Positive speed self-avoiding walks on graphs with more than one end
Introduction
Self-avoiding walks, which are paths on graphs visiting no vertex more than once, were first introduced as a model for long-chain polymers in chemistry ([9], see also [25]). The theory of SAWs impinges on several areas of science including combinatorics, probability, and statistical mechanics. Each of these areas poses its characteristic questions concerning counting and geometry. Despite the simple definition, SAWs have been notoriously difficult to study due to the fact that SAWs are, in general, non-Markovian.
The most natural SAW models are defined on regular graphs, such as the square-grid, the hexagonal lattice, etc; SAWs on these graphs have been studied extensively. In this paper, we consider SAWs on the general quasi-transitive graphs. Let be an infinite, connected graph, and let be the automorphism group for G. We say that G is quasi-transitive, if there exists a subgroup Γ of acting quasi-transitively on G, i.e. the action of Γ on V has only finitely many orbits. More precisely, there exist a finite set of vertices , , such that for any , there exist and with . The set W is called a fundamental domain. A graph is called locally finite, if every vertex has finite degree, i.e., incident to finitely many edges. A subset of vertices is called connected, if for any , there exists such that for , and are adjacent vertices (two vertices joined by an edge).
The connective constant is a fundamental quantity concerning counting the number of SAWs starting from a fixed vertex, and this is the starting point of a rich theory of geometry and phase transition. It is defined, on a quasi-transitive graph, to be the exponential growth rate of the number of n-step SAWs starting from a fixed vertex. More precisely, let be the number of n-step SAWs starting from a fixed vertex , the connective constant μ is defined to be The limit on the right hand side of (1.1) is known to exist by a sub-additivity argument. It is proved in [21] that the connective constant μ defined in (1.1), can be expressed as follows
Although the definition of the SAW is quite simple, a lot of fundamental questions concerning SAWs remain unknown. For example, it is still an open problem to compute the exact value of the connective constant for the 2-dimensional square grid. A recent breakthrough is a proof of the fact that the connective constant of the hexagonal lattice is ; see [7]. See [15], [14], [17], [13], [19] for results concerning bounds of connective constants on quasi-transitive graphs; [18], [16] for results concerning the dependence of connective constants on local structures of graphs; [20] for the continuous dependence of connective constants of weighted SAWs on edge weights of the graph; and [12] for the changes of the connective constant of SAWs under local transformations of the underlying graph.
Another important quantity relating to SAWs is the mean square displacement exponent ν. Let be an n-step SAW on G starting from a fixed vertex v, and let where is the graph distance on G. Let be the expectation taken with respect to the uniform probability measure for n-step SAWs on G starting from a fixed vertex. The mean square displacement exponent ν for SAWs, defined by has been an interesting topic to mathematicians and scientists for long. Here “∼” means that there exist constants , independent of n, such that .
Although the connective constant depends on the local structure of the graph, the mean square displacement exponent ν is believed to be universal in the sense that it depends only on the dimension of the space where the graph is embedded, but independent of the graph. It is conjectured that for SAWs on graphs embedded in the 2-dimensional Euclidean plane (in particular, this means that the square grid, the hexagonal lattice and the triangular lattice share the same exponent , although they obviously have distinct connective constants), for SAWs on with , and that for SAWs on a non-amenable graph with bounded vertex degree. See [12] for the invariance of SAW components under local transformations of cubic (valent-3) graphs.
The conjecture that when was proved in [5], [22]. See [2] for related results when , and [6] for related results for .
It is proved in [28] that if a non-amenable Cayley graph satisfies then SAWs have positive speed. Here Δ is the vertex degree, ρ is the spectral radius for the transition matrix of the simple random walk on the graph, and μ is the connective constant as defined in (1.1). Combining with the results in [27], [4], [29], it is known that for any finitely generated non-amenable group, there exists a locally finite Cayley graph on which SAWs have positive speed.
It is proved in [26] that SAWs have positive speed for certain regular tilings of the hyperbolic plane. An upper bound of the spectral radius for a planar graph with given maximal degree is proved in [8], which, combining with (1.3), can be used to show that SAWs have positive speed on a large class of planar graphs. It is shown in [3] that SAWs on the 7-regular infinite planar triangulation has linear expected displacement.
The main goal of this paper is to study the mean square displacement exponent ν for SAWs on quasi-transitive graphs with more than one end. The number of ends of a connected graph is the supremum over its finite subgraphs of the number of infinite components that remain after removing the subgraph.
