Positive speed self-avoiding walks on graphs with more than one end

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Abstract

A self-avoiding walk (SAW) is a path on a graph that visits each vertex at most once. The mean square displacement of an n-step SAW is the expected value of the square of the distance between its ending point and starting point, where the expectation is taken with respect to the uniform measure on n-step SAWs starting from a fixed vertex. It is conjectured that the mean square displacement of an n-step SAW is asymptotically n2ν, where ν is a constant. Computing the exact values of the exponent ν on various graphs has been a challenging problem in mathematical and scientific research for long.

In this paper we show that on any locally finite Cayley graph of an infinite, finitely-generated group with more than two ends, the number of SAWs whose end-to-end distances are linear in lengths has the same exponential growth rate as the number of all the SAWs. We also prove that for any infinite, finitely-generated group with more than one end, there exists a locally finite Cayley graph on which SAWs have positive speed - this implies that the mean square displacement exponent ν=1 on such graphs.

These results are obtained by proving more general theorems for SAWs on quasi-transitive graphs with more than one end, which make use of a variation of Kesten's pattern theorem in a surprising way, as well as the Stalling's splitting theorem. Applications include proving that SAWs have positive speed on the square grid in an infinite cylinder, and on the infinite free product graph of two connected, quasi-transitive graphs.

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 G=(V,E) be an infinite, connected graph, and let Aut(G) be the automorphism group for G. We say that G is quasi-transitive, if there exists a subgroup Γ of Aut(G) 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 WV, |W|<, such that for any vV, there exist wW and γΓ with w=γv. 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 UV is called connected, if for any p,qU, there exists u0(=p),u1,,un1,un(=q)U such that for 1in, ui1 and ui 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 cn(v) be the number of n-step SAWs starting from a fixed vertex vV, the connective constant μ is defined to beμ:=limn[supvVcn(v)]1n 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 followsμ=limncn(v)1n,vV.

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 2+2; 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 πnv be an n-step SAW on G starting from a fixed vertex v, and letπnv=distG(π(n),π(0)), where distG(,) 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 byπnv2n2ν, has been an interesting topic to mathematicians and scientists for long. Here “∼” means that there exist constants C1,C2>0, independent of n, such that C1n2νπnv2C2n2ν.

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 ν=34 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 ν=34, although they obviously have distinct connective constants), ν=12 for SAWs on Zd with d4, and that ν=1 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 ν=12 when d5 was proved in [5], [22]. See [2] for related results when d=4, and [6] for related results for d2.

It is proved in [28] that if a non-amenable Cayley graph satisfies(Δ1)ρμ1<1, 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 G=(V,E) be an infinite, connected, locally finite, quasi-transitive graph with more than one end. Let ΓAut(G) 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, SV and |S|<, such that

  • (1)

    S is connected;

  • (2)

    GS (the graph obtained from G by removing all the vertices in S and their incident edges) has at least two infinite components;

  • (3)

    for any component A of GS, let AS be the set consisting of all the vertices in S incident to a vertex in A. There exists an infinite component B of GS and a graph automorphism γΓ, such that BA=, γAB; for any vAS, γvBSB, v and γv are joined by a path in G(AγA), whose length is bounded above by a constant N independent of A,v. Denote γ by ϕ(S,A):=γ.

See Fig. 1.1.

Assumption 1.2

There exist a finite set of vertices S, SV and |S|< satisfying Assumption 1.1. Moreover, assume that

  • there exists a finite set of vertices S, such that SS. Let S be the set consisting of all the vertices in S incident to a vertex in GS. For any two distinct vertices u,vS, there exists an SAW luv 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 G=(V,E) be an infinite, connected, locally finite, quasi-transitive graph with more than one end. Let μ be the connective constant of G. Let πnv be an n-step SAW on G starting from a fixed vertex v.

  • A.

    If G satisfies Assumption 1.1, then there exists a(0,1]limsupnsupvV|{πnv:πnvan}|1n=μ.

  • B.

    If G satisfies Assumption 1.2, then πnv has positive speed, i.e., there exist constants C,α,β>0, such thatPn(πnvαn)Cenβ, where Pn 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 n2, i.e.πnv2n2.

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 GS 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 Z2, Z×Zl, where l is a positive integer. This is a graph with two ends. We can choose S={0}×Zl and S={1,0,1}×Z. 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 G1=(V1,E1,o1), G2=(V2,E2,o2) be two connected, locally finite, quasi-transitive, rooted graphs with vertex sets V1, V2; edge sets E1,E2 and roots o1V1,o2V2, respectively. For i{1,2}, assume that

  • (1)

    |Vi|2; and

  • (2)

    Vi×=Vi{oi}; and

  • (3)

    I(x)=i if xVi×.

DefineV:=V1V2={x1x2xn|nN,xkV1×V2×,I(xk)I(xk+1)}{o} We define an edge set E for the vertex set V as follows: if i{1,2} and x,yVi, and (x,y)Ei, then (wx,wy)E for all wV. See [11] for discussions of SAWs on free product graphs of quasi-transitive graphs.

Theorem 1.6

Let G=(V,E) be the free product graph of two connected, locally finite, quasi-transitive, rooted graphs G1=(V1,E1,o1) and G2=(V2,E2,o2) with |Vi|2, for i=1,2, 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 G=(V,E) be a locally finite Cayley graph of Γ. For vV Let πnv be an n-step SAW on G starting from v. Then there exists a(0,1), such thatlimsupn|{πnv:πnvan}|1n=μ.

Theorem 1.8

Let Γ be an infinite, finitely-generated group with more than one end. There exists a locally finite Cayley graph G=(V,E) 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 S 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 G=(V,E) 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 Aut(G) acting quasi-transitively on G. Let ΓS be the set of images of S under Γ. By quasi-transitivity of G, for each γΓ, γS still satisfies Assumption 1.1. We also call γS for γΓ a copy of S.

We shall next introduce events E, Ek and E˜k and their restrictions to a length-2m sub-walks E(m), Ek(

Proof of Theorem 1.3 B

We prove Theorem 1.3 B. in this section. Let G=(V,E) 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 E occurs at the jth step of π if there exists γSΓS such that π(j)γS, and all the vertices of γS are visited by π. For k1, we say that Ek occurs at the jth step of π, if there exists γSΓS, such that π

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 G=(V,E) be an infinite, connected, locally finite graph. Let A,BV be two finite set of vertices of G satisfying AB. Let GA (resp. GB) be the subgraph of G obtained from G by removing all the vertices in A (resp. B) as well as their incident edges. If GA has at least two infinite components, then GB 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 S={o}V. Then GS has at least two infinite components. Indeed, let x,yV satisfyx=x1xn;y=y1ym; where m,n1, xi,yjV1×V2×, I(xi)I(xi+1), I(yj)I(yj+1) (see Definition 1.5 for notations). If x1V1× and y1V2×, then x and y are in two distinct components of GS.

Let A (resp. B) be a component of GS, such that for any xA (resp. yB), x (resp. y) has the form (5.1) (resp. (5.2)) with x1V1×

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.

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