The mass of sites visited by a random walk on an inﬁnite graph

We determine the log-asymptotic decay rate of the negative exponential moments of the mass of sites visited by a random walk on an inﬁnite graph which satisﬁes a two-sided sub-Gaussian estimate on its transition kernel. This provides a new method of proof of the correct decay rate for Cayley graphs of ﬁnitely generated groups with polynomial volume growth. This method also extend known results by determining this decay rate for certain graphs with fractal-like structure or with non-Alfors regular volume growth functions.


Introduction
The behavior of R n , the number of sites visited up to time n by a simple random walk in Z d , was studied by Dvoretsky and Erdos in 1948. [9] They computed the expected value and variance of R n and proved a strong law of large numbers. Later Donsker and Varadhan [8] showed that for an explicitly computed constant c(d, ν). The relationship observed by Varopoulos in [25; 26] (for details see [21]) between (1.1) and the behavior of the return probability of a random walk on F ≀Z d (the wreath product of a finitely generated group F with Z d ) sparked a renewed interest in understanding the extent to which (1.1) might hold for other infinite graphs. After studying these return probabilities using isoperimetric profiles Erschler [10; 11] used this relationship in reverse to show (among other things) that on the Cayley graphs of finitely generated groups whose volume growth functions are polynomial degree d, (By f ≃ g we imply the existance of constants c and C such that cg(n) ≤ f (n) ≤ Cg(n) for large enough n.) More recently, Rau [22] has also used this technique to obtain results similar to (1.2) for random walks on supercritical percolation clusters in Z d .
Here we perform a direct analysis of E [exp {−νR n }] using the coarse graining techniques popularized by Sznitman in the field of random media [5; 24]. The main results are analogous to (1.2), but apply to graphs with less regular volume growth and to fractal-like graphs [3; 4]. In Section 1 we introduce notation and state the main results. Section 2 contains the proof of the asymptotic lower bound statements, and Section 3 contains the proof of the corresponding asymptotic upper bounds. In Section 4 we present several examples of graphs to which these results apply.

Preliminaries.
A time homogeneous random walk X k on a graph Γ may be described by a transition probability function p(x, y) = P [ X k+1 = y| X k = x], the probability that at step k the walk at vertex x will move to vertex y. The future path of the walk depends only on the current position (the Markov property), and this implies that p n (x, y) = P [ X k+n = y| X k = x] may be described inductively by p n (x, y) = z∈Γ p n−1 (x, z)p(z, y).

(1.3)
A walk is called irreducible if for any vertices x and y in Γ, p n (x, y) > 0 for some n. A graph is locally finite if each vertex of the graph has at most finitely many neighbors and is connected if for any vertices x and y in Γ there is some path of edges in Γ which starts at x and ends at y. The distance d(x, y) between two vertices is measured by the number of edges in any shortest path between them. Although not mentioned again, all of the random walks treated here will be irreducible walks on locally-finite, connected, infinite graphs.
A random walk is reversible with respect to a measure m on the vertices of Γ if for any x and y m(x)p(x, y) = m(y)p(y, x). (1.4) Reversibility implies that x∈Γ m(x)p(x, y) = m(y), (1.5) or that m is an invariant or stationary measure for p(x, y). If P is an operator on then P is self-adjoint whenever p(x, y) is reversible with respect to m. The associated Dirichlet form is written for all x ∈ Γ and ρ > 0. With α = log 2 C VD and ρ ≥ σ, we will more often use the alternative formulation (1.8) We say that a random walk on Γ satisfies a Neumann Poincaré inequality with parameter β (PI(β)) if there exists a constant C PI such that on each ball B(z, ρ) The use of f B = x∈B(z,ρ) f (x)m(x) V (z, ρ) −1 in place of ξ to realize the infimum on the left hand side is another common way to state PI(β). An immediate consequence of PI(β) is the bound on (the Neumann eigenvalue of I − P in B(z, ρ)) from below by cρ −β . Our last condition is called GE(β) and is written piecewise as and . (1.11) That GE(β) implies VD and PI(β) is shown in appendix Proposition 5.1.

