Finite Transitive Graphs Research Articles

Supercritical percolation on finite transitive graphs I: Uniqueness of the giant component

Supercritical percolation on finite transitive graphs I: Uniqueness of the giant component

Non-triviality of the phase transition for percolation on finite transitive graphs

We prove that if (G_{n})_{n \geq1}=((V_{n},E_{n}))_{n\geq 1} is a sequence of finite, vertex-transitive graphs with bounded degrees and |V_{n}|\to\infty that is at least (1+\varepsilon) -dimensional for some \varepsilon>0 in the sense that \operatorname{diam} (G_{n})=O(|V_{n}|^{1/(1+\varepsilon)}) \quad \text{as }n\to\infty then this sequence of graphs has a non-trivial phase transition for Bernoulli bond percolation. More precisely, we prove under these conditions that for each 0<\alpha <1 there exists p_{c}(\alpha)<1 such that for each p\geq p_{c}(\alpha) , Bernoulli- p bond percolation on G_{n} has a cluster of size at least \alpha |V_{n}| with probability tending to 1 as n\to \infty . In fact, we prove more generally that there exists a universal constant a such that the same conclusion holds whenever \operatorname{diam} (G_{n})=O\Big(\frac{|V_{n}|}{(\log |V_{n}|)^{a}}\Big) \quad \text{as }n\to\infty. This verifies a conjecture of Benjamini (2001) up to the value of the constant a , which he suggested should be 1 . We also prove a generalization of this result to quasitransitive graph sequences with a bounded number of vertex orbits. A key step in our argument is a direct proof of our result when the graphs G_{n} are all Cayley graphs of Abelian groups, in which case we show that one may indeed take a=1 . This result relies crucially on a new theorem of independent interest stating roughly that balls in arbitrary Abelian Cayley graphs can always be approximated by boxes in \Z^{d} with the standard generating set. Another key step is to adapt the methods of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin [Duke Math. J. 169 (2020)] from infinite graphs to finite graphs. This adaptation also leads to an isoperimetric criterion for infinite graphs to have a non-trivial uniqueness phase (i.e., to have p_{u}<1 ), which is of independent interest. We also prove that the set of possible values of the critical probability of an infinite quasitransitive graph has a gap at 1 in the sense that for every k,n<\infty there exists \varepsilon>0 such that every infinite graph G of degree at most k whose vertex set has at most n orbits under \operatorname{Aut}(G) has either p_c=1 or p_c\leq 1-\varepsilon .

Existence of a Percolation Threshold on Finite Transitive Graphs

Abstract Let $(G_{n})$ be a sequence of finite connected vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters $(p_{n})$ is a percolation threshold if for every $\varepsilon> 0$, the proportion $\left \lVert K_{1}\right \rVert $ of vertices contained in the largest cluster under bond percolation ${\mathbb {P}}_{p}^{G}$ satisfies both $$ \begin{align*} \begin{split}{} \lim_{n \to \infty} {\mathbb{P}}_{(1+\varepsilon)p_{n}}^{G_{n}} \left( \left\lVert K_{1}\right\rVert \geq \alpha \right) &= 1 \qquad \textrm{for some}\ \alpha> 0, \textrm{and}\\ \lim_{n \to \infty} {\mathbb{P}}_{(1-\varepsilon)p_{n}}^{G_{n}} \left( \left\lVert K_{1}\right\rVert \geq \alpha \right) &= 0 \qquad \textrm{for all}\ \alpha > 0. \end{split} \end{align*}$$We prove that $(G_{n})$ has a percolation threshold if and only if $(G_{n})$ does not contain a particular infinite collection of pathological subsequences of dense graphs. Our argument uses an adaptation of Vanneuville’s new proof of the sharpness of the phase transition for infinite graphs via couplings [27] together with our recent work with Hutchcroft on the uniqueness of the giant cluster [15].

On conjugacy of subalgebras in graph C*-algebras. II

AbstractWe apply a method inspired by Popa's intertwining-by-bimodules technique to investigate inner conjugacy of MASAs in graph $C^*$-algebras. First, we give a new proof of non-inner conjugacy of the diagonal MASA ${\mathcal {D}}_E$ to its non-trivial image under a quasi-free automorphism, where $E$ is a finite transitive graph. Changing graphs representing the algebras, this result applies to some non quasi-free automorphisms as well. Then, we exhibit a large class of MASAs in the Cuntz algebra ${\mathcal {O}}_n$ that are not inner conjugate to the diagonal ${\mathcal {D}}_n$.

