Mean field conditions for coalescing random walks

8/9/2019 Mean field conditions for coalescing random walks

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Submitted to the Annals of Probability

arXiv: 1109.5684

MEAN FIELD CONDITIONS FOR COALESCING

RANDOM WALKS

By Roberto Imbuzeiro Oliveira,

IMPA

The main results in this paper are about the full coalescence timeCof a system of coalescing random walks over a finite graph G. Let-ting m(G) denote the mean meeting time of two such walkers, wegive sufficient conditions under which E [C] 2m(G) and C/m(G)has approximately the same law as in the mean field setting of alarge complete graph. One of our theorems is that mean field behav-ior occurs over all vertex-transitive graphs whose mixing times aremuch smaller than m(G); this nearly solves an open problem of Al-

dous and Fill and also generalizes results of Cox for discrete tori ind 2 dimensions. Other results apply to non-reversible walks andalso generalize previous theorems of Durrett and Cooper et al. Slightextensions of these results apply to voter model consensus times,which are related to coalescing random walks via duality.

Our main proof ideas are a strenghtening of the usual approxi-mation of hitting times by exponential random variables, which giveresults for non-stationary initial states; and a new general set of con-ditions under which we can prove that the hitting time of a unionof sets behaves like a minimum of independent exponentials. In par-ticular, this will show that the first meeting time among k randomwalkers has mean m(G)/

k2

.

1. Introduction. Start a continuous-time random walk from each ver-tex of a finite, connected graphG. The walkers evolve independently, exceptthat when two walkers meet ie. lie on the same vertex at the same time, they coalesce into one. One may easily show that there will almost surelybe a finite time at which only one walk will remain in this system. The firstsuch time is called the full coalescence timeforGand is denoted by C.

The main goal of this paper is to show that one can estimate the lawofC for a large family of graphs G, and that this law only depends on Gthrough a single rescaling parameter. More precisely, we will prove resultsof the following form: if the mixing timetGmix ofG (defined in Section 2) issmall, then there exists a parameter m(G)> 0 such that the law C/m(G)

Supported by aUniversalgrant and aBolsa de Produtividade em Pesquisafrom CNPq,Brazil.

AMS 2000 subject classifications: Primary 60K35; secondary 60J27Keywords and phrases: coalescing random walks, voter model, hitting times, exponen-

tial approximation

1


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2 R. I. OLIVEIRA

takes a universal shape. Slight extensions of these results will be used tostudy the so-called voter model consensus timeon G.

The universal shape of C/m(G) comes from a mean field computationover a large complete graph Kn. In this case the distribution ofC can becomputed exactly (cf. [4, Chapter 14]):

C

(n 1)/2=d

ni=2

Zi,

where:

(1.1) The Zis are independent andi 2, t 0 : P (Zi t) =et(i2).

In words, C is a rescaled sum of independent exponential random variables

with means 1/i2, 2 i n.The scaling factor (n 1)/2 is the expected meeting time of two indepen-dent random walks overKn, and we see that

C

(n 1)/2w

i2

Zi and E [C]

(n 1)/22 when ngrows.

This suggests the general problem we address in this paper:

Problem 1.1. Given a graphG, letm(G) denote the expected meetingtime of two independent random walks overG, both started from stationarity.Give sufficient conditons onG under whichC has mean-field behavior, that

is:

(1.2) Law (C/m(G)) Law

i2

Zi

,and

(1.3) E [C] m(G) E

i2

Zi

= 2m(G).A version of this problem was posed in Aldous and Fills 1994 draft [4,

Chapter 14] and much more recently by Aldous [2]. However, as far as weknow there are only two families of examples the problem has been fullysolved. Discrete tori G = (Z/mZ)d with with d 2 fixed and m 1were considered in Coxs 1989 paper [7]. More recently, Cooper, Frieze andRadzik [6] proved mean field behavior in large random d-regular graphs (d


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MEAN FIELD COALESCING R. W. 3

bounded). Partial results were also obtained by Durrett [8, 9] for certainmodels of large networks.

We note that mean-field behaviour is not universal over all large graphs.One counterexample comes from a sequence of growing cycles, where thelimiting law ofC was also computed by Cox [7]. Stars with n vertices arealso not mean field: C is lower bounded by the time the last edge of the staris crossed by some walker, which is about log n, whereas m(G) is uniformlybounded.

1.1. Results for transitive, reversible chains. Our results in this paperaddress (1.2) and (1.3) simultaneously by proving approximation boundsin L1 Wasserstein distance, which implies closeness of first moments (cf.Section 2.2).

The first theorem implies that mean field behaviour occurs whenever G is

vertex-transitive and its mixing time(defined in Section 2) is much smallerthan m(G). This nearly solves a problem posed by Aldous and Fill in [4,Chapter 14]. In their Open Problem 12, they ask for an analogous resultwith the relaxation time replacing the mixing time (more on this below).

The natural setting for this first theorem is that of walkers evolving ac-cording to the same reversible, transitive Markov chain (the definition ofCeasily generalizes to this case), wheretransitivemeans that for any two statesxand y one can find a permutation of the state space mapping x to y andleaving the transition rates invariant. Clearly, the standard continuous-timerandom walk on a vertex-transitive graph is transitive in this sense.

Notational convention 1.1. In this paper we will use b = O (a)in the following sense: there exist universal constants C , > 0 such that|a| |b| C|a|.

Theorem 1.1 (Mean field for transitive, reversible chains). Let Q bethe (generator of a) transitive, reversible, irreducible Markov chain over afinite state spaceV, with mixing time tQmix. Definem(Q) to be the expectedmeeting time of two independent continuous-time random walks overV thatevolve according to Q, when both are started from stationarity. Denote byCthe full coalescence time for walks evolving according to Q. Finally, define{Zi}

+i=2 as in (1.1). Then:

dWLaw C

m(Q)

, Law

i2

Zi

= O (Q) ln 1(Q)

1/6,


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4 R. I. OLIVEIRA

where:

(Q) tQmix

m(Q)anddW denotesL1 Wasserstein distance. In particular,

E [C] =

2 + O

(Q) ln

1

(Q)

1/6m(Q).

This result generalizes Coxs theorem [7] for (Z/mZ)d with d 2 andgrowingm. In this case, for any fixed d, the mixing time grows asm2 whereasm(G) m2 ln mford = 2 andm(G) md for largerd. The original problemposed by Aldous and Fill remains open, but we note that:

For transitive, reversible chains, the mixing time is at most a C ln |V|

factor away from the relaxation time, with C > 0 universal (this istrue whenever the stationary distribution is uniform). This means weare not too far off from a full solution;

Any counterexample to their problem would have to come from avertex-transitive graph with mixing time of the order ofm(G) andrelaxation time asymptotically smaller than the mixing time. To thebest of our knowledge, such an object is not known to exist.

1.2. Results for other chains. We also have results on coalescing randomwalks evolving according to arbitrary generators Q on finite state spaces V.Again, we only require that the mixing time tQmix ofQ be sufficiently small

relative to other parameters of the chain.

Theorem 1.2 (Mean field for general Markov chains). Let Q denote(the generator of) a mixing Markov chain over a finite setV, with uniquestationary distribution. Denote byqmax the maximum transition rate fromanyx V and bymax the maximum stationary probability of an element ofV. Letm(Q)denote the expected meeting time of two random walks evolvingaccording toQ, both started from. Finally, letC denote the full coalescencetime of random walks evolving inV according to Q. Then:

dW

Law

C

m(Q), Law

i2

Zi

= O

(Q) ln

1

(Q)ln4 |V|

1/6

where(Q) = (1 + qmax t

Qmix) max


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anddWagain denotesL1 Wasserstein distance. In particular,

E [C] = 2 + O(Q) ln 1(Q) ln4 |V|1/6 m(Q).

We note that this theorem does not imply Theorem 1.1: for instance, itdoes notwork for two-dimensional discrete tori. However, the well-knownformula for over graphs gives the following corollary:

Corollary 1.1 (Proof omitted). AssumeG is a connected graph withvertex set V, where each vertex x V has degree degG(x). Assume that (|V|1, 1) is such that:

maxxV degG(x)

|V|1xV degG(x) t

Gmix

|V|

ln4

|V| lnln |V|

.

Then

dW

Law Cm(G)

, Law

i2

Zi

= O 1 + ln(1/)lnln |V|

1/6.