Statement of Results.
Although m(x) is constant for the simple random walk in Z d , m(x) will generally vary with x in the graph context. It is therefore more appropriate to consider m (D n ), where D n is the collection of vertices visited by the random walk up to step n. Let (1.14) Theorem 1.1. Let Γ be a locally-finite connected infinite graph, and let p(x, y) be the transition operator of a locally elliptic (η > 0) , bounded range (d < ∞) , reversible random walk on Γ. Suppose that p(x, y) together with its reversible measure m satisfies VD, PI(2), and κ > 0. Then for any ν > 0 and every vertex x for which holds, the random walk starting at x satisfies (1.16) Theorem 1.2. Let Γ be a locally-finite connected infinite graph, and let p(x, y) be the transition operator of a locally elliptic (η > 0) , bounded range (d < ∞) , reversible random walk on Γ. Suppose that p(x, y) together with its reversible measure m satisfies GE(β) and κ > 0. Then for any ν > 0 and every vertex x for which holds, the random walk starting at x satisfies Note that when β = 2 and for strongly irreducible random walks the results of [7] show that Theorem 1.2 is equivalent to Theorem 1.1. However, the equivalence of VD and PI(2) on the one hand and GE(2) on the other (again in [7]) is not at all trivial, and part of the value of Theorem 1.1 is that its proof does not use GE (2). Also note that for any graph with bounded geometry, the simple random walk satisfies η > 0, d < ∞, and κ > 0 automatically. Graphs with super-polynomial volume growth are not able to be treated by the techniques used here due to the heavy dependence of these results on the volume doubling condition.
These results concern only rough rates of decay of the negative exponential moments of the mass of visited sites, and not specific asymptotic values. The letters c and C which appear in several of the proofs below will be used to represent some positive constants that may change as convenient, even line by line and do not depend on the most important parameters, such as x ∈ Γ and the time parameter n. Specific information regarding constants which describe the upper and lower limits implicit in (1.16) and (1.18) may be obtained via more sophisticated methods, see [12]. We do not know whether or not a graph may satisfy the results of these theorems without also satisfying their hypotheses.

Lower Bounds
The lower bounds in (1.16) and (1.18) follow the classical approach as in [8]. If H B(x,ρ) is the first time the random walk starting at x exits B(x, ρ), then It remains only to show that for sufficiently large n, To do this we introduce the following notation. For any set U ⊂ Γ, define the killed random walk (on exit from U ) by the transition function and let P U denote the corresponding L 2 operator. In this context, the smallest non-zero eigenvalue of I − P U satisfies This quantity is also refered to as the principle Dirichlet eigenvalue of I − P in U even though λ(U ) is the bottom of the Dirichlet spectrum and may not be an eigenvalue when U is infinite.
Here the set U will be finite whenever λ(U ) is used, so (2.2) follows from the Rayleigh-Ritz variational characterization of eigenvalues (e.g. [16]).
Lemma 2.1. For σ sufficiently large with respect to ρ, Proof. Let B = B(x, ρ). Since the random walk may be periodic, note that For σ sufficiently large with respect to ρ, take ℓ ∈ {ρ, ρ + 1} such that σ − ℓ is even. Then using But now since ℓ ≥ ρ, there must be some path (not necessarily the shortest) of length either ℓ or ℓ + 1 from x to y for any y ∈ B. The probability that the random walk traverses this path bounds the probability of moving from x to y in ℓ or ℓ + 1 steps from below by All the eigenvalues of P σ−ℓ B are non-negative since σ − ℓ is even, so and a change of constants now completes the proof.
These estimates may be used in ( and that By the spectral theorem By Proposition 5.3, 0 ≤ λ(B) < 1 for large enough ρ, and so (2.5), (2.6) and (2.7) now yield Two applications of GE(β) and (1.8) give On the other hand, use Cauchy-Schwartz and (2.6) again to obtain

