Uniform mixing time for random walk on lamplighter graphs

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www.imstat.org/aihp 2014, Vol. 50, No. 4, 1140–1160

DOI:10.1214/13-AIHP547

Uniform mixing time for random walk on lamplighter graphs

Júlia Komjáthy

^a,1

, Jason Miller

^b

and Yuval Peres

^c

aTechnische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven, The Netherlands. E-mail:j.komjathy@tue.nl

bDepartment of Mathematics, Massachusetts Institute of Technology, Headquarters Office, Building 2, Room 236, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. E-mail:jpmiller@mit.edu

cMicrosoft Research, One Microsoft Way, WA 98052, USA. E-mail:peres@microsoft.com Received 20 September 2011; revised 27 December 2012; accepted 3 February 2013

Abstract. Suppose thatGis a finite, connected graph andXis a lazy random walk onG. The lamplighter chainXassociated with Xis the random walk on the wreath productG=Z₂G, the graph whose vertices consist of pairs(f , x)wheref is a labeling of the vertices ofGby elements ofZ₂= {0,1}andxis a vertex inG. There is an edge between(f , x)and(g, y)inGif and only if xis adjacent toyinGandf_z=g_zfor allz=x, y. In each step,Xmoves from a configuration(f , x)by updatingxtoyusing the transition rule ofXand then sampling bothf_x andf_yaccording to the uniform distribution onZ₂;f_zforz=x, yremains unchanged. We give matching upper and lower bounds on the uniform mixing time ofXprovidedGsatisfies mild hypotheses.

In particular, whenGis the hypercubeZ^d₂, we show that the uniform mixing time ofXisΘ(d2^d). More generally, we show that whenGis a torusZ^d_nford≥3, the uniform mixing time ofXisΘ(dn^d)uniformly innandd. A critical ingredient for our proof is a concentration estimate for the local time of the random walk in a subset of vertices.

Résumé. SoitGun graphe connexe fini etXune marche aléatoire fainéante surG. La chaîne de l’allumeur de réverbèresX associée àXest la marche aléatoire sur le groupe produitG=Z₂G, le graphe dont les sites sont des paires(f , x)oùf est un label des sites deGpar des éléments deZ₂= {0,1}etxest un site deG. Il existe une arête entre(f , x)et(g, y)dansGsi et seulement sixest adjacent àydansGetf_z=g_zpour toutz=x, y. A chaque pas,Xse déplace d’une configuration(f , x) en mettant à jourx versypar la règle de translation deXet ensuite en mettant à jour à la foisf_xetf_y selon la distribution uniforme surZ₂;f_zpourz=x, yrestant inchangé. Nous prouvons des bornes supérieures et inférieures équivalentes sur le temps de mélange uniforme deXsous des hypothèses faibles surG. En particulier quandG est l’hypercubeZ^d₂, nous montrons que le temps de mélange uniforme deX estΘ(d2^d). Plus généralement, nous montrons que quandGest le toreZ^d_n avecd≥3, le temps de mélange uniforme deXestΘ(dn^d)uniformément ennetd. Un ingrédient crucial de notre preuve est une estimation de concentration pour le temps local d’une marche aléatoire dans un sous ensemble de sites.

MSC:60J10; 60D05; 37A25

Keywords:Random walk; Uncovered set; Lamplighter walk; Mixing time

1. Introduction

Suppose thatG is a finite graph with verticesV (G)and edgesE(G), respectively. LetX(G)= {f:V (G)→Z2}be the set of markings ofV (G)by elements ofZ2. The wreath productZ2Gis the graph whose vertices are pairs(f , x) wheref ∈X(G)andx∈V (G), see Fig.1. There is an edge between(f , x)and(g, y)if and only if{x, y} ∈E(G) andf_z=g_z for allz /∈ {x, y}. Suppose thatP is a transition matrix for a Markov chain onG. The lamplighter walk X(with respect to the transition matrixP) is the Markov chain onGwhich moves from a configuration(f , x)by

1Supported by Grant KTIA-OTKA # CNK 77778, funded by the Hungarian National Development Agency (NFÜ) from a source provided by KTIA.

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Fig. 1. A typical configuration of the lamplighter over a 5×5 planar grid. The colors indicate the state of the lamps and the dashed circle gives the position of the lamplighter.

1. pickingyadjacent toxinGaccording toP, then

2. updating each of the values off_xandf_yindependently according to the uniform measure onZ2.

