematics, PhysicsRandom Struct. Algorithms
- 2017

This paper finds some regions for the temperature parameter ensuring that a given TISGM is (non-)extreme in the set of all Gibbs measures, and shows the existence of a temperature interval for which there are at least $2^{q-1} + q$ extremal TISGMs.

Description of the Translation-Invariant Splitting Gibbs Measures for the Potts Model on a Cayley Tree

- Mathematics, Physics
- 2014

For the $$q$$q-state Potts model on a Cayley tree of order $$k\ge 2$$k≥2 it is well-known that at sufficiently low temperatures there are at least $$q+1$$q+1 translation-invariant Gibbs measures…

Boundary Conditions for Translation-Invariant Gibbs Measures of the Potts Model on Cayley Trees

- Mathematics, Physics
- 2015

We consider translation-invariant splitting Gibbs measures (TISGMs) for the q-state Potts model on a Cayley tree of order two. Recently a full description of the TISGMs was obtained, and it was shown…

Tree-hierarchy of DNA and distribution of Holliday junctions

- Physics, MathematicsJournal of mathematical biology
- 2017

A hierarchy of a countable set of DNAs each of which ’lives’ on the same Cayley tree is given, which has property that each vertex of the Cayley Tree belongs only to one of DNA.

Evolutionary trees and the Ising model on the Bethe lattice: a proof of Steel’s conjecture

- Mathematics, Computer ScienceArXiv
- 2005

It is demonstrated that extremality of the free Gibbs measure on the infinite binary tree, which has been studied before in probability, statistical physics and computer science, determines how distinguishable are Gibbs measures on finite binary trees.

On the Phase Diagram of the Random Field Ising Model on the Bethe Lattice

- Physics
- 1998

The ferromagnetic Ising model on the Bethe lattice of degree k is considered in the presence of a dichotomous external random field ξx = ±α and the temperature T≥0. We give a description of a part of…

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

- Mathematics, Computer ScienceSIAM J. Comput.
- 2016

This work presents a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases.

Random Walks in I.I.D. Random Environment on Cayley Trees

- Mathematics
- 2013

We consider the random walk in an \emph{i.i.d.} random environment on the infinite $d$-regular tree for $d \geq 3$. We consider the tree as a Cayley graph of free product of finitely many copies of…

Markov random fields and Markov chains on trees

- Mathematics
- 1981

We consider probability measures on a space S(^A) (where S and A are countable and the σ-field is the natural one) which are Markov random fields with respect to a given neighbour relation ~ on A. In…