Upper Bounds
A different approach to understanding the behavior of E [exp (−νR n )] in Z d emerges from the field of random media [5]. If each vertex is taken independently with probability 1 − exp (−ν) to be a trap, then each time a vertex is visited for the first time there is an exp(−ν) chance that the walk will not encounter a trap. Let P ν be the probability measure on the configurations of traps and P x the measure on the random walk paths starting at vertex x. If T denotes the time at which the random walk first encounters a vertex with a trap -the survival time, then by averaging over all configurations of traps we obtain (3.1) The notation P ν ⊗P x refers to the underlying product of σ-algebras in the spaces of configurations and of paths, respectively. More precisely, if ω ν ∈ Ω ν is a configuration of traps and ω x ∈ Ω x a path of length n starting from x, then By reversing the order of summation, conditioning on the set of vertices in the path up to time n, and considering the total probability of a configuration which has no traps at these vertices, we recover (3.1). This formulation allowed Antal [1] to apply the Method of Enlargement of Obstacles to give a second proof of (1.1). In our context, if the probability of an obstacle at vertex x is 1 − exp(−νm(x)), then Since GE(β) implies VD and PI(β) (Lemma 5.5 in the appendix) we may prove the upper bound portion of both Theorems 1.1 and 1.2 at once. The method of proof roughly follows Lecture 3 of [5]. For convenience take n β V (x, n) = ℓ, and let B ω ρ = B(x, ρ) \ Υ ω , where Υ ω is the collection of traps for a given configuration ω. From the spectral theorem and the Cauchy-Schwartz inequality, Computing the norms and averaging over all ω, Since by (1.8) and (2.4) 2) will now follow if we show that for some c 0 , Therefore, (absorbing the constants) Choosing c 0 sufficiently small with respect to ν and κ and c according to Lemma 5.5, Therefore Proof. In [14] it is shown that when the volume growth of Γ is polynomial of degree d, then (1.10) and (1.11) hold for the simple random walk on Γ. The desired result then follows either by direct application of Theorem 1.2 or of Theorem 1.1 after first using Proposition 5.1. However, these routes utilize the intricate techniques of [14], which may be avoided by verifying the hypotheses of Theorem 1.1 independently.
First, VD follows immediately from the degree d polynomial growth of Γ, as does (1.15) or (1.17). A weak version of PI(2) may be quickly established by taking advantage of the group structure. Specifically, for any x, xy ∈ B (x 0 , ρ) with y = s 1 s 2 . . . s k , by the Cauchy-Schwartz inequality where s 0 is the identity in the group G. For each i, the sum of the right hand side of (4.1) over all x ∈ B (x 0 , ρ) is smaller than the sum over all xs 0 . . . s i−1 ∈ B (x 0 , 2ρ) . Thus, Finally, since we consider the simple random walk, By a covering argument due to [17], (4.2) is known to improve to the full PI(2) whenever VD holds.

An example with
Theorem 1.1 applies to the following example, even though for any x or for any α, i.e. Γ is not Alfors regular. No results similar to those presented here have been shown concerning graphs of this kind. Since the random walk presented here is strongly irreducible, Theorem 1.2 might also be applied, exercising again the equivalence shown in [7] between PI(2) and VD.
For each i ∈ N, let x i = 2 2 2i , y i = 2 2 2i+1 and y 0 = 0, and define for all where α > 1, c 1 = 1 and c i+1 = c i x −α i y α i . Use the reversibility condition to recursively define p(x, x + 1) as a random walk transition probability function on Z + . Since m(x) is increasing, and whenever ρ > |x| if VD is shown to hold then (1.15) will follow for all x, since inf y∈B(x,n 2 V (x,n)) V (y, n) ≥ V (0, n).
This example also serves to demonstrate that (1.15) is a weaker condition than polynomial volume growth. To finish checking the hypotheses one must then prove that p(x, x + 1) is bounded away from one and zero for all x and check that PI(2) holds.
We will use the approximation V (x, n) ≈ x+n x−n m(t)dt, which results in a small relative error that does not affect the results. We will also assume that α ∈ N, although only small modifications are needed to show that all real α ≥ 1 is a sufficient assumption. A useful computation shows that Note that when β < σ, where C α = 1 α+1 α j=0 α+1 j+1 . Also, replacing β by 2β in the above, where C α,2 = 1 α+1 α j=0 α+1 j+1 2 j > 1. One final useful computation is that when y i − x ≤ x, which follows from Proof. Note that if ρ = x, V (x, ρ) = V (0, 2ρ), and V (x, 2ρ) = V (0, 3ρ). So, if we show that VD holds whenever ρ ≤ x with a constant which is independent of x, then it will also hold (for a slightly different constant) for all ρ > 0 for V (0, ρ). But, if ρ > x, then V (x, ρ) > V (0, ρ) and V (x, 2ρ) < V (0, 3ρ), so VD holds for all x ∈ Z + and all ρ > 0 (again for a slightly different constant).
To prove VD for ρ ≤ x, it is necessary to check several cases. It is helpful to keep in mind that if x < x i , (or y i ) then x + ρ < 2x < y i (or x i+1 ).
If x i < x < y i , then when x + ρ < y i , using (4.4) and (4.5), When x + ρ > y i by (4.6) and and if x + 2ρ > x i , then by (4.5) and C α,2 > 1 Next, show that p(x, x + 1) is bounded away from zero and one for all x. It follows that if p * is defined by recursively by p * (0, 0) = p * (0, 1) = 1/2 and p * (x, x) = 0 if x = 0, based on the measure Using the recursive definition of p * we obtain When n is even, since x α has non-decreasing derivative, by the mean value theorem So, and similarly When n is odd, the result follows similarly.
Before verifying PI(2) for this example, consider how this inequality might be proved on any weighted graph. Let B = B(x 0 , ρ), Q(e) = p(e − , e + )m(e − ) for each edge e = (e − , e + ),and let γ xy be a shortest path from x to y. Then (as in [23]) by the Cauchy-Schwartz inequality, Therefore, where m(γ xy ) = m(x)m(y) and Γ * contains exactly one shortest path γ xy for each pair of vertices x, y ∈ B. PI(2) will hold for any graph for which Γ * can be chosen in such a way that Proof. For Z, there is only one shortest path between any pair of vertices. Furthermore, for B = B(x 0 , n) at most n 2 paths are required to form Γ * , so each edge may appear in at most n 2 paths. Let γ + indicate the last vertex of the path, and γ − the first vertex. Since m(x) is non-decreasing, when e ∈ γ which together with p(n, n − 1) ≥ 2 −α−1 by (4.7) yields To complete the proof it remains only to show that V (x 0 , n) ≥ cnm(x 0 + n).
and when x 0 < y i−1 , by (4.6) Note that this example is recurrent (lim inf n −2 V (x, n) < ∞), but is both not Ahlfors regular and not strongly recurrent in the sense mentioned below. Although somewhat more complicated, when α ∈ N it is relatively straightforward to construct a non-weighted sub-graph of Z α for which the simple random walk mimics the behavior of the example constructed here on Z.