The lamp states at all other vertices inGremain fixed. It is easy to see that ifPis ergodic and reversible with stationary distributionπ_P then the unique stationary distribution ofXis the product measure

π (f , x)

=πP(x)2^−|G|,

andXis itself reversible. In this article, we will be concerned with the special case thatP is the transition matrix for thelazy random walkonGin order to avoid issues of periodicity. That is,P is given by

P (x, y)= 1

2 ifx=y,

1

2d(x) if{x, y} ∈E(G) (1.1)

forx, y∈V (G)and whered(x)is the degree ofx.

1.1. Main results

LetP be the transition kernel for the lazy random walk on a finite, connected graphGwith stationary distributionπ. The-uniform mixing timeofGis given by

tu(G, ε)=min

t≥0: max

x,y∈V (G)

P^t(x, y)−π(y) π(y)

≤

. (1.2)

Throughout, we lett_u(G)=t_u(G,_2e¹). The main result of this article is a general theorem which gives matching upper and lower bounds oftu(G)providedGsatisfies several mild hypotheses. One important special case of this result is the hypercubeZ^d₂and, more generally, toriZ^d_nford≥3. These examples are sufficiently important that we state them as our first theorem.

Theorem 1.1. There exists constantsC1, C2>0such that C₁≤t_u((Z^d₂))

d2^d ≤C₂ for alld.

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More generally, C1≤t_u((Z^d_n))

dn^d⁺² ≤C2 for alln≥2andd≥3.

Prior to this work, the best known bound [11] fort_u((Z^d₂))was C₁d2^d≤t_u

Z^d₂

≤C₂log(d)d2^d forC₁, C₂>0.

In order to state our general result, we first need to review some basic terminology from the theory of Markov chains. Therelaxation timeofP is

t_rel(G)= 1 1−λ2

, (1.3)

whereλ₂is the second largest eigenvalue ofP. Themaximal hitting timeofP is t_hit(G)= max

x,y∈V (G)Ex[τ_y], (1.4)

whereτ_y denotes the first timet thatX(t )=y andEx stands for the expectation under the law in whichX(0)=x.

The Green’s functionG(x, y)forP is

G(x, y)=Ex

_t_u₍_G₎

t=0

1_{X(t )=y}

=

tu(G)

t=0

P^t(x, y), (1.5)

i.e. the expected amount of timeXspends atyup to the uniform mixing timetugivenX(0)=x. For each 1≤n≤ |G|, we let

G^∗(n)= max

S⊆V (G)

|S|=n

maxz∈S y∈S

G(z, y). (1.6)

This is the maximal expected timeXspends in a setS⊆V (G)of sizenbefore the uniform mixing time. This quantity is related to the hitting time of subsets ofV (G).

Finally, recall thatG is said to be vertex transitive if for everyx, y∈V (G)there exists an automorphismϕofG withϕ(x)=y. Our main result requires the following hypothesis.

Assumption 1.2. G is a finite,connected,vertex transitive graph andX is a lazy random walk on G.There exists constantsK₁, K₂, K₃>0such that

(A) thit(G)≤K1|G|,

(B) 2K2(5/2)^K²(maxy:y∼xG(x, y))^K²≤exp(−_t^t_rel^u⁽₍^G_G⁾₎), (C) G^∗(n^∗)≤K3(trel(G)+log|G|)/(logn^∗)

wheren^∗=4K2t_u(G)/miny:y∼xG(x, y)forx, y∈V (G)adjacent.

Remark 1.3. Note that such constants always exists for a given fixed graphG.However,to get the Theorem1.5to hold for a givenfamily of graphs,the constantK1, K2, K3need to be uniform in the family of graphs under consideration.

Thus,to get Theorem1.1,the constants need to be uniform innandd.

Remark 1.4. We need to writemaxy:y∼xG(x, y)in(B)andminy:y∼xG(x, y)in the definition ofn^∗since the graph G is only assumed to be vertex transitive and not edge-transitive,thusG(x, y)may vary along the neighbors of a vertexx.

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These assumptions are rather technical:(B)and(C)express some sort of local transience criterion. The gap left by Peres–Revelle in [11] is only there iftrel(G)=o(tmix(G))and log|G| =o(tmix(G))both hold. Thus, the right hand side of criterion(B)is tending to 0 as|G| → ∞in cases of our main interest, and the criterion says that the Green function should behave in a similar manner. The criterion connects two error terms later in the proofs. Assumption (C)will serve to be able to establish a Binomial domination argument in the lemmas below for covering the last few left-out vertices, and establishes that the extra time we will give for the Binomial domination is still less then the total time in which we want to mix the chain.

