Crossing speeds of random walks among “sparse” or “spiky” potentials

Crossing speeds of random walks among “sparse” or “spiky” Bernoulli potentials on integers

Elena Kosygina Department of Mathematics, Baruch College, One Bernard Baruch Way, New York, NY 10010
Abstract.

We consider a random walk among i.i.d. obstacles on under the condition that the walk starts from the origin and reaches a remote location . The obstacles are represented by a killing potential, which takes value with probability and value with probability , , independently at each site of . We consider the walk under both quenched and annealed measures. It is known that under either measure the crossing time from to of such walk, , grows linearly in . More precisely, the expectation of converges to a limit as . The reciprocal of this limit is called the asymptotic speed of the conditioned walk. We study the behavior of the asymptotic speed in two regimes: (1) as for fixed (“sparse”), and (2) as for fixed (“spiky”). We observe and quantify a dramatic difference between the quenched and annealed settings.

2000 Mathematics Subject Classification. Primary: 60K37, 60F05, 60J80. Secondary: 60J60.
Key words: random walk, random potential, speed, quenched, annealed.

1. Introduction

We shall start with a model description and the necessary notation. After stating our results we offer an informal discussion, references, and describe a conjecture which makes our results a part of a more general picture.

1.1. Model description and main results

Let , be i.i.d. random variables on a probability space such that

 (1.1) P(V(0,⋅)≥0)=1,P(V(0,⋅)>0)>0,andE[V(0,⋅)]<∞.

These random variables represent a random potential on . Given a realization of the potential, i.e. for each fixed , we consider a Markov chain on with transition probabilities:

 (1.2) p(x,y,ω)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩1−e−V(x,ω),if y=†, x≠†;12e−V(x,ω),if y=x±1, x≠†;1,if y=†, x=†;0,otherwise.

Informally, this is a so called “killed random walk”: at each site the walk either gets killed with probability (and moves to the absorbing state ) or survives and moves to one of the two neighboring sites with equal probabilities . The corresponding path measure and the expectation with respect to it will be denoted by and respectively. Unless stated otherwise, the killed random walk starts from .

Denote by a path on and set

 (1.3) τx:=inf{n>0:Sn=x}∈N∪{∞}.

We shall put different measures on the set of nearest neighbor paths. Measure , and the corresponding expectation , will always refer to the simple symmetric random walk on starting from .

Next, fix and consider the conditional measure on nearest neighbor paths starting from , which is defined by

 (1.4) Qωy(⋅):=Pω(⋅|τy<∞).

Measure is called the quenched path measure. Notice that it can be equivalently defined as follows:

 (1.5) Qωy(A): =(Zωy)−1E(1{τy<∞}1Ae−∑τy−1n=0V(Sn,ω)), % where (1.6) Zωy :=Pω(τy<∞)=E(1{τy<∞}e−∑τy−1n=0V(Sn,ω)).

Finally, we define the annealed measure , which is a measure on the product space of and the nearest-neighbor paths starting from . We let

 (1.7) Qy(B):=E(Pω(B∩{τy<∞}))E(Pω(τy<∞)).

Whenever the starting point of a process is different from , we shall indicate it with a superscript, for example, will denote a quenched path measure of a killed random walk, which starts at and is conditioned to hit , .

The quenched and annealed asymptotic speeds, and , are the deterministic quantities, defined by

 (1.8) 1vqueV:=limy→∞EQωy(τy)y(P-a.s.);1vannV:=limy→∞EQy(τy)y.

By Proposition 2.1 below and [KM12, Theorem 1.2] respectively these limits exist and are finite.

In this paper we consider i.i.d. Bernoulli potentials , , , ,

 (1.9) P(Vp,M(x,⋅)=M)=1−P(Vp,M(x,⋅)=0)=p,

and study the behavior of the corresponding quenched and annealed asymptotic speeds and in two regimes: “sparse” (, is fixed) and “spiky” (, is fixed). Our main results are contained in the following two theorems. By as we mean that .

Theorem 1.1.