This corollary suffices to prove mean field behavior over a variety of ex-amples, such as:

all graphs with bounded ratio of maximal to average degree and mixingtime at most of the order|V|/ ln5 |V|: this includes expanders [6] andsupercritical percolation clusters in (Z/mZ)d with d 3 fixed [5, 15];

all graphs with maximal degree |V|1 ( >0 fixed) and mixing timethat is polylogarithmic in |V|: this includes the giant component of atypical Erdos-Renyi graph Gn,d/n with d > 1 [10] and the models oflarge networks considered by Durrett [8, 9].

Let us briefly comment on the case of large networks. Durrett has es-timated m(G) in these models, and has proven results similar to ours fora bounded number of walkers. We do not attempt to compute m(G) here,which in general is a model-specific parameter. However, we do show thatmean field behavior for C follows from generic assumptions about net-works that hold for many different models. This is important because recentmeasurements of real-life social networks [11] suggest that known models oflarge networks are very innacurate with respect to most network character-istics outside of degree distributions and conductance. In fairness, coalescingrandom walks and voter models over large networks are not particularly re-alistic either, but at the very least we know that mean field behavior is not


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6 R. I. OLIVEIRA

an artifact of a particular class of models. We also observe that our Theo-rem 1.2 also works for non-reversible chains, eg. random walks on directed

graphs.

1.3. Results for the voter model. The voter model is a very well-knownprocess in the Interacting Particle Systems literature [13]. The configurationspace for the voter model is the power set OV of functions : V O, whereVis some non-empty set andO is a non-empty set of possible opinions. Theevolution of the process is determined by numbers q(x, y) (x, y V , x=y)and is informally described as follows: at rate q(x, y), node x copies ysopinion. That is, there is a transition at rate q(x, y) from any state : V O to the corresponding state xy, where:

xy(z) = (y), ifz = x;(z), for all otherz V\{x}.A classical duality result relates this voter model to a system of coalescingrandom walks with transition rates q(, ) and corresponding generator Q.More precisely, suppose thatV = {x(1), . . . , x(n)} and that (Xt(i))t0,1inis a system of coalescing random walks evolving according toQwithX0(i) =x(i) for each 1 i n.

Proposition 1.1 (Duality [4]). Choose 0 OV. Then the configura-tion

t: x(i) V 0(Xt(i)) O (1 i n)

has the same distribution as the statet of the voter model at timet, whenthe initial state is0. In particular, theconsensus time for the voter model:

inf{t 0 : i, j V, t(i) =t(j)}

satisfiesE [] E [C]< +.

Now assume that the initial state0 OV is random and that the randomvariables {0(x)}xV are iid and have common law which is not a pointmass. In this case one can show via duality that the law of the consensustimeis that ofCKn, whereK is aN-valued random variable independentof the coalescing random walks, defined by:

K= min{i N : Ui+1=U1}, where U1, U2, U3 . . . , are iid draws from

and for each 1 k n

Ck min{t 0 : |{Xt(i) : 1 i n}|= k}.


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Thus the key step in analyzing the voter model via our techniques is toprove approximations for the distribution ofCk. Theorems 1.1 and 1.2 imply

mean-field behavior forC = C1. A quick inspection of the proofs reveals thatthe same bounds for Wasserstein distance can be obtained for Ck for any1 k n. It follows that:

Theorem 1.3 (Proof omitted). Let V, O and be as above, and con-sider the voter model defined byV,O and by the generatorQ correspondingto transition rates q(x, y). Assume that the sequence {Zi}i2 is defined asin (1.1), and also that K has the law described above and is independentfrom theZi. Define(Q) and(Q) as in Theorems 1.1 and 1.2. Then theconsensus timefor this voter model satisfies:

dWLaw m(Q) , Lawi>KZi= O ((Q) ln(1/(Q)))1/6ifQ is reversible and transitive, and

dW

Law

m(Q)

, Law

i>K

Zi

= O

(Q) ln(1/(Q)) ln4 |V|

1/6,

otherwise.

1.4. Main proof ideas. Our proofs of Theorems 1.1 and 1.2 both startfrom the formula (1.1) for the terms in the distribution ofC over Kn. Cru-cially, each termZ

ihas a specific meaning: Z

iis the time it takes for a system

with i particles to evolve to a system with i 1 particles, rescaled by theexpected meeting time of two walkers. For i = 2, this is just the (rescaled)meeting time of a pair of particles, which is an exponential random variablewith mean 1. For i >2, we are looking at the first meeting time amoung

i2

pairs of particles. It turns out that these pairwise meeting times are inde-pendent; since the minimum ofk independent exponential random variableswith mean is an exponential r.v. with mean /k, we deduce that Zi isexponential with mean 1/

i2

.

The bulk of our proof consists of proving something similar for moregeneral chains Q. Fix some such Q, with state space V, and let Ci denotethe time it takes for a system of coalescing random walks evolving according

to Q to have i uncoalesced particles. Clearly, M C1 C2 is the meetingtime of a pair of particles, which is the hitting time of the diagonal set:

{(x, x) : x V}


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8 R. I. OLIVEIRA

by the Markov chain Q(2) given by a pair of independent realizations ofQ.More generally, M(i+1) =Ci Ci+1 is the hitting time of

(i+1) ={(x(1), . . . , x(i + 1)) : 1 i1 < i2 i + 1, x(i1) =x(i2)}.

The mean-field picture suggests that each M(i+1) should be close in distri-bution to Zi. Indeed, it is known that:

General principle:LetHA be the hitting time of a subset A of states.If the mixing timetQmix is small relative toE [HA], thenHAis approximatelyexponentially distributed.

This is a general meta-result for small subsets of the state space of aMarkov chain; precise versions (with different quantitative bounds) are proven

in [3, 1] when the chain starts from the stationary distribution. However, weface a few difficulties when trying to use these off-the-shelf results.

1. For each i, M(i+1) is the first hitting time of (i+1) after time Ci+1.The random walkers are not stationary at this random time, so weneed to do exponential approximation from non-stationary startingpoints.

2. In order to get Wasserstein approximations, we need better control ofthe tail ofM(i+1).

3. To prove thatZiandM(i+1)/m(Q) are close, we must show something

like that E

M(i+1)

E [M] /i+12

, ie that M(i+1) behaves like the

minimum of i+12 independent exponentials.4. Finally, we should not expect the exponential approximation to hold

when (i+1) is too large. That means that the big bang phase (to useDurretts phrase) at the beginning of the process has to be controlledby other means.

It turns out that we can deal with points 1 and 2 via a different kindexponential approximation result, stated as Theorem 3.1. This result willgive bounds of the following form:

(1.4) Px(HA > t) = (1 + o (1)) exp

t

(1 + o (1))E [HA]

as long as

tQmix = o (E [HA]) and Px

HA tQmix

= o (1) .

Notice that this holds even for non stationarystarting points x if the chainstarted fromx is unlikely to hit A before the mixing time. This is discussed


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in Section 3 below. We also take some time in that section to develop aspecific notion of near exponential random variable. Although this takes

up some space, we believe it provides a useful framework for tackling otherproblems. We note that a version of Theorem 3.1 for stationary initial statesresult is implicit in [1].

We now turn to point 3. The key difficulty in our setting is that, unlikeCox [7] or Cooper et al. [6], we do not have a good local description of

the graphs under consideration which we could use to compute E

M(i+1)

directly. We use instead a simple general idea, which we believe to be new,to address this point. Clearly, M(i+1) is a minimum of

i+12

hitting times.

Let us consider the generalproblem of understanding the law of:

HB = min1i

HBi where B=

i=1Bi,under the assumption thatE [HBi ] =does not depend oniwhen the initialdistribution is stationary (this covers the case ofM(i+1)). Assume also that(1.4) holds for all A {B, B1, B2, . . . , B}. Then the following holds for ina suitable range:

A {B, B1, B2, . . . , B} : P (HA E [HA]) .

Morally speaking, this means that E [HA] is the -quantile ofHA for all Aas above; this is implicit in [1] and is made explicit in our own Theorem 3.1.Now apply this to A= B, with replaced by /E [HB ], and obtain:

E [HB ]P (HB ) =P

i=1

{HBi }

.

If we can show that the pairwise correlations between the events {HBi }are sufficiently small, then we may obtain:

E [HB ]P

i=1

{HBi }

i=1

P (HBi ) =,

This gives:

E [HB ]

,

as if the times HB1, . . . , H B were independent exponentials. The reasoningpresented here is made rigorous and quantitative in Theorem 3.2 below.