The general theorem is:

Theorem 1.5. Let G be any graph satisfying Assumption 1.2. There exists constants C₁, C₂ depending only on K₁, K₂, K₃such that

C₁≤ t_u(G)

|G|(trel(G)+log|G|)≤C₂. (1.7)

The lower bound is proved in [11], Theorem 1.4. The proof of the upper bound is based on the observation from [11] that the uniform distance to stationarity can be related toE[2^|U^(t)^|]whereU(t)is the set of vertices inGwhich have not been visited byXby timet. Indeed, suppose thatf is any initial configuration of lamps, letF (t)be the state of the lamps at time t, and letgbe an arbitrary lamp configuration. LetW be the set of vertices wheref =g. Let C(t)=V (G)\U(t)be the set of vertices which have been visited byXby timet. WithP(f ,x)the probability under whichX(0)=(f , x), we have that

P(f ,x)

F (t)=g|C(t)

=2^−|C^(t)^|1_{_W_⊆C_(t)_}.

Since the probability of the configurationgunder the uniform measure is 2^−|G|, we therefore have P(f ,x)[F (t)=g]

2^−|G| =E(f ,x)

2^|U^(t)^|1_{_W_⊆C_(t)_}

. (1.8)

The right hand side is clearly bounded from above by E[2^|U^(t)^|](the initial lamp configuration and position of the lamplighter no longer matters). On the other hand, we can bound (1.8) from below by

P(f ,x)

W⊆C(t)

≥PU(t)=0

≥1− E

2^|U^(t)^|

−1 . Consequently, to boundt_u(G, ε)it suffices to compute

min

t≥0:E 2^|U^(t)^|

≤1+

(1.9) since the amount of time it requires forXto subsequently uniformly mix after this time is negligible.

In order to establish (1.9), we will need to perform a rather careful analysis of the process by which U(t) is decimated byX. The key idea is to break the process of coverage into two different regimes, depending on the size ofU(t). The main ingredient to handle the case whenU(t)is large is the following concentration estimate of the local time

LS(t)=

t

s=0

1_{X(s)∈S}

forXinS⊆V (G).

Proposition 1.6. Assume that the second largest eigenvalue ofP,λ2≥¹₂and fixS⊆V (G).Then forC0=1/50,we have that

Pπ

LS(t)≤tπ(S) 2

≤exp

−C₀t π(S) t_rel(G)

. (1.10)

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Proposition1.6is a corollary of [8], Theorem 1; we consider this sufficiently important that we state it here. By invoking Green’s function estimates, we are then able to show that the local time is not concentrated on a small subset ofS. Since the total time of ours is limited byC|G|(t_rel(G)+log|G|), these estimates give us a small error probability only as long as the set which is not yet covered is large enough. Thus, we will handle the case whenU(t)is small separately, via an estimate (Lemma3.5) of the hitting timeτ_S=min{t≥0: X(t )∈S}ofS.

1.2. Previous work

Suppose thatμ, νare probability measures on a finite measure space. Recall that thetotal variation distancebetween μ, νis given by

μ−νTV=max

A

μ(A)−ν(A)=1

2 _x μ(x)−ν(x). (1.11)

The-total variation mixing timeofP is t_mix(G, ε)=min

t≥0: max

x∈V (G)

P^t(x,·)−π

TV≤

. (1.12)

Lett_mix(G)=t_mix(G,_2e¹). It was proved [11], Theorem 1.4, by Peres and Revelle that ifGis a regular graph such that t_hit(G)≤K|G|, there exists constantsC₁, C₂depending only onKsuch that

C₁|G|

t_rel(G)+log|G|

≤t_u G

≤C₂|G|

t_mix(G)+log|G|

.

These bounds fail to match in general. For example, for the hypercubeZ^d₂,trel(Z^d₂)=Θ(d)[9], Example 12.15, while t_mix(Z^d₂)=Θ(dlogd)[9], Theorem 18.3. Theorem1.5says that the lower from [11], Theorem 1.4, is sharp.

Before we proceed to the proof of Theorem1.5, we will mention some other work on mixing times for lamplighter chains. The mixing time ofGwas first studied by Häggström and Jonasson in [6] in the case of the complete graph Kn and the one-dimensional cycleZn. Their work implies a total variation cutoff with threshold ¹₂tcov(Kn)in the former case and that there is no cutoff in the latter. Here,tcov(G)for a graphGdenotes the expected number of steps required by the lazy random walk to visit every site inG. The connection betweentmix(G)andtcov(G)is explored further in [11], in addition to developing the relationship between the relaxation time ofGandthit(G), andE[2^|U^(t)^|] andtu(G). The results of [11] include a proof of total variation cutoff forZ²_n with thresholdtcov(Z²_n). In [10], it is shown thatt_mix((Z^d_n))∼¹₂t_cov(Z^d_n)when d≥3 and more generally thatt_mix(Gn)∼¹₂t_cov(Gn)whenever(Gn)is a sequence of graphs satisfying some uniform local transience assumptions.