With as above, the quenched speed, , satisfies

 (1.10) vquep,M∼3p2 as p→0 % uniformly in M∈[M0,∞), ∀M0>0; (1.11) vquep,M∼3p2−p+2p2 as M→∞ locally uniformly in p∈(0,1].
Theorem 1.2.

With as above, the annealed speed, , satisfies

 (1.12) −logvannp,M∼2(eM−1)p as p→0 uniformly in M∈[M0,∞), ∀M0>0; (1.13) −logvannp,M∼2(1−p)eMp as M→∞ locally uniformly in p∈(0,1).
Remark 1.3.

Observe that in the “sparse” regime both the quenched and annealed speeds vanish but the latter does so at a dramatically higher rate. In the “spiky” regime the quenched speed converges to a positive constant while the annealed one vanishes extremely fast. This striking difference is a purely one-dimensional phenomenon. In dimensions two and higher the existence of the annealed asymptotic velocity is known ([IV12b, Theorem C]) but the existence of the quenched asymptotic velocity is still a largely open problem (see [IV12a, Section 1.2] and references therein). Thus the comparison question might seem a bit premature. Nevertheless, even when both speeds are well-defined, we do not expect to see anything like this in higher dimensions.

Remark 1.4.

Our results can also be interpreted in terms of a closely related model of killed biased random walks conditioned to survive up to time . The latter model exhibits a first order phase transition (in all dimensions) as the size of the bias increases (see, for example, [MG02], [Fl07], [IV12b], and references therein). In dimension 1, there is a critical bias () such that these random walks have the zero asymptotic speed when the bias is less than (resp., ) and a strictly positive asymptotic speed when the bias is greater or equal to (resp., ). The model of crossing random walks considered in the present article informally corresponds to the critical bias case. In particular, (1.13) describes how the first order transition gap closes in on the second order transition in the pure trap model ().

1.2. Discussion and an open problem.

There are many papers concerning the relationship of the quenched and annealed Lyapunov exponents of a random walk in a random potential on , (see [Fl08], [Zy09], [KMZ11], [IV12c], [Zy12], and references therein). Lyapunov exponents represent the exponential decay rates of the quenched and annealed survival probabilities, and respectively. The equality or non-equality of quenched and annealed Lyapunov exponents determines whether the disorder introduced by the random environment is “weak” or “strong”. According to this classification, any non-trivial disorder in the one-dimensional case is strong (as well as in dimensions 2 and 3 under mild additional conditions on , see Theorem 1 and a paragraph after it in [Zy12]). Our results provide more refined information about differences between quenched and annealed behavior.

In dimensions 4 and higher one expects a transition from weak to strong disorder for i.i.d. potentials of the form , , for some . Such result is known to hold when is bounded away from ([Fl08], [Zy09], [IV12c]).

Let us discuss the case of a “small” potential, i.e. the potential of the form where , in more detail. We shall restrict ourselves to dimension 1 but we believe that a similar result holds in all dimensions (when the quenched speed is well defined, see Remark 1.3 above).

Let , , be i.i.d. random variables satisfying (1.1) and and be the quenched and annealed speeds as defined in (1.8). Using our methods it should be not difficult to show that as

 (1.14) vqueγV, vannγV∼√2γE(V).

This result would complement the results of this paper. The relation (1.14) is suggested by the following two facts.

(i) The speeds can be equivalently defined as follows (Proposition 2.1 below and [KM12, Theorem 1.2]):

 (1.15) 1vqueV=ddλαλ+V(1)∣∣λ=0+,1vannV=ddλβλ+V(1)∣∣λ=0+,

where for each the non-random quantities

 (1.16) αλ+V(1) :=−limy→∞1ylogE(1{τy<∞}e−∑τy−1n=0(λ+V(Sn,ω)))(P-a.s.); (1.17) βλ+V(1) :=−limy→∞1ylogE[E(1{τy<∞}e−∑τy−1n=0(λ+V(Sn,ω)))]

are the quenched and annealed (respectively) Lyapunov exponents of a random walk in the potential . Under our assumptions, both limits are positive and finite. For more details about the existence and properties of Lyapunov exponents see [Ze98], [Fl07], and [Mou12].

