Transitive Graphs Research Articles

Locality of percolation for graphs with polynomial growth

Schramm’s Locality Conjecture asserts that the value of the critical parameter pc of a graph satisfying pc<1 depends only on its local structure. In this paper, we prove this conjecture in the particular case of transitive graphs with polynomial growth. Our proof relies on two recent works about such graphs, namely supercritical sharpness of percolation by the same authors and a finitary structure theorem by Tessera and Tointon.

A new infinite family of star normal quotient graphs of twisted wreath type

We construct the first infinite families of locally 2-arc transitive graphs with the property that the automorphism group has two orbits on vertices and is quasiprimitive on exactly one orbit, of twisted wreath type. This work contributes to Giudici, Li and Praeger’s program for the classification of locally 2-arc transitive graphs by showing that the star normal quotient twisted wreath category also contains infinitely many graphs.

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).

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 ${\mathcal {D}}_n$.

Mean field behavior during the Big Bang regime for coalescing random walks

In this paper we consider coalescing random walks on a general connected graph $G=(V,E)$. We set up a unified framework to study the leading order of the decay rate of $P_t$, the expectation of the fraction of occupied sites at time $t$, particularly for the `Big Bang' regime where $t\ll t_{\text{coal}}:=\mathbb{E}[\inf\{s:\text{There is only one particle at time }s\}]$. Our results show that $P_t$ satisfies certain mean field behavior, if the graphs satisfy certain transience-like conditions. We apply this framework to two families of graphs: (1) graphs given by the configuration model with a degree distribution supported in $[3,\bar d]$ for some $\bar d\geq 3$, and (2) finite and infinite vertex-transitive graphs. In the first case, we show that for $1 \ll t \ll |V|$, $P_t$ decays in the order of $t^{-1}$, and $(tP_t)^{-1}$ is approximately the probability that two particles starting from the root of the corresponding unimodular Galton-Watson tree never collide after one of them leaves the root, which is also roughly $|V|/(2t_{\text{meet}})$, where $t_{\text{meet}}$ is the mean meeting time of two walkers. By taking the local weak limit, for the unimodular Galton-Watson tree we prove the convergence of $tP_t$ as $t\to\infty$. For the second family of graphs, if we take a sequence of finite graphs $G_n=(V_n, E_n)$, such that $t_{\text{meet}}=O(|V_n|)$ and the inverse of the spectral gap $t_{\text{rel}}$ is $o(|V_n|)$, then for $t_{\text{rel}}\ll t\ll t_{\text{coal}}$, $(tP_t)^{-1}$ is approximately the probability that two random walks never meet before time $t$, and also $|V|/(2t_{\text{meet}})$. In addition, we define a natural uniform transience condition, and show that it implies the above for all $1\ll t\ll t_{\text{coal}}$. Such estimates of $tP_t$ are also obtained for all infinite transient transitive unimodular graphs, in particular, all transient transitive amenable graphs.

Slightly supercritical percolation on non‐amenable graphs I: The distribution of finite clusters

We study the distribution of finite clusters in slightly supercritical ( p ↓ p c $p \downarrow p_c$ ) Bernoulli bond percolation on transitive non-amenable graphs, proving in particular that if G $G$ is a transitive non-amenable graph satisfying the L 2 $L^2$ boundedness condition ( p c < p 2 → 2 $p_c<p_{2\rightarrow 2}$ ) and K $K$ denotes the cluster of the origin and then there exists δ > 0 $\delta >0$ such that if p ∈ ( p c − δ , p c + δ ) $p\in (p_c-\delta ,p_c+\delta )$ , then P p ( n ⩽ | K | < ∞ ) ≍ n − 1 / 2 exp − Θ | p − p c | 2 n \begin{align*} \hskip1.5pc\mathbf {P}_p(n \leqslant |K| < \infty ) &\asymp n^{-1/2} \,{\exp} {\left[ -\Theta {\left(|p-p_c|^2 n\right)} \right]}\hskip-1.5pc \end{align*} and P p ( r ⩽ Rad ( K ) < ∞ ) ≍ r − 1 exp [ − Θ ( | p − p c | r ) ] \begin{align*} \hskip1.5pc\mathbf {P}_p(r \leqslant \operatorname{Rad}(K) < \infty ) &\asymp r^{-1} \exp {[ -\Theta (|p-p_c| r) ]}\hskip-1.5pc \end{align*} for every n , r ⩾ 1 $n,r\geqslant 1$ , where all implicit constants depend only on G $G$ . We deduce in particular that the critical exponents γ ′ $\gamma ^{\prime }$ and Δ ′ $\Delta ^{\prime }$ describing the rate of growth of the moments of a finite cluster as p ↓ p c $p \downarrow p_c$ take their mean-field values of 1 and 2, respectively. These results apply in particular to Cayley graphs of non-elementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius ρ < 1 / 2 $\rho <1/2$ . In particular, every finitely generated non-amenable group has a Cayley graph to which these results apply. They are new for graphs that are not trees. The corresponding facts are yet to be understood on Z d $\mathbb {Z}^d$ even for d $d$ very large. In a second paper in this series, we will apply these results to study the geometric and spectral properties of infinite slightly supercritical clusters in the same setting.