Let be an infinite, connected, locally finite, quasi-transitive graph with more than one end. Let be a subgroup of the automorphism group of G acting quasi-transitively on G. Since G has more than one end, there exists a finite subset of V (which is called a “cut set”), such that after removing all the vertices as well as incident edges of the set, the remaining graph has at least two infinite components. If distinct components of the remaining graph have certain “symmetry” under the action of Γ, one may map certain portions of an SAW from one component to another component of the remaining graph and form a new SAW, such that the end-to-end distance of the new SAW is linear in its length. Then the number of n-step SAWs with end-to-end distance linear in n, when n is large, may be compared with the total number of n-step SAWs. To that end, we may make the following assumptions on the graph G concerning the “symmetry” of different components after removing the finite “cut set”.
Assumption 1.1 There exist a finite set of vertices S, and , such that S is connected; (the graph obtained from G by removing all the vertices in S and their incident edges) has at least two infinite components; for any component A of , let be the set consisting of all the vertices in S incident to a vertex in A. There exists an infinite component B of and a graph automorphism , such that , ; for any , , v and γv are joined by a path in , whose length is bounded above by a constant N independent of . Denote γ by .
See Fig. 1.1.
Assumption 1.2 There exist a finite set of vertices S, and satisfying Assumption 1.1. Moreover, assume that there exists a finite set of vertices , such that . Let be the set consisting of all the vertices in incident to a vertex in . For any two distinct vertices , there exists an SAW joining u and v and visiting every vertex in S.
See Fig. 1.2.
Here are the main results of the paper.
Theorem 1.3 Let be an infinite, connected, locally finite, quasi-transitive graph with more than one end. Let μ be the connective constant of G. Let be an n-step SAW on G starting from a fixed vertex v. If G satisfies Assumption 1.1, then there exists If G satisfies Assumption 1.2, then has positive speed, i.e., there exist constants , such that where is the uniform measure on the set of n-step SAWs on G starting from a fixed vertex.
For a graph satisfying Assumption 1.2, Theorem 1.3 implies that the mean square displacement of SAWs on the graph is of the order , i.e.
The approach to prove Theorem 1.3 is to consider a finite “cut set” S as given by Assumption 1.1, such that SAWs, once crossing this “cut set”, will move to another component of and most of them may never come back again. The analysis involves arguments and technical details inspired by the pattern theorem ([23]), see also ([25], [6], [14], [32], [1]). The proofs of Part A. and Part B. are similar; note that under the stronger Assumption 1.2, not only the number of n-step SAWs whose end-to-end distance is linear in n has the same exponential growth rate as the total number of n-step SAWs starting from a fixed vertex, but the number of n-step SAWs whose end-to-end distance is not linear in n is actually exponential small compared to the total number of n-step SAWs starting from a fixed vertex.
Applications of Theorem 1.3 include a proof that SAWs on an infinite cylindrical square grid have positive speed, and that SAWs on an infinite free product graph of two quasi-transitive, connected graphs have positive speed.
Example 1.4 (Cylinder) Consider the quotient graph of the square grid , , where l is a positive integer. This is a graph with two ends. We can choose and . Then Assumption 1.2 is satisfied and SAWs have positive speed. See also [10] for discussions about SAWs on a cylinder.
Definition 1.5 (Free product of graphs) Let , be two connected, locally finite, quasi-transitive, rooted graphs with vertex sets , ; edge sets and roots , respectively. For , assume that ; and ; and if .
Define We define an edge set E for the vertex set V as follows: if and , and , then for all . See [11] for discussions of SAWs on free product graphs of quasi-transitive graphs.
Theorem 1.6 Let be the free product graph of two connected, locally finite, quasi-transitive, rooted graphs and with , for , as defined in 1.5. Then SAWs on G have positive speed.
Cayley graphs in the group theory, whose vertices correspond to elements in a group and edges correspond to a set of generators, form a large class of vertex-transitive graphs. The ends of a finitely generated group are defined to be the ends of the corresponding Cayley graph; this definition is insensitive to the choice of the finite generating set. It is well known that every finite-generated infinite group has either 1, 2, or infinitely many ends. Stalling's splitting theorem tells us that in the latter two cases the group decomposes either as a non-trivial amalgamated free product over a finite subgroup, or as an HNN extension over a finite base (edge group), which makes its Cayley graph in some sense a “treelike graph”, in either case the “cut set” can be constructed naturally from the finite base, and symmetries of the group action make it possible to construct a lot of SAWs whose end-to-end distances are linear in lengths through local manipulations across the “cut set”.