The mixing time ofX=(F , X)is typically dominated by the first coordinateF since the amount of time it takes forXto mix is negligible compared to that required byX. We can sample fromF (t)by:

1. sampling the rangeC(t)of the lazy random walk run for timet, then 2. marking the vertices ofC(t)by i.i.d. fair coin flips.

Determining the mixing time ofXis thus typically equivalent to computing the thresholdtwhere the corresponding marking becomes indistinguishable from a uniform marking ofV (G)by i.i.d. fair coin flips. This in turn can be viewed as a statistical test for the uniformity of the uncovered setU(t)ofX– ifU(t)exhibits any sort of non-trivial systematic geometric structure thenX(t)is not mixed. This connects this work to the literature on the geometric structure of the last visited points by the random walk [1–3,10].

1.3. Outline

The remainder of this article is structured as follows. In Section2, we will give the proof of Theorem1.1by checking the hypotheses of Theorem1.5. Next, in Section3we will collect a number of estimates regarding the amount ofX spends in and requires to cover sets of vertices inGof various sizes. Finally, in Section4, we will complete the proof of Theorem1.5.

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2. Proof of Theorem1.1

We are going to prove Theorem1.1by checking the hypotheses of Theorem1.5. We begin by noting that by [9], Corollary 12.12, and [9], Section 12.3.1, we have that

t_rel Z^d_n

=Θ dn²

. (2.1)

By [4], Example 2, p. 2155, we know thatt_u(Zn)=O(n²). Hence by [5], Theorem 2.10, we have that t_u

Z^d_n

=O

(dlogd)n²

. (2.2)

The key to checking parts (A)–(C) of Assumption 1.2are the Green’s function estimates which are stated in Proposition2.2(low dimension) and Proposition2.6(high dimension). In order to establish these we will need to prove several intermediate technical estimates. We begin by recording the following facts about the transition kernel P for the lazy random walk on a vertex transitive graphG. First, we have that

P^t(x, y)≤P^t(x, x) for allx, y. (2.3)

To see this, we note that fort even, the Cauchy–Schwarz inequality and the semigroup property imply P^t(x, y)=

z

P^{t /2}(x, z)P^{t /2}(z, y)≤

P^t(x, x)P^t(y, y)=P^t(x, x).

The inequality and final equality use the vertex transitivity ofGso thatP (x, z)=P (z, x)andP (x, x)=P (y, y). To get the same result fortodd, one just applies the same trick used in the proof of [9], Proposition 10.18(ii). Moreover, by [9], Proposition 10.18, we have that

P^t(x, x)≤P^s(x, x) for alls≤t. (2.4)

The main ingredient in the proof of Proposition2.2, our low dimension Green’s function estimate, is the following bound for the return probability of the lazy random walk onZ^d.

Lemma 2.1. LetP (x, y;Z^d)denote the transition kernel for the lazy random walk onZ^d.For allt≥1,we have that P^t

x, x;Z^d

≤√ 2

4d π

d/2

1

t^d/2+e⁻^{t /8}. (2.5)

Proof. To prove the lemma we first give an upper bound on the transition probabilities for a (non-lazy) simple random walkY onZ^d. One can easily give an exact formula for the return probability ofY to the origin ofZ^d in 2t steps by counting all of the possible paths from 0 back to 0 of length 2t (here and hereafter,P_NL(x, y;Z^d)denotes the transition kernel ofY):

P_NL^2t

x, x;Z^d

=

n1+···+nd=t

(2t )!

(n₁!)²(n₂!)²· · ·(n_d!)² · 1 (2d)^2t

= 1 (2d)^2t

2t t

n₁+···+n_d=t

t! n₁!n₂! · · ·n_d!

2

.

We can bound the sum above as follows, using the multinomial theorem in the second step:

P_NL^2t

x, x;Z^d

≤ 1 (2d)^2t

2t

t max

n1+···+nd=t

t! n₁! · · ·n_d!

n1+···+nd=t

t! n₁! · · ·n_d!

≤ 1 (2d)^2t

2t t

t!

[(t /d)!]^d ·d^t.