(ii) It was shown in [KMZ11] (see also [Wa01], [Wa02]) that as . Since

a formal differentiation of with respect to and division by lead to the conjecture (1.14). We remark that in the case of a constant potential (without loss of generality we set ) it is easy to compute the Lyapunov exponent explicitly: , , and obtain from (1.15)

 vγ:=vqueγ=vannγ=√1−e−2γ,γ≥0.

Thus, and as . The latter is (1.14) in this special case.

The relation (1.14) further supports an informal statement that when the potential is “small” (but not “sparse”!) both the quenched and annealed behavior in such a potential are well approximated by the behavior of the walk in a constant potential .

An open problem. Theorems 1.1, 1.2, and asymptotics (1.14) provide information about and when or or . A more interesting and challenging question is to study surfaces formed by the speeds when .

As the first step, fix and consider and as functions of . It does not seem surprising that the annealed environment will typically be more sparse than the quenched one and thus the random walker will feel less need to quickly navigate towards the goal . One expects that is strictly increasing in for a fixed and that has a single maximum as changes from to (see Figure 1).

Is this really the case? These questions are presumably hard to resolve analytically. We are not aware even of any simulations done on this problem (see though [MG02] as well as [GP82]).

1.3. Organization of the paper.

In Section  2 we prove Theorem 1.1. The proof of Theorem 1.2 is given in Section 3, which is subdivided into 3 subsections. Subsection 3.1 gives heuristics and an outline of the proof. Subsection 3.2 is the technical core of the proof. There, after providing an informal calculation, we study the environment under the annealed measure. Subsection 3.3 uses the estimates obtained in the previous subsection and completes the proof of Theorem 1.2. The Appendix contains several auxiliary results and proofs of several lemmas used in the main part of the paper.

1.4. Terminology.

Sites at which the potential is equal to will be called occupied sites or obstacles. Vacant sites are unoccupied sites. An interval is empty if all its sites are unoccupied. We reserve the term vacant interval for maximal empty intervals. A gap between two occupied sites is the length of the vacant interval between these two sites.

2. Quenched speed

The results of Theorem 1.1 are rather straightforward. The fact that the speed is at most of order comes immediately from the fact that no matter what the value of , the time for a conditioned random walk to traverse an empty interval of size will be of order , corresponding to speed of order . The upper bound on speed follows since most sites lie in vacant intervals of size of order .

We shall need two facts. The first is a very basic fact about the standard random walk but we do not have a reference at hand and, thus, give a proof in the Appendix.

Proposition 2.1.

Let be the simple symmetric random walk, , and . Then

 Ek(τn;τn<τ0)=k(n−k)(n+k)3n.

The second is mostly a consequence of the ergodic theorem (see [Sz94, (1.30) and Theorem 2.6] for a treatment of Brownian motion among Poissonian obstacles). The proof is given in the Appendix.

Proposition 2.2.

Let , be i.i.d. random variables, which satisfy (1.1). Then there exists limit

 1vque 1vque =ddλαλ+V(1)∣∣λ=0+.

The key idea for calculation of the quenched speed is an observation that the main contribution to comes from paths which hit before entering where at some positive time. Our first step is to compute the main term.

Lemma 2.3.

For every and

 E[Eω(τ1|τ1<τ−a1)]=2−p+2p23p.
Proof.

Notice that our definition of (see (1.3)) implies . By Proposition 2.1 we have

 E[Eω(τ1|τ1<τ−a1)]=p+p∞∑a=12a1+13(1−p)a1=p+1−p3(2p+1)=2−p+2p23p.

Given , let be the occupied sites in , , , and . Then

 (2.1) E[Eω(τ1|τ1<∞)]=E[Eω(τ11{τ1<τ−a1}|τ1<∞)]+E[Eω(1{τ1>τ−a1}τ1−1∑n=01{Sn∈I}|τ1<∞)]+∞∑j=1E[Eω(1{τ1>τ−a1}τ1−1∑n=01{Sn∈Ij}|τ1<∞)].