The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set

We prove the rather counterintuitive result that there exist finite transitive graphs H and integers k such that the Free Uniform Spanning Forest in the direct product of the k-regular tree and H has infinitely many trees almost surely. This shows that the number of trees in the FUSF is not a quasi-isometry invariant. Moreover, we give two different Cayley graphs of the same virtually free group such that the FUSF has infinitely many trees in one, but is connected in the other, answering a question of Lyons and Peres (Probability on Trees and Networks (2016) Cambridge Univ. Press) in the negative. A version of our argument gives an example of a nonunimodular transitive graph where WUSF≠FUSF, but some of the FUSF trees are light with respect to Haar measure. This disproves a conjecture of Tang (Electron. J. Probab. 26 (2021) Paper No. 141).

All finite transitive graphs admit a self-adjoint free semigroupoid algebra

AbstractIn this paper we show that every non-cycle finite transitive directed graph has a Cuntz–Krieger family whose WOT-closed algebra is $B(\mathcal {H})$. This is accomplished through a new construction that reduces this problem to in-degree 2-regular graphs, which is then treated by applying the periodic Road Colouring Theorem of Béal and Perrin. As a consequence we show that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras.

Planar Transitive Graphs

We prove that the first homology group of every planar locally finite transitive graph $G$ is finitely generated as an $\Aut(G)$-module and we prove a similar result for the fundamental group of locally finite planar Cayley graphs. Corollaries of these results include Droms's theorem that planar groups are finitely presented and Dunwoody's theorem that planar locally finite transitive graphs are accessible.

On the critical probability in percolation

For percolation on finite transitive graphs, Nachmias and Peres suggested a characterization of the critical probability based on the logarithmic derivative of the susceptibility. As a first test-case, we study their suggestion for the Erdős–Rényi random graph $G_{n,p}$, and confirm that the logarithmic derivative has the desired properties: (i) its maximizer lies inside the critical window $p=1/n+\Theta (n^{-4/3})$, and (ii) the inverse of its maximum value coincides with the $\Theta (n^{-4/3})$–width of the critical window. We also prove that the maximizer is not located at $p=1/n$ or $p=1/(n-1)$, refuting a speculation of Peres.

Finite s-Geodesic Transitive Graphs Which Are Locally Disconnected

It was previously shown that there is a bijection between the family of locally disconnected 2-geodesic transitive graphs $$\Gamma $$ and a certain family of partial linear spaces $${\mathcal {S}}(\Gamma )$$ . In this paper, we first determine a relationship between the 2-geodesic transitivity of $$\Gamma $$ and the local s-arc transitivity of the incidence graph of $${\mathcal {S}}(\Gamma )$$ . Next, we give a reduction theorem for the family of locally disconnected s-geodesic transitive graphs.

Accessibility in Transitive Graphs

We prove that the cut space of any transitive graph G is a finitely generated Aut(G)-module if the same is true for its cycle space. This confirms a conjecture of Diestel which says that every locally finite transitive graph whose cycle space is generated by cycles of bounded length is accessible. In addition, it implies Dunwoody’s conjecture that locally finite hyperbolic transitive graphs are accessible. As a further application, we obtain a combinatorial proof of Dunwoody’s accessibility theorem of finitely presented groups.

Two-geodesic transitive graphs of valency six

For a positive integer s less than or equal to the diameter of a graph Γ, an s-geodesic of Γ is a path (v0,v1,…,vs) such that the distance between v0 and vs is s. The graph Γ is said to be s-geodesic transitive, if Γ contains an s-geodesic and its automorphism group is transitive on the set of t-geodesics for all t≤s. In particular, if Γ is s-geodesic transitive with s equal to the diameter of Γ, then Γ is called geodesic transitive. In this paper, we classify the family of finite 2-geodesic transitive graphs of valency 6. Then we completely determine such graphs which are geodesic transitive.

A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model

We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime $\beta<\beta_c$, and the mean-field lower bound $\mathbb{P}_\beta[0\longleftrightarrow\infty]\ge (\beta-\beta_c)/\beta$ for $\beta>\beta_c$. For finite-range models, we also prove that for any $\beta<\beta_c$, the probability of an open path from the origin to distance $n$ decays exponentially fast in $n$. For the Ising model, we prove finiteness of the susceptibility for $\beta<\beta_c$, and the mean-field lower bound $\langle \sigma_0\rangle_\beta^+\ge \sqrt{(\beta^2-\beta_c^2)/\beta^2}$ for $\beta>\beta_c$. For finite-range models, we also prove that the two-point correlations functions decay exponentially fast in the distance for $\beta<\beta_c$.