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Applying Stirling’s formula to each term above, we consequently arrive at P_NL^2t

x, x;Z^d

≤

√2

(2π)^d/2·d^d/2

t^d/2. (2.6)

We are now going to deduce from (2.6) a bound on the return probability for the lazy random walkXonZ^d. We note that we can coupleXandY so thatXis a random time change ofY:X(t )=Y (N_t)whereN_t=_t

i=0ξ_i and the(ξ_i)are i.i.d. withP[ξ_i=0] =P[ξ_i =1] =¹₂ and are independent ofY. Note thatN_t is distributed as a binomial random variable with parameterst and 1/2. Thus,

P^t

x, x;Z^d

=

t /2

i=0

P_NL²ⁱ

x, x;Z^d

P(Nt =2i)

≤P(Nt < t /4)+√ 2

4d π

d/2

1 t^d/2,

where in the second term we used the monotonicity of the upper bound in (2.6) int. The first term can be bounded from above by using the Hoeffding inequality. This yields the term e⁻^{t /8}in (2.5).

Throughout the rest of this section, we let|x−y|denote theL¹distance betweenx, y∈Z^d_n.

Proposition 2.2. LetG(x, y)denote the Green’s function for the lazy random walk onZ^d_n.For eachδ∈(0,1),there exists constantsC₁, C₂, C₃>0independent ofn, dford≥3such that

G(x, y)≤ C₁ d

4d π

d/2

|x−y|¹⁻^d/2+C2(dlogd) 4d

π d/2

n²⁻^d(1⁻^δ/2) +C3

d²logd

n²e⁻ⁿ^δ^/2 for allx, y∈Z^d_ndistinct.

Proof. Fixδ∈(0,1). We first observe that the probability that there is a coordinate in which the random walk wraps around the torus withint < n²steps can be estimated by using Hoeffding’s inequality and a union bound by

d·P

Z(t ) > n

=de⁻ⁿ²^{/(2t )},

whereZ(t )is a one dimensional simple random walk onZ. Letk= |x−y|. Applying (2.3) and (2.4) in the second step, and estimating the probability of wrapping around in timen²⁻^δin the third term, we see that

G(x, y)=

tu

t=k

P^t(x, y)≤

n²⁻^δ

t=k

P^t

x, x;Z^d

+tuPⁿ²⁻^δ

x, x;Z^d

+dtue⁻ⁿ^δ^/2. (2.7)

We can estimate the first term on the right hand side above using Lemma2.1, yielding the first term in the assertion of the lemma. Applying Lemma2.1again, we see that there exists a constantC2which does not depend onn, dsuch that the second term in the right side of (2.7) is bounded by

C2(dlogd) 4d

π d/2

n²⁻^d(1⁻^δ/2). (2.8)

Indeed, the factor(dlogd)n²comes from (2.2) and the other factor comes from Lemma2.1. Combining proves the

lemma.

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Proposition2.2is applicable whennis much larger thand. We now turn to state some preliminary estimates for Proposition2.6. This proposition will give us an estimate for the Green’s function that we will use whend is large.

Here comes the first ingredient:

Lemma 2.3. Suppose thatXis a lazy random walk onZ^d_nford≥8and that|X(0)| =k≤^d₈.For eachj ≥0,letτ_j be the first timetthat|X(t )| =j.There exists a constantC_k>0depending only onksuch thatP[τ0< τ2k] ≤C_kd⁻^k. If,instead,|X(0)| =1,then there exists a universal constantp >0such thatP[τ₀< τ₂] ≥p.

Proof. It clearly suffices to prove the result whenXis non-lazy. Assume that|X(t )| =j ∈ {k, . . . ,2k}. It is obvious that the probability that|X|moves toj+1 in its next step is at least 1−^2k_d. The reason is that the probability that the next coordinate to change is one of the coordinates ofX(t )whose value is 0 is at least 1−^2k_d. Similarly, the probability that|X|next moves toj−1 is at most ^2k_d. Consequently, the first result of the lemma follows from the Gambler’s ruin problem (see, for example, [9], Section 17.3.1). The second assertion of the lemma follows from the

same argument.

Lemma 2.4. Assume thatk∈Nand thatd=2k∨3.Suppose thatXis a lazy random walk onZ^dand that|X(0)| = 2k.Letτ_k be the first timetthat|X(t )| =k.There existsp_k>0depending only onksuch thatP[τ_k= ∞] ≥p_k>0.

Proof. This is an easy corollary of the transience of the random walk onZ^dford≥3.

The next lemma is the corresponding version for the torus.

Lemma 2.5. Assume thatk∈Nandd≥2k∨3.Suppose thatXis a lazy random walk onZ^d_nand that|X(0)| =2k.Let τ_kbe the first timet that|X(t )| =k.There existsp_k, c_k>0depending only onksuch thatP[τ_k> c_kdn²] ≥p_k>0.

Proof. We first assume thatd=2k∨3. It follows from Lemma2.4that there exists a constantpk,1>0 depending only on k such that P[τ_k > τ_n/4] ≥p_k,1. The central limit theorem (see [7], Chapter 2) implies that there exists constantsc_k,1, p_k,2>0 such that the probability that a random walk onZ^d_nmoves more than distanceⁿ₄ in timec_k,1n² is at most 1−pk,2. Combining implies the result ford=2k∨3.