On the Derivation of Mean-Field Percolation Critical Exponents from the Triangle Condition

We give a new derivation of mean-field percolation critical behaviour from the triangle condition that is quantitatively much better than previous proofs when the triangle diagram \(\nabla _{p_c}\) is large. In contrast to earlier methods, our approach continues to yield bounds of reasonable order when the triangle diagram \(\nabla _p\) is unbounded but diverges slowly as \(p \uparrow p_c\), as is expected to occur in percolation on \({\mathbb {Z}}^d\) at the upper-critical dimension \(d=6\). Indeed, we show in particular that if the triangle diagram diverges polylogarithmically as \(p \uparrow p_c\) then mean-field critical behaviour holds to within a polylogarithmic factor. We apply the methods we develop to deduce that for long-range percolation on the hierarchical lattice, mean-field critical behaviour holds to within polylogarithmic factors at the upper-critical dimension. As part of the proof, we introduce a new method for comparing diagrammatic sums on general transitive graphs that may be of independent interest.

Two-distance transitive normal Cayley graphs

In this paper, we construct an infinite family of normal Cayley graphs, which are $2$-distance-transitive but neither distance-transitive nor $2$-arc-transitive. This answers a question raised by Chen, Jin and Li in 2019 and corrects a claim in a literature given by Pan, Huang and Liu in 2015.

What are the limits of universality?

It is a central prediction of renormalization group theory that the critical behaviours of many statistical mechanics models on Euclidean lattices depend only on the dimension and not on the specific choice of lattice. We investigate the extent to which this universality continues to hold beyond the Euclidean setting, taking as case studies Bernoulli bond percolation and lattice trees. We present strong numerical evidence that the critical exponents governing these models on transitive graphs of polynomial volume growth depend only on the volume-growth dimension of the graph and not on any other large-scale features of the geometry. For example, our results strongly suggest that percolation, which has upper-critical dimension 6, has the same critical exponents onZ4and the Heisenberg group despite the distinct large-scale geometries of these two lattices preventing the relevant percolation models from sharing a common scaling limit. On the other hand, we also show that no such universality should be expected to hold on fractals, even if one allows the exponents to depend on a large number of standard fractal dimensions. Indeed, we give natural examples of two fractals which share Hausdorff, spectral, topological and topological Hausdorff dimensions but exhibit distinct numerical values of the percolation Fisher exponentτ. This gives strong evidence against a conjecture of Balankinet al.(2018Phys. Lett. A382, 12–19 (doi:10.1016/j.physleta.2017.10.035)).

Homology-changing percolation transitions on finite graphs

We consider homological edge percolation on a sequence (Gt)t of finite graphs covered by an infinite (quasi)transitive graph H and weakly convergent to H. In particular, we use the covering maps to classify 1-cycles on graphs Gt as homologically trivial or non-trivial and define several thresholds associated with the rank of thus defined first homology group on the open subgraphs generated by the Bernoulli (edge) percolation process. We identify the growth of the homological distance dt, the smallest size of a non-trivial cycle on Gt, as the main factor determining the location of homology-changing thresholds. In particular, we show that the giant cycle erasure threshold pE0 (related to the conventional erasure threshold for the corresponding sequence of generalized toric codes) coincides with the edge percolation threshold pc(H) if the ratio dt/ln nt diverges, where nt is the number of edges of Gt, and we give evidence that pE0&amp;lt;pc(H) in several cases where this ratio remains bounded, which is necessarily the case if H is non-amenable.

Return probabilities on nonunimodular transitive graphs

Consider simple random walk (Xn)n≥0 on a transitive graph with spectral radius ρ. Let un=P[Xn=X0] be the n-step return probability and fn be the first return probability at time n. It is a folklore conjecture that on transient, transitive graphs un∕ρn is at most of the order n−3∕2. We prove this conjecture for graphs with a closed, transitive, amenable and nonunimodular subgroup of automorphisms. We also conjecture that for any transient, transitive graph fn and un are of the same order and the ratio fn∕un even tends to an explicit constant. We give some examples for which this conjecture holds. For a graph G with a closed, transitive, nonunimodular subgroup of automorphisms, we prove a weaker asymptotic behavior regarding to this conjecture, i.e., there is a positive constant c such that fn≥uncnc.