Finally, we need to take care of point 4, ie. the big bang phase. Inthe setting of Theorem 1.2, we simply use our results on the coalescence


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10 R. I. OLIVEIRA

times for smaller number of particles, which seems wasteful but is enoughto prove our results. For the reversible/transitive case, we use a bound from

[14] which is of the optimal order. Incidentally, the differences in the boundsof the two theorems come from this better bound for the big bang phaseand from a more precise control of the correlations between meeting timesof different pairs of walkers.

1.5. Outline. The remainder of the paper is organized as follows. Sec-tion 2 contains several preliminaries. Section 3 contains a general discussionof random variables with nearly exponential distribution and our generalapproximation results for hitting times. In Section 4 we apply these resultsto the first meeting time among k particles, after proving some technicalestimates. Section 5 contains the formal definition of the coalescing randomwalks process and proves mean field behavior for a moderate initial number

of walkers. Finally, Section 6 contains the proofs of Theorems 1.1 and 1.2.Related results and open problems are discussed in the final sections.

2. Preliminaries.

2.1. Basic notation. We write N for non-negative integers and [k] ={1, 2, . . . , k} for any k N\{0}. Given a set S, we let |S| denote its cardi-nality. Moreover, for k N, we let:

S

k

{A S : |A|= k}.

Notice that with this notation:

|S|finite

S

k

=

|S|

k

|S|!

k! (|S| k)!.

We will often speak of universal constantsC >0. These are numbers thatdo not depend on any of the parameters or mathematical objects underconsideration in a given problem. We will also use the notation a= O (b)in the universal sense prescribed in Notational convention 1.1. In this waywe can write down expressions such as:

eb = 1 + b + O

b2

and ln

1

1 b

= b + O

b2

= O (b) .

Given a finite set S, we let M1(S) denote the set of all probability measuresoverS. Givenp, q M1(S), their total variation disance is defined as follows.

dTV(p, q)1

2

sS

|p(s) q(s)|= supAS

[p(A) q(A)],


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where p(A) =

aAp(a). For S not finite, M1(S) will denote the set ofall probability measures over the natural -field over S. For instance,

for S = R we consider the Borel -field, and for S = D([0, +), V) (seeSection 2.3.1 for a definition) we use the -field generated by projections.IfXis a random variable taking values over S, we let Law (X) M1(S)

denote the distribution (or law) ofX. Here we again assume that there is anatural-field to work with.

2.2. Wasserstein distance. TheL1 Wasserstein distanceis a metric overprobability measures overR with finite first moments, given by:

dW(1, 2) =

R

|1(x, +] 2(x, +]| dx (1, 2 M1(R)).

A classical duality result gives:

dW(1, 2) = supf:RR 1-Lipschitz

R

f(x)1(dx) R

f(x)2(dx)

.

Notational convention2.1. Whenever we compute Wasserstein dis-tances, we will assume that the distributions involved have first moments.This can be checked in each particular case.

Remark 2.1. IfZ1, Z2 are random variables, we sometimes write

dW(Z1, Z2) instead ofdW(Law (Z1) , Law(Z2)).

Note that:dW(Z1, Z2) =

R

|P (Z1 t) P (Z2 t) | dt.

Also notice that:|E [Z1] E [Z2] | dW(Z1, Z2).

This is an equality ifZ1 0 a.s. andZ2= C Z1 for some constantC >0:

(2.1) C R, dW(Z1, C Z1) =|C 1|E [Z1] ,

since|f(C Z1) f(Z1)| |C 1| Z1 for every1-Lipschitz functionf :R R.

We note here three useful lemmas on Wasserstein distance. These areprobably standard, but we could not find references for them, so we provideproofs for the latter two lemmas in Section A of the Appendix. The firstlemma is immediate.


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12 R. I. OLIVEIRA

Lemma 2.1 (Sum lemma for Wasserstein distance; proof omitted). Forany two random variablesX, Ywith finite first moments and defined on the

same probability space:dW(X, X+ Y) E [|Y|] .

For the next lemma, recall that, given two real-valued random variablesX, Y, we say that Xis stochastically dominated by Y, and write Xd Y,ifP (X > t) P (Y > t) for all t R.

Lemma 2.2 (Sandwich lemma for Wasserstein distance). Let Z, Z,Z+ and W be real-valued random variables with finite first moments andZ d ZdZ+. Then:

dW(Z, W) dW(Z, W) + dW(Z+, W).

Lemma2.3 (Conditional Lemma for Wasserstein distance). LetW1,W2,Z1, Z2 be real-valued random variables with finite first moments. AssumethatZ1 and Z2 independent and that W1 is G-measurable for some sub--fieldG. Then:

dW(Law (W1+ W2) , Law(Z1+ Z2))

dW(Law (W1) , Law(Z1)) + E [dW(Law (W2| G) , Law (Z2))] .

Remark2.2. Here we are implicitly assuming thatLaw (W2| G)is givenby some regular conditional probability distribution.

2.3. Continuous-time Markov chains.

2.3.1. State space and trajectories. LetV be some non-empty finite set,called the state space. We write D D([0, +), V) for the set of all paths:

: t 0 t V

for which there exist 0 = t0 < t1 < t2


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We also define the time-shift operators:

T :() D ( + T) D (T 0).

2.3.2. Markov chains and their generators. Let q(x, y) be non-negativereal numbers for each pair (x, y) V2 with x=y. Define a linear operatorQ: RV RV, which maps fRV to QfRV satisfying:

(Qf)(x)

yV\{x}

q(x, y)(f(x) f(y)) (x V).

It is a well-known result that there exists a unique family of probabilitymeasures{Px}xV with the properties listed below:

1. for all x V,Px(X0= x) = 1;

2. for all distinct x, y V, lim0Px(X=y)

=q(x, y);

3. Markov property: for any x V and T 0, the conditional law ofX T given FT under measure Px is given by PXT.

The family {Px}xV satisfying these properties is the Markov chain withgeneratorQ. We will often abuse notation and omit any distinction betweena Markov chain and its generator in our notation.

For M1(V), P denotes the mixture:

PxV

(x) Px.

This corresponds to starting the process from a random state distributedaccording to. Forx V or M1(V) andY :D Sa random variable,

we let Lawx(Y) or Law(Y) denote the law ofY under Px or P (resp.).

2.3.3. Stationary measures and mixing. Any Markov chain Q as abovehas at least one stationary measure M1(V); this is a measure such thatfor any T 0,

Law(X T) = Law(X) .

We will be only interested in mixing Markov chains, which are thoseQ witha unique stationary measure that satisfy the following condition:

(0, 1),T 0, x V : dTV(Law (x) XT, ) .

The smallest such T is called the -mixing time of Q and is denoted bytQmix(). By the Markov property and the definition of total-variation dis-

tance, we also have that for all (0, 1), all t tQmix(), all x V and alleventsS,

|Px(X t S) P(X S) | .


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14 R. I. OLIVEIRA

The specific value tQmix tQmix(1/4) is called themixing time ofQ. We note

that for all (0, 1/2):

(2.2) tQmix() C ln(1/) tQmix

whereC >0 is universal (this is proven in [12, Section 4.5] for discrete timechains, but the same argument works here).

2.3.4. Product chains. LettingQ be as above, we may consider the jointtrajectory ofk independent realizations ofQ:

X(k)t = (Xt(1), . . . , X t(k)) (t 0)

where each (Xt(i))t0 has lawPx(i). It turns out that this corresponds to a

Markov chain Q(k) on Vk with transition probabilities:

q(k)(x(k), y(k)) = q(x(i), y(i)), ifx(i)=y(i) j [k]\{i}, x(j) =y(j);0, otherwise.

Remark 2.3. In what follows we will always denote elements of Vk

(resp. M1(Vk) by symbols like x(k), y(k), . . . (resp. (k), (k), . . . .) We will

then denote the distribution of Q(k) started from x(k) or (k) by Px(k) orP(k). This is a slight abuse of our convention for the Q chain, but theinitial state/distribution will always make it clear that we are referring tothe product chain.

The following result on Q(k) will often be useful.

Lemma 2.4. AssumeQ is mixing and has (unique) stationary distribu-tion. ThenQ(k) is also mixing, and the product measurek is its (unique)stationary distribution. Moreover, the mixing times ofQ(k) satisfy:

(0, 1/2), tQ(k)

mix () tQmix(/k) C ln(k/) t

Qmix

withC >0 universal.