The case β = 2, Fractal examples
Since we are considering only cases in which β = 2 in this section, only Theorem 1.2 may be applied to these examples. One must first verify GE(β) in order to apply Theorem 1.2 to any specific example. This has proved to be a difficult task in most venues for which it has been accomplished, so the applications presented here will be restricted to those cases where this hypothesis is already known to hold.
Historically, the first graph for which the simple random walk was known to witness GE(β) with β > 2 was the Sierpinski gasket pre-fractal graph as pictured in figure 2 of [18]. Later this example was shown [4,Propsition 5.4] to fit into a class of strongly recurrent graphs for which GE(β) holds, namely graphs based on the pre-fractal structure of finitely ramified fractals for which the resitance metric (see section 2 of [13]) is a polynomial scaling of the graph distance. The class of generalized Sierpinski carpet pre-fractal graphs (e.g. figure 1.2 of [3]) also satisfies GE(β) [3]. Since the volume growth in all of these examples is polynomial, Theorem 1.2 applies in each case to give the asymptotic rate of decay of the negative exponential moments of the number of visited sites for the simple random walk.
A similar analysis of survival time for the Brownian motion on the Sierpinski gasket appears in [19] and [20], no results of this type have previously been explored for the discrete graph context.
As shown in [4], any nearest neighbor random walk on a tree which satisfies

Appendix
The following results are included for completeness When β = 2, the following result and its converse for strongly irreducible nearest neighbor random walks is found in [7]. Here it is used to simplify the statement of Theorem 1.2, since PI(β) and VD are used directly in the proof of that theorem. The proof below is only slightly modified from [7], and like all the results in this section, are included for the sake of completeness.
Proof. Only the lower estimate is needed to prove VD, since Let Q = P n + P n+1 /2, i.e.
let ρ = n 1/β , and set B = B (x 0 , ρ) . One can easily check that Q is reversible with respect to m, since m > 0. When x ∈ B by (1.11) and (1.7) is chosen to minimize the previous expression. Having obtained the left hand side of PI(β), we now work toward the right hand side. Starting with (5.1) and using properties of Q, We need only control each of these terms by E(f, f ) to complete the lemma since Now since P is a contraction, Finally, Another consequence of GE(β) from [14] is used to obtain a stronger upper bound than Proposition 5.3 on the Dirichlet eigenvalue for the proof of the lower bound portion of Theorem 1.2.
Proof. Note that the result is void if σ β /n is small. Let B j = B y, 2 j σ be the ball of radius 2 j σ about y. We first prove that .
Indeed, P y X n / ∈ B 0 = x / ∈B 0 p n (y, x), so that by (1.10), (1.8), and a change of constants we obtain The necessary size of σ β /n only depends on the size of the constants from GE(β) and C VD .
Only VD is required to obtain the cρ −2 upper bound on λ (B(x, ρ)) used in the proof of the lower bound portion of Theorem 1.1. The following general argument is commonly known, see for example [6].
The following lemma (nearly identical to Lemma 3.3 of [5]) is used in the upper bound proof to take full advantage of the Bernoulli distribution of traps.
For each x ∈ Γ, let Y x be an independant Bernoulli(1 − exp (−νm(x))) random variable. Let S U = x∈U Y x m(x). The following simple large deviation result is used in the upper bound.