Concerning groups with more than one end, Theorem 1.3 has the following corollaries.
Theorem 1.7 Let Γ be an infinite, finitely-generated group with more than two ends. Let be a locally finite Cayley graph of Γ. For Let be an n-step SAW on G starting from v. Then there exists , such that
Theorem 1.8 Let Γ be an infinite, finitely-generated group with more than one end. There exists a locally finite Cayley graph of Γ, such that SAWs on G have positive speed.
The proofs of Theorem 1.7, Theorem 1.8 make use of the Stalling's splitting theorem (see [30]), which gives explicit presentations for groups with more than one end; as well as constructions of sets S and satisfying Assumption 1.1, Assumption 1.2.
The organization of the paper is as follows. In Section 2, we prove Theorem 1.3 A. In Section 3, we prove Theorem 1.3 B. Theorem 1.7, Theorem 1.8 are proved in Section 4. In Section 5, we prove Theorem 1.6.
Section snippets
Proof of Theorem 1.3 A
This section is devoted to prove Theorem 1.3 A.
Let be a graph satisfying the assumption of Theorem 1.3. Let S be a finite set of vertices satisfying Assumption 1.1. Recall that Γ is a subset of acting quasi-transitively on G. Let ΓS be the set of images of S under Γ. By quasi-transitivity of G, for each , γS still satisfies Assumption 1.1. We also call γS for a copy of S.
We shall next introduce events , and and their restrictions to a length-2m sub-walks ,
Proof of Theorem 1.3 B
We prove Theorem 1.3 B. in this section. Let be a graph satisfying Assumption 1.2. Since any graph satisfying Assumption 1.2 must satisfy Assumption 1.1 as well, all the results proved in Section 2 also apply to graphs satisfying Assumption 1.2.
Let π be an n-step SAW on G. Recall that occurs at the jth step of π if there exists such that , and all the vertices of γS are visited by π. For , we say that occurs at the jth step of π, if there exists , such that
Groups with more than one end
In this section, we prove Theorem 1.7. The proof is based on the stalling's splitting theorem, and an explicit construction of the set S satisfying Assumption 1.1.
Lemma 4.1 Let be an infinite, connected, locally finite graph. Let be two finite set of vertices of G satisfying . Let (resp. ) be the subgraph of G obtained from G by removing all the vertices in A (resp. B) as well as their incident edges. If has at least two infinite components, then has at least two infinite
Free product graph of two quasi-transitive graphs
In this section, we prove Theorem 1.6.
Proof Obviously G is an infinite, connected, quasi-transitive graph. Let . Then has at least two infinite components. Indeed, let satisfy where , , , (see Definition 1.5 for notations). If and , then x and y are in two distinct components of . Let A (resp. B) be a component of , such that for any (resp. ), x (resp. y) has the form (5.1) (resp. (5.2)) with
Acknowledgements
The author thanks Yuval Peres, Geoffrey Grimmett for helpful discussions. The author's research is partially supported by National Science Foundation grant #1608896. The author thanks anonymous reviewers for valuable comments on improving the readability of the paper.
References (32)
- et al.
Spectral radius of finite and infinite planar graphs and of graphs of bounded genus
J. Comb. Theory, Ser. B
(2010) - et al.
Counting self-avoiding walks on free products of graphs
Discrete Math.
(2017) - et al.
Cubic graphs and the golden mean
Discrete Math.
(January 2020) - et al.
Locality of connective constants
Discrete Math.
(2018) Isoperimetric inequalities, growth, and spectrum of graphs
Linear Algebra Appl.
(1988)- et al.
On non-uniqueness of percolation on non-amenable Cayley graphs
C. R. Acad. Sci., Sér. 1 Math.
(2000) - et al.
Stretched polygons in a lattice cube
J. Phys. A
(2009) - et al.
Lectures on self-avoiding walks
Self-avoiding walk on the 7-regular triangulation
- et al.
Percolation beyond : many questions and a few answers
Electron. Commun. Probab.
(1996)
Self-avoiding walk in 5 or more dimensions
Commun. Math. Phys.
Self-avoiding walk is sub-ballistic
Commun. Math. Phys.
The connective constant of the honeycomb lattice equals
Ann. Math.
Principles of Polymer Chemistry
Two-dimensional self-avoiding walks on a cylinder
Phys. Rev. E
Self-avoiding walks and the Fisher transformation
Electron. J. Comb.
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