Proof sketch. Notice that the law ofX(k)T has a product form:

Lawx(k)

X(k)T

= Lawx(1)(XT) Lawx(2)(XT) Lawx(k)(XT) .

It is well-known (and not hard to show) that the total-variation distance

between product measures is at most the sum of the distances of the factors.This gives:

dTV(Lawx(k)

X(k)T

, k)

ki=1

dTV(Lawx(i)(XT) , ).


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The RHS is if each term in the sum is less than /k. This is achievedwhenT tQmix(/k); (2.2) then finishes the proof.

3. Nearly exponential hitting times.

3.1. Basic definitions. We first recall a standard definition: the expo-nential distribution with mean m > 0, denoted by Exp(m), is the uniqueprobabilty dstribution M1(R) such that, ifZis a random variable withlaw,

P (Z t) =et/m (t 0).

We write Z =d Exp(m) when Z is a random variable with Law (Z) =Exp(m).

Similarly, given m >0 as above and parameters >0, (0, 1), we say

that a measure M1(R) has distributionExp(m,,) if it is the law of arandom variableZ withZ0 almost surely and for all t >0:(1 ) e

t(1)m P

Z t (1 + )e t(1+)mWe will write = Exp(m,,) orZ=d Exp(m,,) as a shorthand for this.Notice that Exp(m,,) does not denote a single distribution, but rathera family of distributions that obey the above property, but we will mostlyneglect this minor issue.

Random variables with lawExp(m,,) will naturally appear in our studyof hitting times of Markov chains. We compile here some simple results aboutthem. The first proposition is trivial and we omit its proof.

Proposition3.1 (Proof omitted). If M1(R) satisfies

= Exp(m,,)

andm >0, (0, 1) are such that+ +


16/46


17/46


3.2. Hitting times are nearly exponential. In this section we consider amixing continuous-time Markov chain {Px}xV with generator Q, taking

values over a finite state space V, with unique stationary distribution .Given a nonemptyA V with(A)> 0, we define the hitting time ofA tobe:

HA() inf{t 0 : (t) A} ( D([0, +), V)).

The condition (A)> 0 ensures that Ex[HA]< +for all x V.Our first result in this section presents sufficient conditions on A and

M1(V) that ensure thatHA is approximately exponentially distributed.

Theorem3.1. In the above Markov chain setting, assume that0< <


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18 R. I. OLIVEIRA

Theorem 3.2. Assume that the setA considered above can be writtenas:

A=

i=1

Ai

where the setsA1, . . . , A are non-empty and

m:= E[HA1] =E[HA2 ] = = E[HA] .

Assume0<


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In particular, this implies:

(3.1) M1(V), E[HA] = +

0 P(HA t)dt (1 + O ())

t(A)

.

It turns out that the upper bound in the above Lemma can be nearly reversedif we start from some distribution that is far from A.

Lemma 3.3 (Proven in Section 3.3.2). With the assumptions of Theo-rem 3.1, if2 + r < 1/2:

t 0, P(HA t) (1 O () r)+ e(1+O()) t

t(A) .

Notice that the combination of these two lemmas already implies the firststatement in the proof, as it shows that for all t 0,

P(HA t) [(1 O () 2r) e t

(1+O()) t(A) , (1 + O ()) e t

(1+O()) t(A) ].

To see this, notice that the upper bound is always valid by Lemma 3.2. Forthe lower bound, we use Lemma 3.3 if 2 + r 1/2, and note that the lowerbound is 0 if 2 + r>1/2 and the constant in the O () term is at least 4.

We now prove the assertion about expectations in the Theorem. We useLemma 3.1 and deduce:E[HA] t(A)

dW(Law(HA) ,Exp(1 t(A))) O (+ r)

t(A)

and the assertion follows from dividing by 1 t(A) and noting that

r =P

HA tQmix()

by assumption. The final assertion in the Theorem then follows from Propo-sition 3.1.

3.3.1. Proof of Lemma 3.2.

Proof. Set T = tQmix(). We note for later reference that T < t(A),since

P(HA T) < = P(HA t(A)) .

Our main goal will be to show the following inequality:

(3.2) k N,P(HA> (k+ 1)t(A)) (1 + 2) P(HA > kt(A)) .


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20 R. I. OLIVEIRA

Once established, this goal will imply:

k N,P

(HA

kt(A)) (1 + 2)k

and

t 0,P(HA t) e(1+O())

tt(A)

= (1 + O ()) e

t(1+O())t(A) ,

which is the desired result. To achieve the goal, we fix some k N and useT t(A) to bound:

P(HA > (k+ 1)t(A)) P

HA > kt(A),HA kt(A)+T > t(A) T

(Markov prop.) = P(HA > kt(A)) P(HA > t(A) T)

where is the law ofXkt(A)+Tconditioned on {HA > kt(A)}. Since this

event belongs to Fkt(A)and T =tQmix(), is-close to in total variation

distance. We deduce:

(3.3) P(HA > (k+ 1)t(A))

P(HA> kt(A)) P(HA > t(A) T) + .

Now observe that:

P(HA > t(A) T) P(HA > t(A))

+P(HA (t(A) T, t(A)])

P(HA > t(A))

+P

HA t(A)T T

(defn. oft(A)) = 1 + P

HA t(A)T T

( stationary) = 1 + P(HA T)

(T =tQmix() + assumption) 1 + .

and plugging this into (3.3) gives:

P(HA > (k+ 1)t(A))

P(HA > kt(A)) (1 (1 2)),

as desired.


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3.3.2. Proof of Lemma 3.3.

Proof. The general scheme of the proof is similar to that of Lemma 3.2,but we will need to be a bit more careful in our estimates. In particular, wewill need that (1 + 5)


22/46

22 R. I. OLIVEIRA

where is the conditional law ofXkt(A)+T given HA kt(A). Since T =

tQmix(), is within distance from . We deduce:

(3.6) (I) P(HA kt(A)) (P(HA t(A)) ) =f(k) (1 ).

We now upper bound term (II) in (3.5). Notice that (again because of theMarkov property):

(II) P

HA (k 1)t(A), HA kt(A) < T

= f(k 1) P(HA< T)

where is the law ofXkt(A)conditioned on{HA (k 1)t(A)}. Recalling

thatt(A) T =tQmix(), we see that

is-close to . Since we have alsoassumed that P(HA T) , we deduce:

(II) f(k 1) (P(HA< T) + ) 2 f(k 1).

We combine this with (3.6) and (3.5) to obtain:

k N\{0, 1} f(k+ 1) f(k)(1 ) f(k 1)(2).

One can argue inductively that f(k)/f(k 1) 1/2 for all k 1. Indeed,this holds for k 2 by the Claim applied to k 1. For k = 1 we may use(3.4) and the assumption on 2+ r to deduce the same result. Applyingthis to the previous inequality we obtain

k N\{0} f(k+ 1) f(k)(1 5),

which finishes the proof of the Claim and of the Lemma.

3.4. Hitting times of a union of sets: proofs. We present the proof ofTheorem 3.2 below.

Proof of Theorem 3.2. There are three main steps in the proof, hereoutlined in a slightly oversimplified way.

1. We show that Theorem 3.1 is applicable to the hitting times of A1,. . . ,A. In particular, this shows that P(HAi m) .

2. We show that

P(HA m)

i=1

P(HAi m) ,

so that t(A) m.


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3. Finally, we apply Theorem 3.1 to HA and deduce that this randomvariable is approximately exponential with mean:

E[HA] t(A)/ m/.

The actual proof is only slightly more complicated than this outline. Webegin with a claim corresponding to step 1 above.

Claim3.2. For all1 i ,

i P(HAi m) = (1 + O ()) .

Proof of the Claim. Consider some [/2, 2]. Notice that

tQ

mix

() tQ

mix

(/2)

and therefore:

P

HAi tQmix(

)

2 .

This shows that Theorem 3.1 is applicable with Ai replacing A and re-placing. We deduce in particular that:

2 2,

E[HAi ]t(Ai) 1 O + = O () .

In particular, there exists a universal constant c > 0 such that if (1 c) , thent(Ai)< E[HAi ], whereas if

>(1 +c) ,t(Ai)> E[HAi ].

In other words,

(1 c) P(HAi E[HAi ]) (1 + c).

We now come to the second part of the proof.

Claim3.3. Letbe as in the statement of Theorem 3.2. Then:

P(HA m) = (1 + O (+ )) .

In particular, there exists a number = (1 + O (+ )) with m= t(A).