Now we suppose thatd≥2k∨3. Let(X₁(t), . . . , X_d(t))be the coordinates ofX(t ). By re-ordering if necessary, we may assume without loss of generality thatX_2k₊₁(0), . . . , Xd(0)=0. LetY (t )=(X₁(t), . . . , X_2k(t)). ThenY is a random walk onZ^2k_n . Clearly,|Y (0)| =2kbecauseX(0)cannot have more than 2knon-zero coordinates. For eachj, letτ_j^Y be the first timet that|Y (t )| =j. Thenτ_k^Y ≤τk. For eacht, letNt denote the number of steps thatXtakes in the time interval{1, . . . , t}in which one of its first 2kcoordinates is changed (in other words,N_t is the number of steps taken byY). The previous paragraph implies thatP[N_τY

k ≥c_k,1n²] ≥p_k,3>0 for a constantp_k,3>0 depending only onk. Since the probability that the first 2kcoordinates are changed in any step isk/d(recall thatXis lazy), the

final result holds from a simple large deviations estimate.

Now we are ready to state and prove our estimate ofG(x, y)whendis large.

Proposition 2.6. Suppose thatd≥8.LetG(x, y)denote the Green’s function for the lazy random walk onZ^d_n.For eachk∈Nwithk≤ ^d₈,there exists a constantCk>0which does not depend onn, dsuch that

G(x, y)≤Ck

d^k for allx, y∈Z^d_nwith|x−y| ≥k.

Proof. See Fig.2for an illustration of the proof. By translation, we may assume without loss of generality thaty=0;

letk= |x|. Letτ₀be the first timetthat|X(t )| =0. The strong Markov property implies that G(x, y)≤P[τ0≤tu])G(x, x).

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Fig. 2. Assume thatd≥8 and thatk∈Nwithd≥8k. LetXbe a lazy random walk onZ^d_nand thatX(0)=xwith|x−y| =k. In Proposition2.6, we show thatG(x, y)≤C_kd^−kwhereC_k>0 is a constant depending only onk. By translation, we may assume without loss of generality that

|x| =kandy=0. The idea of the proof is to first invoke Lemma2.3to show thatXescapes to∂B(0,4k)with probability at least 1−C_k,1d^−k. We then decompose the path ofXinto successive excursions{X(σ_2k^j ), . . . , X(τ_4k^j ), . . . , X(σ_2k^j⁺¹)}between∂B(0,2k)back to itself through∂B(0,4k).

By Lemma2.3, we know that each excursion hits 0 with probability bounded byC_2k,1d⁻^2kand Lemma2.5implies that each excursion takes length c_kdn²with probability at leastp_k>0. Consequently, the result follows from a simple stochastic domination argument.

Consequently, it suffices to show that for eachk∈N, there exists constantsC_k, C₀>0 such that P[τ0≤tu] ≤C_k

d^k and (2.9)

G(x, x)≤C₀. (2.10)

We will first prove (2.9); the proof of (2.10) will be similar.

Let N be a geometric random variable with success probability C_2kd⁻^2k where C_2k is the constant from Lemma2.3. Let(ξ_j)be a sequence of independent random variables with P[ξ_j =c_2kdn²] =p_2k andP[ξ_j =0] = 1−p2kwherec2k, p2kare the constants from Lemma2.5independent ofN. We claim thatτ0is stochastically dominated from below byN ζ−1

j=1 ξ_j whereζ is independent of N and(ξ_j)withP[ζ =0] =C_kd⁻^k=1−P[ζ =1].

Indeed, to see this we will decompose the trajectory of the walk to excursions between the boundaries of theL₁-balls B(0,2k)andB(0,4k)as follows: letσ_k⁰=0 and letτ_4k⁰ be the first timetthat|X(t )| =4k. For eachj ≥1, we induc- tively letσ_2k^j be the first timetafterτ_4k^j⁻¹that|X(t )| =2kand letτ_4k^j be the first timetafterσ_2k^j that|X(t )| =4k. Let Ft be the filtration generated byX. Lemma2.3implies that the probability thatXhits 0 in{σ_2k^j, . . . , τ_4k^j }givenF_σ^j

2k

is at mostC_2kd⁻^2k for eachj ≥1 whereC_2k>0 only depends on 2k. This leads to the success probability in the definition ofN above. The factorζ is to take into account the probability thatXreaches distance 2kbefore hitting 0.