A Feasible Algorithm for Solubility of Transitive Graphs

Transitive, weighted graphs, with the weights coming from an Abelian group and satisfying a transitivity property, were studied in [1] and were referred to as the transitive systems. Of central importance is the question of completing a given system to a transitive system where every ordered pair of the vertices can be assigned a weight. It was proven in [1] that, in the finite case, the question can be reduced to the situation when the Abelian group is a finite cyclic group. Nevertheless, in practical applications, the most common assignments of the weights are coming from the groups of numbers. For this reason, we consider the case when the underlying Abelian group G is torsion-free. We show that within the class of all torsion-free Abelian groups, the underlying transitive and unweighted graph predetermines the possibility of completion of a given transitive system. Such a graph is G-soluble. The question of G-solubility of a graph is equivalent to its ℝ-solubility. This, in turn, is equivalent to solvability of a certain system of linear equations in ℝ. As a result, G-solubility of a transitive graph can be determined by means of a feasible algorithm.

Connectedness of the Free Uniform Spanning Forest as a function of edge weights

Let G be the Cartesian product of a regular tree T and a finite connected transitive graph H. It is shown in [4] that the Free Uniform Spanning Forest (FSF) of this graph may not be connected, but the dependence of this connectedness on H remains somewhat mysterious. We study the case when a positive weight w is put on the edges of the H-copies in G, and conjecture that the connectedness of the FSF exhibits a phase transition. For large enough w we show that the FSF is connected, while for a wide family of H and T, the FSF is disconnected when w is small (relying on [4]). Finally, we prove that when H is the graph of one edge, then for any w, the FSF is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.

Vapnik–Chervonenkis dimension and density on Johnson and Hamming graphs

VC-dimension and VC-density are measures of combinatorial complexity of set systems. VC-dimension was first introduced in the context of statistical learning theory, and is tightly related to the sample complexity in PAC learning. VC-density is a refinement of VC-dimension. Both notions are also studied in model theory, in the context of dependent theories. A set system that is definable by a formula of first-order logic with parameters has finite VC-dimension if and only if the formula is a dependent formula.In this paper we study the VC-dimension and the VC-density of the edge relation Exy on Johnson graphs and on Hamming graphs. On a graph G, the set system defined by the formula Exy is the vertex set of G along with the collection of all open neighborhoods of G. We show that the edge relation has VC-dimension at most 4 on Johnson graphs and at most 3 on Hamming graphs and these bounds are optimal. We furthermore show that the VC-density of the edge relation on the class of all Johnson graphs is 2, and on the class of all Hamming graphs the VC-density is 2 as well. Moreover, we show that our bounds on the VC-dimension carry over to the class of all induced subgraphs of Johnson graphs, and to the class of all induced subgraphs of Hamming graphs, respectively. It also follows that the VC-dimension of the set systems of closed neighborhoods in Johnson graphs and Hamming graphs is bounded.Johnson graphs and Hamming graphs are well known examples of distance transitive graphs. Neither of these graph classes is nowhere dense nor is there a bound on their (local) clique-width. Our results contrast this by giving evidence of structural tameness of the graph classes.

A 2-arc Transitive Hexavalent Nonnormal Cayley Graph on A119

A Cayley graph Γ=Cay(G,S) is said to be normal if the base group G is normal in AutΓ. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 and it plays a vital role in determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group A119. Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to A120.

Finite-energy infinite clusters without anchored expansion

Hermon and Hutchcroft have recently proved the long-standing conjecture that in Bernoulli(p) bond percolation on any nonamenable transitive graph G, at any p>pc(G), the probability that the cluster of the origin is finite but has a large volume n decays exponentially in n. A corollary is that all infinite clusters have anchored expansion almost surely. They have asked if these results could hold more generally, for any finite energy ergodic invariant percolation. We give a counterexample, an invariant percolation on the 4-regular tree.

Two-ended quasi-transitive graphs

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation [Formula: see text] or an HNN-extension [Formula: see text], where [Formula: see text] is a finite group and [Formula: see text] and [Formula: see text]. In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs.

A Case Study on Stochastic Games on Large Graphs in Mean Field and Sparse Regimes

We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph. We find a semiexplicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph’s normalized Laplacian matrix. This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail. In particular, the mean field game is the correct limit only in the dense graph case, that is, when the degrees diverge in a suitable sense. Although equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence.

A Combinatorial Approach to the Computation of the Fractional Edge Dimension of Graphs

E. Yi recently introduced the fractional edge dimension of graphs. It has many applications in different areas of computer science such as in sensor networking, intelligent systems, optimization, and robot navigation. In this paper, the fractional edge dimension of vertex and edge transitive graphs is calculated. The class of hypercube graph Qn with an odd number of vertices n is discussed. We propose the combinatorial criterion for the calculation of the fractional edge dimension of a graph, and hence applied it to calculate the fractional edge dimension of the friendship graph Fk and the class of circulant graph Cn(1,2).

On Finite Subnormal Cayley Graphs

In this paper we introduce and study a type of Cayley graph – subnormal Cayley graph. We prove that a subnormal 2-arc transitive Cayley graph is a normal Cayley graph or a normal cover of a complete bipartite graph $\mathbf{K}_{p^d,p^d}$ with $p$ prime. Then we obtain a generic method for constructing half-symmetric (namely edge transitive but not arc transitive) Cayley graphs.