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24 R. I. OLIVEIRA

Proof. To see this, we note that:

{HA m}=

i=1

{HAi m}.

The union bound gives:

P(HA m)

i=1

P(HAi m) (1 + O ()) .

A lower bound can be obtained via the Bonferroni inequality:

P(HA m)

i=1

P(HAi m)

1i


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4. Meeting times of multiple random walks. We now put our twoexponential approximation results to use, showing that the meeting times

we are interested in are well-approximated by exponential random variables.Much of the work needed for this is contained in technical estimates whoseproofs can be safely skipped in a first reading.

4.1. Basic definitions. For the remainder of this section, V is a finiteset and Q is the generator of a mixing Markov chain over V with mixingtimes tQmix() and stationary measure . For each kN\{0, 1} we will alsoconsider the Markov chains Q(k) over Vk that correspond to k independentrealizations ofQ from prescribed initial states, as defined in Section 2.3.4.We will also follow the notation from that section.

For k= 2, we define the first meeting time:

(4.1) Minf{t 0 : Xt(1) =Xt(2)}

and the parameters:

m(Q) E2[M] ,(4.2)

(Q) tQmixm(Q)

.(4.3)

We also define an extra prametererr(Q) which will appear as an error termat several different points in the paper. Ths parameter err(Q) is defined as

(4.4) err(Q) =c0(Q)ln(1/(Q)) ifQ is reversible and transitive.For otherQ, we define it as:(4.5) err(Q) =c1

(1 + qmaxtQmix) max ln

1

(1 + qmaxtQmix) max

.

The numbers c0, c1 > 0 are universal constants that we do not specify ex-plicitly. We choose them so as to satisfy Propositions 4.1, 4.4 and 4.5 below.

We now take k >2 and consider the process Q(k), with trajectories

(X(k)t = (Xt(1), Xt(2), . . . , X t(k)))t0

corresponding to k independent realizations of Q (cf. Section 2.3.4). Thishas stationary distribution k.

We write M(k) for the first meeting time among these random walks:

(4.6) M(k) inf{t 0 :1 i < j k, Xt(i) =Xt(j)} .


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26 R. I. OLIVEIRA

One may note thatM(k) = min

{i,j}([k]2)Mi,j

where[k]2

was defined in Section 2 and, for 1 i < j k:

(4.7) Mi,j =Mj,i inf{t 0 : Xt(i) =Xt(j)}

is distributed as Mfor a realization ofQ(2) starting from (X0(i), X0(j)).

4.2. Technical estimates for reversible and transitive chains. In this sub-section we collect the estimates that we will use in the case of chains thatare reversible and transitive.

Proposition 4.1. Assume Q is reversible and transitive and define

err(Q) accordingly. Iferr(Q) 1/4, then:

P2

MtQmix(err(Q)2)

err(Q)2.

Remark 4.1. The proof is entirely general, but we will only use thisestimate in the transitive/reversible case.

Proof. We will prove a result in contrapositive form: if 0 < ,

then < c20(Q)ln(1/(Q)) for some universal c0> 0.Notice that for any x(2) V2,

Px(2)

M > tQmix(/4) + tQmix()

Px(2)

M

tQmix(/4)> tQmix()

P2

M > tQmix()

+ /2

< (1 /2),

where the middle inequality follows from the fact that tQmix(/4) is an upperbound for the /2-mixing time ofQ(2) (cf. Lemma 2.4). A standard argumentusing the Markov property implies that for any k N,

Px(2) M > k (tQmix(/4) + tQmix())< (1 /2)k,so that:

m(Q) =E2[M] CtQmix() + t

Qmix(/4)

.


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SincetQmix() Cln(1/) tQmix, we deduce that:

c ln(1/) < (Q),

with c >0 universal, which implies the desired result.

We now prove an estimate on correlations.

Proposition 4.2. Assume Q is transitive. Then for all t, s 0 and{i, j}, {, r} V with{i, j} ={r, },

Pk(Mi,j t, M,r s) 2P2(Ms) P2(Mt) .

Proof. If{i, j} {, r} = , the events {Mi,j t} and {M,r t} are

independent. Since the laws of both Mi,j and M,r under k are equal tothe law ofM under 2, we obtain:

Pk(Mi,j t, M,r s) P2(Ms) P2(Mt)

in this case. Assume now {i, j} {, r} has one element. Without loss ofgenerality we may assume k = 3, {i, j} = {1, 2} and {, r} = {1, 3}. Wehave:

P3(M1,2 t, M1,3 s) P3

M1,2 t, M1,3 M1,2 s

+P3 M1,3 s, M1,2 M1,3 t .(4.8)Consider the first term in the RHS. By the Markov property:

P3

M1,2 t, M1,3 M1,2 s

= P2(Mt)P(2)(Ms)

where(2) is the law ofXM1,2(1), XM1,2(3) conditionally on M1,2 t. Since

(Xt(3))tis stationary and independent from this event, (2) = for some

M1(V) which is the law ofXM1,2(1) under P2. The transitivity ofQ(which implies that is uniform) implies that = and therefore:

P3

M1,2 t, M1,3 M1,2 s

= P2(Mt) P2(Ms) .

The same bound can be shown for the other term in the RHS of (4.8), andthis implies the Proposition.


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28 R. I. OLIVEIRA

4.3. Technical estimates for the general case. We will need the followinggeneral result:

Proposition4.3. For any M1(V) andT 0

P(MT) (1 + 2T qmax) max.

Proof. Let (Xt)t be a single realization ofQ. One may imagine that thetrajectory of (Xt)t0 is sampled as follows. First, let Pbe a Poisson processwith intensity qmax independent from the initial state X0. At each timet P, one updates the value ofXt as follows: ifXs =x for s immediatelybefore t, one sets:

Xt= y with probability q(x, y)

qmax

(y V\{x})

andXt = x with the remaining probability. This implies that, at the pointsof the Poisson process, Xt is updated as in the discrete-time Markov chainwith matrixP = (I+ Q/qmax), and it is easy to see that is stationary forthis chain.

Now letXt(1), Xt(2) be independent trajectories ofQ, withXt(1) startedfromandXt(2) started from the stationary distribution . We will imaginethat each Xt(i) has its own Poisson process P(i) and was generated in theway described above. It then follows that:

P(MT | P(1), P(2)) P(X0(1) =X0(2))

+tP

P(Xt(1) =Xt(2)| P(1), P(2)) ,

where P = P(1) P(2), since the processes can only change values at thetimes of the two Poisson processes. At time 0 we have:

P(X0(1) =X0(2)) =xV

(x)(x)xV

(x)max max.

For t P, the law ofXt(1), Xt(2) equals:

( Pk1) ( Pk2)

where ki = |P(i) (0, t]| (i = 1, 2). Crucially, is stationary for P, hence Pk2 = and we obtain:

P(Xt(1) =Xt(2)| P(1), P(2)) =xV

( Pk1)(x)(x) max


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as for t = 0. We deduce:

P

(MT | P(1), P(2)) (1 + |(P(1) P(2)) (0, T]|) max

.

The Proposition follows from taking expectations on both sides and noticingthat:

E [|(P(1) P(2)) (0, T]|] = 2T qmax.

We now prove an estimate corresponding to Proposition 4.1 in this generalsetting.

Proposition4.4. Assumeerr(Q) is as defined in (4.5). Then:

P2 MtQmix(err(Q)2) err(Q)2.Proof. The previous proposition implies:

P2

MtQmix(err(Q)2)

(1 + 2tQmix(err(Q)) qmax) max

C(1 + 2tQmix qmax) max ln(1/err(Q)).

This is err(Q)2 by definition of this quantity, if we choose c1 in (4.5) tobe large enough.

We now prove an estimate on correlations that is similar to Proposi-

tion 4.2, but with an extra term. Recall that [k]2 was defined in Section 2.1.Proposition4.5. For any mixing Markov chainQ, if one defineserr(Q)

as in (4.5), we have the following inequality fork 3 and all distinct pairs{i, j}, {, r}

[k]2

:

Pk(Mi,j t, M,rs) 2P2(Mt) P2(Ms) + Oerr(Q)2

.