Moreover, Lemma2.5implies thatP[σ_2k^j −τ_4k^j⁻¹≥c_2kdn²|F_τ^j

4k] ≥p_2k. This leads to the definition of the(ξ_j)above.

This implies our claim.

To see (2.9) from our claim, an elementary calculation yields that P

N ζ≤C⁻_2k¹d^k

≤P

N≤C_2k⁻¹d^korζ=0

≤2d⁻^k+Ckd⁻^k.

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We also note that P

_m

j=1

ξ_j≤p_2kc_2kmn² 2

≤e⁻^cm

for some constantc >0. Combining these two observations along with a union bound implies (2.9) (with a larger constantC_kthan in the one in Lemma2.3). To see (2.10), we apply a similar argument using the second assertion of

Lemma2.3.

Now that we have proved both Proposition2.2and Proposition2.6, we are ready to check the criteria of Assump- tion1.2.

2.1. Part(A)

By [9], Proposition 1.14, withτ_x⁺=min{t≥1: X(t )=x}, we have thatEx[τ_x⁺] = |Z^d_n|. Applying Proposition2.6, we see that there exists constantsd₀, r >0 such that ifd≥d₀, then

G(x, y)≤1/2 for all|x−y| ≥r. (2.11)

Proposition2.2implies that there existsn₀such that ifn≥n₀and 3≤d < d₀then (2.11) likewise holds, possibly by increasing r (clearly, part (A)holds when d ≤d0 and n≤n0; note also that we may assume without loss of generality thatd₀, n₀are large enough so that the diameter of the graph is at least 2r). Letτ_r be the first timet that

|X(t )−X(0)| =r. We observe that there existsρ₀=ρ₀(r) >0 such that Px

τr< τ_x⁺

≥ρ0 (2.12)

uniform in n, d since in each time step there ared directions in which X(t )increases its distance fromX(0). By combining (2.11) with (2.12), we see thatPx[τ_x⁺≥tu(G)] ≥ρ1>0 uniformd≥d0. LetFtbe the filtration generated byX. We consequently have that

Ex

τ_x⁺

≥Ex

τ_x⁺1_{_τ+ x≥tu(G)}

=Ex

Ex

τ_x⁺|Ftu(G)

1_{_τ+ x ≥tu(G)}

≥Ex

EX(tu(G))[τx]1_{_τ_x⁺_≥_t_u_(G)_}

≥ρ1

1− 1

2e

Eπ[τx].

That is, withρ2:=ρ1(1−_2e¹) >0, there exists aρ2uniform ind≥d0such that|Z_n^d| =Ex[τ_x⁺] ≥ρ2Eπ[τ_x]. Com- bining this fact with [9], Lemma 10.2, stating thatthit(G)≤2 maxwEπ(τw)holds for any irreducible Markov chain, we arrive att_hit(Z^d_n)≤K₁|Z^d_n|whereK₁=2/ρ2is a uniform constant.

Remark 2.7. There is another proof of Part(A)which is based on eigenfunctions.In particular,we know that t_hit

Z^d_n

≤2Eπ[τ_x] =4

i

1 1−λ_i,

where theλi are the eigenvalues of the simple random walk onZ^d_n distinct from1;the extra factor of2in the final equality accounts for the laziness of the chain.Theλ_i can be computed explicitly using[9],Lemma12.11,and the form of theλ_i whend=1which are given in[9],Section12.3.The assertion follows by performing the summation which can be accomplished by approximating it by an appropriate integral.

2.2. Part(B)

It follows from Proposition2.6that there exist constantsC >0 andd₀≥3 such that G(x, y)≤C

d forx, y∈Z^d_nwith|x−y| =1 (2.13)

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providedd≥d₀. Consequently, there existsK∈Nwhich does not depend ond≥d₀such that 2K(5/2)^KG^K(x, y)=O

2K

5/2 d

K

. (2.14)

It follows by combining (2.1) and (2.2) that we have that t_u(Z^d_n)

t_rel(Z^d_n)=O(logd). (2.15)

Combining (2.14) with (2.15) shows that part (B)of Assumption 1.2is satisfied provided we takeK₂=K large enough. Moreover, (2.15) clearly holds if 3≤d < d0by Proposition2.2.

2.3. Part(C)

We first note that it follows from (2.1), (2.2), Proposition2.2, and Proposition2.6that there exists constantsC >0 such thatn^∗forZ^d_nis at mostCd²n²logd for alld≥3. Thus, to check Assumption(C), we need to show that there existsK3>0 such that

G^∗ n^∗

≤K₃

dn²+dlogn logd+logn

. (2.16)

We are going to prove the result by considering the regimes ofd≤√

lognandd >√

lognseparately.

Case1:d <√ logn.