Finite 2‐Geodesic Transitive Graphs of Prime Valency

AbstractWe classify noncomplete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of 2‐geodesics. We prove that either Γ is 2‐arc transitive or the valency p satisfies , and for each such prime there is a unique graph with this property: it is a nonbipartite antipodal double cover of the complete graph with automorphism group and diameter 3.

Mean-Field Conditions for Percolation on Finite Graphs

Let {G n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G n | = n. Denote by p t(v, v) the return probability after t steps of the non-backtracking random walk on G n . We show that if p t(v, v) has quasi-random properties, then critical bond-percolation on G n behaves as it would on a random graph. More precisely, if $$\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$$then the size of the largest component in p-bond-percolation with \({p =\frac{1+O(n^{-1/3})}{d-1}}\) is roughly n 2/3. In Physics jargon, this condition implies that there exists a scaling window with a mean-field width of n −1/3 around the critical probability \({p_c =\frac{1}{d-1}}\).A consequence of our theorems is that if {G n } is a transitive expander family with girth at least \({\left( {\frac{2}{3} + \epsilon} \right) {\rm log}_{d - 1}n}\) then {G n } has the above scaling window around \({p_c =\frac{1}{d-1}}\). In particular, bond-percolation on the celebrated Ramanujan graph constructed by Lubotzky, Phillips and Sarnak [LuPS] has the above scaling window. This provides the first examples of quasi-random graphs behaving like random graphs with respect to critical bond-percolation.

The spectrum of the random environment and localization of noise

We consider random walk on a mildly random environment on finite transitive d-regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized. The graphs of the noise covariance structure for d = 4, 3, 2.1 from above.

K-CS-transitive infinite graphs

A graph Γ is k-CS-transitive, for a positive integer k, if for any two connected isomorphic induced subgraphs A and B of Γ, each of size k, there is an automorphism of Γ taking A to B. The graph is called k-CS-homogeneous if any isomorphism between two connected induced subgraphs of size k extends to an automorphism. We consider locally-finite infinite k-CS-homogeneous and k-CS-transitive graphs. We classify those that are 3-CS-transitive (respectively homogeneous) and have more than one end. In particular, the 3-CS-homogeneous graphs with more than one end are precisely Macpherson's locally finite distance transitive graphs. The 3-CS-transitive but non-homogeneous graphs come in two classes. The first are line graphs of semiregular trees with valencies 2 and m, while the second is a class of graphs built up from copies of the complete graph K 4 , which we describe.

Percolation on finite graphs and isoperimetric inequalities

Consider a uniform expanders family Gn with a uniform bound on the degrees. It is shown that for any p and c>0, a random subgraph of Gn obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least c|Gn|, with probability going to one, uniformly in p. The method from Ajtai, Komlós and Szemerédi [Combinatorica 2 (1982) 1–7] is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth, as well as a simple proof of a result of Kesten about the critical probability for bond percolation in high dimensions. Several problems and conjectures regarding percolation on finite transitive graphs are presented.

Real functions incrementally computable by finite automata

This is an investigation into exact real number computation using the incremental approach of Potts (Ph.D. Thesis, Department of Computing, Imperial College, 1998), Edalat and Potts (Electronic Notes in Computer Science, Vol. 6, Elsevier Science Publishers, Amsterdam, 2000), Nielsen and Kornerup (J. Universal Comput. Sci. 1(7) (1995) 527), and Vuillemin (IEEE Trans. on Comput. 39(8) (1990) 1087) where numbers are represented as infinite streams of digits, each of which is a Möbius transformation. The objective is to determine for each particular system of digits which functions R→ R can be computed by a finite transducer and ultimately to search for the most finitely expressible Möbius representations of real numbers. The main result is that locally such functions are either not continuously differentiable or equal to some Möbius transformation. This is proved using elementary properties of finite transition graphs and Möbius transformations. Applying the results to the standard signed-digit representations, we can classify functions that are finitely computable in such a representation and are continuously differentiable everywhere except for finitely many points. They are exactly those functions whose graph is a fractured line connecting finitely many points with rational coordinates.

Characterizing bisimilarity of value-passing parametrised processes

Abstract Symbolic transition graphs of Hennessy and Linextended with assignments are used to represent parametrisedvalue-passing processes. Semantics appealing to thenotion of state is introduced for the symbolic transition graphsand their extended version. Showing strong early bisimulation between finite symbolic transition graphs extended with assignments isreduced to solving sets of equations involving first-order state formulas.

Finite two-are transitive graphs admitting a ree simple group

Finite two-are transitive graphs admitting a ree simple group