Proof. The case {i, j} {, r} = follows as in the proof of Proposi-tion 4.2. In case {i, j} {, r} has one element, we may again assume thati = = 1, j = 2 and k = 3. Equation (4.8) still applies, so we proceed to

bound: P3

M1,2 t, M1,3 M1,2 s

,


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30 R. I. OLIVEIRA

which is upper bounded by:

P3 M1,2 t, M1,3 M1,2 s

P3

M1,2 t, M1,3 M1,2 tQmix()

+ P3

M1,2 t, M1,3 M1,2+tQmix()

s

= (I) + (II)

for some (0, 1/4) to be chosen later.Term (I) is equal to:

P3(M1,2 t) P(2)

MtQmix()

where (2) is the law of (XM1,2(2), XM1,2(3)) conditionally on {M1,2 t}.As in the previous proof, (Xt(3))t is stationary and independent from the

conditioning, hence(2) =for some M1(V). We use Proposition 4.3to deduce:

(I) P3(M1,2 t)O

(1 + tQmix() qmax) max

The analysis of term (II) is simpler: we have

(II) =P3(M1,2 t) P(Ms)

for some M1(V) which is the law of XM1,2+tQmix() conditionally on

{M1,2 t}. The time shift bytQmix() implies thatis-close to stationary,

hence:

(II) P3

(M1,2 t) (+ P2

(Ms)).We deduce that:

P3

M1,2 t, M1,3 M1,2 s

P2(Mt) P2(Ms)

+ P2(Mt) O

+ (1 + tQmix() qmax) max

.

RecalltQmix() C tQmix ln(1/) for some universalC >0. If

0 (1 + qmax tQmix) max 1/2

we may take = 0 to obtain:P3

M1,2 t, M1,3 M1,2 s

P2(Mt) P2(Ms)

+P2(Mt) O

ln

1

.


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The case of0 1/2 is covered automatically by the big-oh notation.An analogous bound can be obtained with the roles of (t, 2) and (s, 3)

reversed. Plugging these into (4.8) gives the desired bound.

4.4. Exponential approximation for a pair of particles. We now comeback to the setting of Section 4.1 and show Mis approximatelly exponen-tially distributed.

Lemma 4.1. Defineerr(Q)as in (4.4) (ifQ is reversible and transitive)or as in (4.5) (if not). Then(2) M1(V(2)):

Law(2)(M) =Exp(m(Q), O (err(Q)) + 2r(2), O (err(Q)))

wherer(2) =P(2) Mt

Qmix(err(Q)

2) .Proof. This is a direct application of Theorem 3.1 to the hitting time

of the diagonal set {(x, x) : x V} V2

by the chain with generator Q(2) defined in Section 2 and with = err(Q),= 2err(Q). All we need to show is that:

P2

MtQ(2)

mix()

wheretQ(2)

mix() denotes the mixing times ofQ(2). This inequality follows from

tQ(2)

mix(2err(Q)2

) tQ

mix(err(Q)2

) (Lemma 2.4)

andP2

MtQmix(err(Q)

2)

err(Q)2 < ,

which follows from Proposition 4.1 in the reversible/transitive case andProposition 4.4 in the general case.

4.5. Exponential approximation for many random walkers. We now con-sider the more complex problem of bounding the meeting times among k 2particles. We take the notation in Section 4.1 for granted.

Lemma 4.2. Let = k2> 0 and assume that the quantityerr(Q)definedin (4.4) (if Q is reversible and transitive) or as in (4.5) (if not) satisfies

err(Q) 1/10. Then for all(k) M1(Vk),

Law(k)

M(k)

= Exp

m(Q)

, O

k2err(Q)

+ 2r(k) , O

k2err(Q)


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32 R. I. OLIVEIRA

wherer(k) =P(k)

M(k) tQmix(err(Q)2)

.

Proof. M(k)

is the hitting time of a union of sets.(k)

{i,j}(k2)

{i,j} where {i,j} {x(k) Vk : x(k)(i) =x(k)(j)}.

We will apply Theorem 3.2 applied to the product chain Q(k) to show thatthis hitting time is approximatelly exponential. We set = 2err(Q), =err(Q) and verify the conditions of the Theorem:

0 < < 1/5, 0 < < /2: These conditions follow from err(Q) t, where j is the first time t at which Xt(i) =Xt(j) for somei < j with i At.

Intuitively,Atis the set of all walkers that are alive at timet 0, and awalker dies at the first time it meets an alive walker with smaller index. Onemay check that coalescing random walks is equivalent to the killed processin the following sense.

Proposition5.2 (Proof omitted). We havej =Tj for allj [k]\{1}.Moreover, for allt 0 we have:

{Xt(j) : j At}= {Xt(j) : j [k]}.

Finally, for alli [k 1]

Ci= inf{t 0 : |At| i}.

Recall that Mi,j is the meeting time between walkers i and j (cf. (4.7)).We have the following simple proposition.

Proposition5.3 (Proof omitted). Assume that the initial state

x(k) = (x(1), x(2), . . . , x(k))

is such thatx(i)=x(j) for all1 i < j k. Then for each1 p k 1,

Cp Cp+1= min{i,j}ACp+1

Mi,j Cp+1.

Moreover, each timeCp equalsMi,j for some{i, j} [k]

2

.

5.2. Mean-field behavior for moderately largek. We now prove a mean-field-like result for an initial number of particles k that is not too large,assuming that meeting times of up to k walkers satisfy our exponentialappoximation property.

Lemma 5.1. Assume that Q, err(Q) and k satisfy the assumptions ofLemma 4.2. Letx(k) Vk. Then for allp [k 1]

dW

Lawx(k) Cpm(Q)

, Law

ki=p+1

Zi

= O k2err(Q) + 12 (x(k))p

,


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where

(x

(k)

) =Px(k) M(k) tQmix(err(Q)2)+ Px(k)

{i, j}, {, r}

[k]2

:

{, r} ={i, j} but Mi,j M,r tQmix(err(Q)

2)

and theZi are the random variables described in (1.1).

Proof. Write x(k) = (x(1), . . . , x(k)). We will prove the similar bound:

(5.1) 1 i < j k : x(i)=x(j)

dW

Lawx(k)

Cp

m(Q)

, Law

k

i=p+1Zi

=O

k2err(Q)

+ 4 (x(k))

p .

To see how this implies the general result, consider somex(k) such that someof its coordinates are equal, so that in particular (x(k)) 1. One still hasthe trivial bound:

dW

Lawx(k) Cpm(Q)

, Law

ki=p+1

Zi

Ex(k) Cpm(Q)

+ E

ki=p+1

Zi

.The second term in the RHS is 2/p. For the first term, letj be the numberof distinct coordinates ofxand

y(j) = (y(1), . . . , y(j)) Vj

have distinct coordinates with:

{y(1), . . . , y(j)}= {x(1), . . . , x(k)}.

Then clearly:

Ex(k)

Cp

m(Q)

= Ey(j)

Cp

m(Q)

.

Ifp j the RHS is 0. If not, it can be upper bounded using the bound in(5.1):

Ey(j) Cpm(Q) E k

i=p+1Zi+ dWLawy(j) Cpm(Q) , Law

ji=p+1

Zi

2 + 4(y(j)) + O

k2err(Q)

p .


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36 R. I. OLIVEIRA

Since(x(k)) 1 (y(j))/2 in this case, we obtain

dWLawx(k) Cpm(Q) , Lawk

i=p+1Zi 12 (x(k)) + O k2err(Q)p

for such x(k) with repetitions, which gives the Lemma in general.We prove (5.1) by reverse induction on p. The case p = k 1 is trivial:

Ck1 is simply M(k) and (x(k)) is an upper bound for r

x(k), so we may

apply Lemma 4.2 to deduce the desired bound.For the inductive step, consider p0< k 1 and assume the result is true

for all p0 < p k 1. We will use the easily proven fact that Cp0+1 is a

stopping time for the process (X(k)t )t0 process. Consider the corresponding

-field FCp0+1. We will apply Lemma 2.3 with:

Z1 =k

i=p0+2

Zi,

Z2 = Zp0+1,

W1 = Cp0+1

m(Q),

W2 = Cp0 Cp0+1

m(Q) =

min{i,j}ACp0+1Mi,j Cp0+1

m(Q) ,

G = FCp0+1.

(We used Proposition 5.3 to obtain the second expression for W2 above.)

Applying Lemma 2.3 in conjunction with the induction hypothesis gives:

(5.2) dW

Lawx(k) Cp0m(Q)

,k

i=p0+1

Zi

O k2err(Q) + 4(x(k))p0+ 1

+ Ex(k)

dW

Lawx(k)

min{i,j}ACp0+1

Mi,j Cp0+1

m(Q)

FCp0+1

,Zp0+1

.