From (2.16) it is enough to show thatG^∗(n^∗)≤Kdn²/logn. We can boundG^∗(n^∗)in this case as follows. Let D=(dlogdlogn)^1/((1/2)d⁻¹⁾. By Proposition 2.2, we can bound from above the expected amount of time thatX starting at 0 inZ^d_nspends in theL¹ball of radiusDby summing radially as follows:

D

k=1y:|y|=k

G(0, y)≤

D

k=1

C₁ d

4d π

d/2

k¹⁻^d/2·2d(2k)^d⁻¹

≤C₁ 16d

π

d/2 D k=1

k^d/2≤C2

d 16d

π d/2

D¹⁺^d/2≤C₃n(dlogdlogn)⁵

for constantsC₁, C₂, C₃>0, where we used thatd^d/2≤n. We also note that 2d(2k)^d⁻¹is the size of theL^∞ball of radiusk. The exponent 5 comes from the inequality

(1/2)d+1

(1/2)d−1≤5 for alld≥3.

We can estimateG^∗(n^∗)by dividing between the set of points which have distance at mostDto 0 and those whose distance to 0 exceedsD, and then using the previous estimate on the first set and Proposition2.2on the latter:

G^∗ n^∗

≤C3n(dlogdlogn)⁵+C4D¹⁻^(1/2)dn^∗

≤C₃n(dlogdlogn)⁵+C₄·Cd²n²logd dlogdlogn ,

whereC₄>0 is a constant and we recall thatC >0 is the constant from the definition ofn^∗. This implies the desired result.

Case2:d≥√ logn.

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In this case, we are going to employ Proposition2.6to boundG^∗(n^∗). The number of points which have distance at mostkto 0 is clearly 1+(2d)^k. Consequently, by Proposition2.6, we have that

G^∗ n^∗

≤

C₀+

3

k=1

C_kd⁻^k(2d)^k

+C₄d⁻⁴n^∗

≤C5+C₆(logd)n² d²

for some constants C5, C6>0. Since d²≥logn, this is clearly dominated by the right hand side of (2.16) (with a

large enough constant), which completes the proof in this case.

3. Coverage estimates

Now we turn to collect some preliminary lemmas for the proof of the general Theorem1.5.

Throughout, we assume thatG is a finite, connected, vertex transitive graph andX is a lazy random walk onG with transition matrix P and stationary measure π. For S⊆V (G), we letCS(t)be the set of vertices inS visited byX by timet and letUS(t)=S\CS(t)be the subset ofS whichX has not visited by timet. Recall the notation C(t)=CV (G)(t)andU(t)=UV (G)(t). As before, we will usePx,Exto denote the probability measure and expectation under whichX(0)=x. Likewise, we letPπ,Eπcorrespond to the case thatXis initialized at stationarity. The purpose of this section is to develop a number of estimates which will be useful for determining the amount of time required byXin order to cover subsetsSofV (G). We consider two different regimes depending on the size ofS. IfSis large, we will estimate the amount of time it takes forXto visitt_u(G)distinct vertices inS. IfSis small, we will estimate the amount of time it takes forXto visit 1/2 of the vertices inS.

3.1. Large sets

In this subsection, we will prove that the amount of time it takes forXto visitt^∗=tu(G)distinct elements of a large set of verticesS⊆V (G)by timet=constantt^∗/π(S)is stochastically dominated by a geometric random variable whose parameter depends ont^∗/t_rel(G). The main result is:

Proposition 3.1. AssumeXsatisfies part(B)of Assumption1.2with constantsK₂.LetS⊆V (G)consist of at least 2K2tu(G)/miny:y∼xG(x, y)elements forx, y∈V (G)adjacent and let

t=2(K2+2)tu(G)

π(S) .

There exists a universal constantC >0such that for everyx∈V (G),we have that Px

CS(t)≤t_u(G)

≤exp

−C tu(G) trel(G)

.

Recall that LS(t)=_t

s=01_{X(s)∈S} is the amount of time thatX spends inS up to timet. The proof of Propo- sition3.1consists of several steps. First we show thatLS(t)is large with high probability. This is done in Proposi- tion1.6, which we will deduce from [8], Theorem 1, shortly, and states that the probability thatLS(t)is less than 1/2 its mean is exponentially small int. Then, in order to show thatXvisits many vertices inS, we need to rule out the possibility ofXconcentrating most of its local time in a small subset ofS. This is accomplished in Lemma3.2. We now proceed to the proof of Proposition1.6.

Proof of Proposition1.6. We rewrite the event

LS(t)≤tπ(S) 2

= _t

s=0

f (Xs) > t

1−π(S)+π(S) 2

, (3.1)