The vertex-transitive TLF-planar graphs

We consider the class of the topologically locally finite (in short TLF) planar vertex-transitive graphs, a class containing in particular all the one-ended planar Cayley graphs and the normal transitive tilings. We characterize these graphs with a finite local representation and a special kind of finite state automaton named labeling scheme. As a result, we are able to enumerate and describe all TLF-planar vertex-transitive graphs of any given degree. Also, we are able decide to whether any TLF-planar transitive graph is Cayley or not.Comment: Article : 23 pages, 15 figures Appendix : 13 pages, 72 figures Submitted to Discrete Mathematics The appendix is accessible at http://www.labri.fr/~renault/research/research.htm

Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank

We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds) such that each model can be obtained from the dual of the other after freezing k spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with k being the rank of the first homology group. Our focus is on the case where k is extensive, that is, scales linearly with the number of bonds n. Flipping any of these additional spins introduces a homologically nontrivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects has no effect on the free energy density f(T) in the thermodynamical limit and of a high-temperature region where an extensive homological defect does not affect f(T). We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting f(T) at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all { f, d} tilings of the infinite hyperbolic plane, where df/(d + f) \u3e 2. Namely, for these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, Tc(f)T(f)

The planar Cayley graphs are effectively enumerable I: consistently planar graphs

We obtain an effective enumeration of the family of finitely generated groups admitting a faithful, properly discontinuous action on some 2-manifold contained in the sphere. This is achieved by introducing a type of group presentation capturing exactly these groups. Extending this in a companion paper, we find group presentations capturing the planar finitely generated Cayley graphs. Thus we obtain an effective enumeration of these Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al.Comment: To appear in Combinatorica. The second half of the previous version is arXiv:1901.0034

Accessibility, planar graphs, and quasi-isometries

We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to planar graphs. In particular, such groups are virtually free products of free and surface groups.Comment: 44 pages, 8 figures. Comments welcome. This revision: Minor corrections, and new remark (4.19

Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank

We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds), such that each model can be obtained from the dual of the other after freezing $k$ spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with $k$ being the rank of the first homology group. Our focus is on the case where $k$ is extensive, that is, scales linearly with the number of bonds $n$. Flipping any of these additional spins introduces a homologically non-trivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects have no effect on the free energy density $f(T)$ in the thermodynamical limit, and of a high-temperature region where in the ferromagnetic case an extensive homological defect does not affect $f(T)$. We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting $f(T)$ at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all $\{f,d\}$ tilings of the hyperbolic plane, where $df/(d+f)>2$. Namely, for these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, $T_c^{(\mathrm{f})}<T_c^{(\mathrm{w})}$.Comment: 18 pages, 6 figure

Phase Transitions on Nonamenable Graphs

We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees

Splitting groups with cubic Cayley graphs of connectivity two

A group $G$ splits over a subgroup $C$ if $G$ is either a free product with amalgamation $A \underset{C}{\ast} B$ or an HNN-extension $G=A \underset{C}{\ast} (t)$. We invoke Bass-Serre theory and classify all infinite groups which admit cubic Cayley graphs of connectivity two in terms of splittings over a subgroup