Note thatACp0+1 is FCp0+1-measurable. The strong Markov property forQ(k)

implies that

Lawx(k)

min{i,j}ACp0+1Mi,j Cp0+1

m(Q) FCp0+1is the same as

LawX

(k)Cp0+1

min{i,j}ACp0+1

Mi,j

m(Q)

.


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Now define Y(p0+1) as the vectors whose coordinates are the p0+ 1 distinctpoints XCp0+1(i) with i Ap0+1 (the order of the coordinates does not

matter). Clearly,

(5.3) LawX

(k)Cp0+1

min{i,j}ACp0+1

Mi,j

m(Q)

= LawY(p0+1)

M(p0+1)

m(Q)

.

By Lemma 4.2, this last law is approximately exponential,

Exp

1p0+12

, O k2err(Q) + 2 rY(p0+1)

, O

k2 err(Q)

,

and Lemma 3.1 gives:

dW

LawY(p0+1)

M(p0+1)

m(Q)

,Zp0+1

O

k2err(Q)

+ 4rY(p0+1)

p0(p0+ 1) .

Using the definition ofrY(p0+1)

, we obtain from (5.2) the following inequal-ity:

(5.4) dW

Lawx(k) Cp0m(Q)

,k

i=p0+1

Zi

O k2err(Q) + 4(x(k))p0+ 1

+O

k2err(Q)

+ 4Ex(k)

PY(p0+1)

M(p0+1) tQmix(err(Q)2)

p0(p0+ 1) .

To finish, we need to show that the expected value in the RHS is (x(k)).For this we recall (5.3) to note that

PY(p0+1) M(p0+1) tQmix(err(Q)2)= P

X(k)Cp0+1

min

{i,j}ACp0+1

Mi,j tQmix(err(Q)

2)

= Px(k)Cp0 Cp0+1 t

Qmix(err(Q)

2)| FCp0+1

where the last line uses Proposition 5.3 and the strong Markov property.Averaging shows that the expectation in the RHS of (5.4) is:

Px(k)Cp0 Cp0+1 t

Qmix(err(Q)

2)

,

and Proposition 5.3 implies that this is at most:

Px(k) {i,j}={,r}

{Mi,j M,r tQmix(err(Q)

2)} .

Since the RHS is (x(k)), we are done.


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38 R. I. OLIVEIRA

6. Proofs of the main theorems.

6.1. The full coalescence time in the transitive case. In this section weprove Theorem 1.1.

Proof of Theorem 1.1. Recall that C= C1 by definition. Lemma 5.1gives the following bound for any k

1/4err(Q) |V| andx(k) Vk

dW

Lawx(k)

C

m(Q)

,

ki=2

Zi

12 (x(k)) + O

k2err(Q)

.

Notice that:

dW

k

i=2Zi,

+

i=2Zi

E

jk+1

Zj

= 2

k+ 1,

hence:

dW

Lawx(k)

C1

m(Q)

,+i=2

Zi

= 12 (x(k)) + O

k2err(Q) +

1

k

.

Convexity ofdW implies:

Proposition6.1. Under the assumptions of Theorem 1.1, the followingholds forerr(Q) 1/4, 1 k

1/4err(Q) |V| and(k) M1(V

k):

dWLaw(k) C1

m(Q) ,

+

i=2 Zi 12 (x(k)) d(k)(x(k))(6.1)

+O

k2err(Q) +

1

k

.

Notice that our control ofC1gets worse ask increases, and we cannot usethe above bound to approximate the law ofC1 started with one particle ateach vertex ofV. What we use instead is a truncation argument combinedwith the Sandwich Lemma for dW(Lemma 2.2 above). For this we need tofind two random variables

Cd C1d C+

such that both C/m(Q) and C+/m(Q) are close to+i=2 Zi. More specifi-cally, we will show that:dW

Cm(Q)

,i2

Zi

= O k2 err(Q) + k4err(Q)2 + 1k+ 1

+ (Q) ln(1/(Q))

.


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Before we continue, let us show how this last bound implies our result. TheSandwich Lemma 2.2 gives:

dW C1m(Q)

,i2

Zi= O k2 err(Q) + k4err(Q)2 +1

k+ (Q) ln

1

(Q)

.

Since(Q)ln(1/(Q)) =O (err(Q)), we may choosek = (err(Q))1/3 (whichworks for err(Q) sufficiently small) to obtain:

dW

C1m(Q)

,i2

Zi

= O err(Q)1/3 ,and this is precisely the bound we seek because

err(Q) =O (Q)ln(1/(Q)) .We now construct C,C+ and prove that they have the required properties.

Construction of C: pick x(1), . . . , x(k) V from distribution , in-dependently and with replacement. Let C denote the full coalescence timefor k walkers started from these positions. This might be degenerate: theremight be more than one walker starting from some element ofV, but thisonly means those particles will coalesce instantly.

Clearly, Cd C1. Moreover,

Law

C

m(Q)

= Lawk

C1

m(Q)

.

Therefore by Proposition 6.1,

dW

Law

C

m(Q)

,+i=2

Zi

= dW

Lawk

C1

m(Q)

,+i=2

Zi

= O

(x(k)) dk + k2 err(Q) +

1

k

.

Notice that the integral in the RHS is at most: (x(k)) dk

{i,j}([k]2)

Pk

Mi,j tQmix(err(Q)

2)

(6.2)

+ {i,j},{,r}([k]2) :{i,j}={,r}

Pk Mi,j M,rtQmix(err(Q)2) = O

k4err(Q)2

.


40/46

40 R. I. OLIVEIRA

as can be deduced from the proofs of Propositions 4.2 and 4.1. We concludethat:

dW Law Cm(Q) , Lawi2Zi(6.3)= O

k4err(Q)2 + k2 err(Q) + 1k

.

Construction ofC+: we will use the following simple stochastic domi-nation result, which we describe in the language of the process with killings.Let be stopping times for the X(k) process. If all killings are supressedbetween time and , the resulting full coalescence time C+ stochasticallydominatesC1. We will use this result, whose proof we omit, with the follow-ing choice of and :

=Ck

and= Ck

+ tQ

mix(err(Q)2).

Lemma 2.1 implies:

dW

C+

m(Q),C1 m(Q)

E []

m(Q)=

E [Ck]

m(Q) +

tQmix(err(Q)2)

m(Q) .

Since Q is transitive, m(Q), can be bounded from below in terms of themaximal hitting time in Q [4, Chapter 14]. Theorem 1.2 in [14] implies:

E [Ck]Cm(Q)

k + C tQmix

for some universalC >0. Recalling the definition of(Q) in (4.3), we obtain:

E [Ck]

m(Q) =O

1

k+ (Q)

.

Moreover, we also have

tQmix(err(Q)2) =O

ln(1/err(Q)) tQmix

= O

tQmixln(1/(Q))

,

hence:

dW

C+

m(Q),C1 m(Q)

= O

1

k+ (Q) ln(1/(Q))

.

This shows:

dW

C+

m(Q),

ki=2

Zi

= O

1

k+ (Q) ln(1/(Q))

+ dW

C1 m(Q)

,k

i=2

Zi

.


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Now consider the time C1 . Since all killings were supressed betweentimes=Ck and = Ck+ t

Qmix(err(Q)

2), there are k alive particles at time

. Letting (k)

denote their law, we have:

Law

C1 m(Q)

= Law(k)

C1

m(Q)

and Proposition 6.1 implies:

dW

C1 m(Q)

,k

i=2

Zi

= O

(x(k)) d(k)(x(k)) + k2err(Q) +

1

k

Now observe that

tQmix(err(Q)2) tQ

(k)

mix (kerr(Q)2) (cf. Lemma 2.4),

hence the law of the k particles at time Ck+ tQmix(err(Q)

2) is kerr(Q)2-closeto stationary, irrespective of their states at time Ck. We deduce that

(k) iskerr(Q)2-close to stationary, and:

dW

C1 m(Q)

,k

i=2

Zi

= O

(x(k)) dk + k2err(Q) + kerr(Q)2 +

1

k

.

The integral in the RHS was estimated in (6.2), and we deduce:

dW

C1 m(Q)

,k

i=2Zi

= O

k2 err(Q) + k4err(Q)2 +

1

k

and we deduce:

dW

C+

m(Q),

ki=2

Zi

= O


1

k+ (Q) ln(1/(Q))

.

6.2. The general setting. We now come to the proof of Theorem 1.2.

Proof of Theorem 1.2. The proof is essentially the same as in thereversible/transitive case, but with the definition oferr(Q) given in (4.5). In

particular, we can still use the same definition ofC used in that proof toobtain

(6.4) dW

C

m(Q),+i=2

Zi

= O


1

k

.


42/46

42 R. I. OLIVEIRA

We will need a different strategy in the analysis ofC+, where we need tobound E [Ck] by different means. Note that Ck t if and only if there exist

distincty(1), . . . , y(k) V such that there isno coalescenceamong the walk-ers started from these vertices. The probability of this no coalescence event

for a given choice of y(i)s is Py(k)

M(k) t

for y(k) = (y(1), . . . , y(k)).

Therefore,

P (Ck t)

y(k)Vk

Py(k)

M(k) t 1.

By Lemma 4.2, each term in the RHS satisfies:

Py(k)

M(k) t

C e

t(k2)(1+O(k2err(Q)))m(Q)

for some universalC >0. Since there are |V|k terms in the sum, we have:

P (Ck t)

C|V|k e t (k2)(1+O(k2err(Q)))m(Q) 1.

Integrating the RHS gives:

E [Ck]

m(Q) C

ln |V|

k

for a potentially different, but still universal C. Going through the previousproof, we see that this gives:(6.5)

dW

C+

m(Q),

ki=2

Zi

= O


ln |V|

k +

tQmix(err(Q)2)

m(Q)

.

To continue, we bound the term containingtQmix(err(Q)2) in terms oferr(Q)

(this was easier before because of the different definition oferr(Q)). Recallfrom Proposition 4.4 that:

P2

MtQmix(err(Q)2)

err(Q)2.

Therefore, for all j N,

P2

Mj tQmix(err(Q)2)

j

i=1 P2

M (i1)tQmix(err(Q)

2)tQmix(err(Q)

2)

jerr(Q)2.


43/46


On the other hand, taking

j= 2E2[M]tQmix(err(Q)2) ,we obtain

P2

Mj tQmix(err(Q)2)

1 E2[M]

j tQmix(err(Q)2)

1

2.

Combining these two inequalities gives:

tQmix(err(Q)2)

E2[M] =O

err(Q)2

.

This implies that the term containing tQmix(err(Q)2) in the RHS of (6.5)can be neglected. Combining that equation with (6.4) and the SandwichLemma 2.2, we obtain:

dW

C1m(Q)

,i2

Zi

= O k2 err(Q) + k4err(Q)2 +ln |V|k

.

We choose k = (ln |V|/err(Q))1/3 to finish the proof, at least if this issmaller than 1/5

err(Q). But the bound in the Theorem is trivial if that is

not the case, so we are done.

7. Final remarks.

Cooper et al. [6] consider many other processes besides coalescing ran-dom walks. It is not hard to modify our analysis to study those pro-cesses over more general graphs, at least when the initial number ofrandom walks is not too large (this restriction is also present in [6]).

Our Theorems 3.1 and 3.2 can be used to study other problems relatedto hitting times. Alan Prata and the present author [16] have usedthese results to prove the Gumbel law for the fluctuations of covertimes for a large family of graphs, including all examples where it waspreviously known. We have also used extensions of these results tocompute the asymptotic distribution of the k last points to be visited,for any constant k : those are uniformly distributed over the graph, asconjectured by Aldous and Fill [4].


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44 R. I. OLIVEIRA

APPENDIX A: PROOFS OF TECHNCAL RESULTS ON L1WASSERSTEIN DISTANCE

A.1. Proof of Sandwich Lemma (Lemma 2.2). Notice that for allt R,

P (Z t) P (Z t) P (Z+ t) .

By convexity, this implies:

|P (Z t) P (W t) | |P (Z t) P (W t) |

+|P (Z+ t) P (W t) |.

Integrate both sides to obtain the result.

A.2. Proof of Conditional Lemma (Lemma 2.3). First notice that

the sigma field (W1) generated byW1 is contained in G. This implies thatfor all t R:

E [|P (W2 t | G) P (Z2 t) |]= E [E [|P (W2 t | G) P (Z2 t)]|(W1)] E [|P (W2 t |(W1)) P (Z2 t) |] .

Integrating both sides in tand applying Fubini-Tonelli gives:

E [dW(Law (W2| G) , Law (Z2))] E [dW(Law (W2| (W1)) , Law (Z2))] .

Therefore it suffices to prove the Theorem in the case G = (W1). For

simplicity, we will assume that (Z1, Z2, W1, W2) are all defined in the sameprobability space, with (Z1, Z2) independent from (W1, W2). Letf :R Rbe 1-Lipschitz. We have:

E [f(W1+ W2)| W1= w1] =

f(w1+ w2) P (W2 dw2| W1= w1) .

By the duality version ofdW, we have: f(w1+ w2) P (W2 dw2| W1=w1)

f(w1+ z2) P (Z2 dz2) + dW(Law (W2| W1= w1) , Law(Z2)).

Integrating over W1 = w1 and using the fact that Z2 is independent fromW1, we obtain:

E [f(W1+ W2)] E [f(W1+ Z2)] + dW(Law (W2| W1) , Law (Z2)).


45/46


But we also have:

E [f(W1+ Z2)| Z2= z2] =E [f(W1+ z2)] E [f(Z1+ z2)] + dW(W1, Z1),

and the independence ofZ1, Z2 implies:

E [f(W1+ Z2)] E [f(Z1+ Z2)] + dW(W1, Z1).

We conclude:

E [f(W1+ W2)] E [f(Z1+ Z2)] + dW(W1, Z1)

+dW(Law (W2| W1) , Law (Z2)).

Sincefis an arbitrary 1-Lipschitz function, we are done.

ACKNOWLEDGEMENTS

We warmy thank the anonymous referee for pointing out several typos inprevious versions of this paper.

REFERENCES

[1] David Aldous. Markov chains with almost exponential hitting times. StochasticProcesses and Applications, 13(3):305310, 1982.

[2] David Aldous. Mixing times and hitting times. AIM talk slides available fromhttp://www.stat.berkeley.edu/~aldous/Talks/., 2010.

[3] David Aldous and Mark Brown. Inequalities for rare events in time-reversible Markovchains. i. In Stochastic Inequalities (Seattle, WA, 1991), pages 116. IMS LectureNotes Series, 1992.

[4] David Aldous and James Allen Fill. Reversible Markov Chains and Random Walks

on Graphs. Book draft, http://www.stat.berkeley.edu/~aldous/RWG/book.html.[5] Itai Benjamini and Elchanan Mossel. On the mixing time of a simple random walk

on the super critical percolation cluster. Probability Theory and Related Fields,125(3):408420, 2003.

[6] Colin Cooper, Alan Frieze, and Tomasz Radszik. Multiple random walks in randomregular graphs. SIAM Journal of Discrete Mathematics, 23(4):17381761, 2009/10.

[7] J. Theodore Cox. Coalescing random walks and voter model consensus times on thetorus in Zd. Annals of Probability, 17:13331366, 1989.

[8] Rick Durrett. Random Graph Dynamics. Cambridge University Press, 2006.[9] Rick Durrett. Some features of the spread of epidemics and information on a random

graph. Proceedings of the National Academy of Sciences, 107(10):44914498, 2010.[10] Nikolaos Fountoulakis and Bruce Reed. The evolution of the mixing rate of a simple

random walk on the giant component of a random graph. Random Structures andAlgorithms, 33(1):6886, 2008.

[11] Jure Leskovec, Kevin Lang, Anirban Dasgupta, and Michael Mahoney. Communitystructure in large networks: Natural cluster sizes and the absence of large well-definedclusters. Internet Mathematics, 6(1):29123, 2009.

[12] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov Chains and MixingTimes. American Mathematical Society, 2008.


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46 R. I. OLIVEIRA

[13] Thomas M. Liggett. Interacting Particle Systems, volume 276 of Grundlehren derMathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].Springer-Verlag, 1985.

[14] Roberto Imbuzeiro Oliveira. On the coalescence time of reversible random walks. Toappear in Transactions of the American Mathematical Society, 2010.

[15] Gabor Pete. A note on percolation on Zd: isoperimetric profile via exponential clusterrepulsion. Electronic Communications in Probability, 13:37, 2008.

[16] Alan Prata. Stochastic processes over finite graphs. PhD thesis in Mathematics,IMPA, Rio de Janeiro, Brazil, 2012.

Estrada Dona Castorina, 110

Rio de Janeiro, RJ - Brazil

E-mail: [email protected]

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