Observe also that the first term in the right hand side of (2.1) equals

 (2.2) E[Eω(τ1|τ1<τ−a1)Pω(τ1<τ−a1|τ1<∞)]=E[Eω(τ1|τ1<τ−a1)]−E[Eω(τ1|τ1<τ−a1)Pω(τ1>τ−a1|τ1<∞)].

We shall need the following three elementary lemmas.

Lemma 2.4.

For

 Pω(τ1>τ−a1|τ1<∞)≤e−M∧((2a1(eM−1)+1)(1+a1))−1.
Lemma 2.5.

There is a constant such that

 Eω(1{τ1>τ−a1}τ1−1∑n=01{Sn∈I}|τ1<∞)≤C1(eM−1)−1.
Lemma 2.6.

There is a constant such that for every

 Eω(1{τ1>τ−a1}τ1−1∑n=01{Sn∈Ij}|τ1<∞)≤C2e−Mj(1−e−M)2|Ij||I|2,

where is the Lebesgue measure of the set .

Let us assume these facts (see Appendix for proofs) and derive Theorem 1.1.

Proof of Theorem 1.1.

The right hand side of the inequality in Lemma 2.4 does not exceed . Therefore,

 E[Eω(τ1|τ1<τ−a1)Pω(τ1>τ−a1|τ1<∞)]≤e−M+pe−M∞∑a1=1(1−p)a12a1+13(a1+1)

This immediately gives

 limp→0pE[Eω(τ1|τ1<τ−a1)Pω(τ1>τ−a1|τ1<∞)]=0; limM→∞E[Eω(τ1|τ1<τ−a1)Pω(τ1>τ−a1|τ1<∞)]=0.

Lemma 2.5 takes care of the second term in the right hand side of (2.1). By Lemma 2.6 and the independence of the values of the potential at distinct sites, the last term in (2.1) is bounded by (we defined the function to be at by continuity)

 C2(1−e−M)2∞∑j=1e−MjE[(aj+1−aj)]E[(a1+1)−2]≤C2(1−e−M)2∞∑j=1e−Mjln(1/p)1−p=C2e−Mln(1/p)(1−p)(1−e−M)3.

This expression clearly vanishes as locally uniformly in . After multiplication by it converges to as uniformly on every interval . The only term left in the right hand side of (2.1) and (2.2) is the main term, , which has the claimed asymptotics by Lemma 2.3. ∎

3. Annealed speed

We start by introducing additional notation. Let

 Q0,y(⋅):=Z−10,yE[Pω( ⋅ ;τy<τ0,τy<∞)],Z0,y:=E[Pω(τy<τ0,τy<∞)].

The corresponding quenched path measure is given by

 Qω0,y(⋅)=(Zω0,y)−1Pω( ⋅ ;τy<τ0,τy<∞),Zω0,y:=Pω(τy<τ0,τy<∞).

3.1. Heuristics and goals.

Our first observation is that we can replace the measure with (see Proposition 3.1, (3.4), and the proof of (1.7) in [KM12]). The key ingredient of the proof of Theorem 1.2 is the study of environments under . We show that under the distribution of gaps between occupied sites is “comparable” to a product of log-series distributions111Random variable is said to have a log-series distribution with parameter if , , where . with the average gap , where for fixed and small or fixed and large

 (3.1) loggann∼K(p,M):=2p−1(1−p)(eM−1).

For i.i.d. Bernoulli potentials the gap distribution is geometric, and we already have the result that the reciprocal of the quenched speed is proportional to the average gap between two occupied sites, which is now . This observation together with (3.1) leads to the limits (1.12) and (1.13).

We shall give a detailed proof of (1.12). The proof of (1.13) is very similar but easier and is omitted but we shall write all steps in such a way that they can be readily adapted to the case when and is fixed. An informal derivation of the formula for is given in the next subsection right after Corollary 3.3.

Our goal will be to construct subsets of environments that are essential and on which the walk has the claimed speed behavior. More precisely, to obtain a lower bound on we shall restrict to environments with the following properties: for every there is such that for each there is such that for all

• and

• for every

 EQω0,y(τy)≥C1ye(1−ε)K(p,M),

where does not depend on . Then

 EQ0,y(τy)y≥(yZ0,y)−1E(EQω0,y(τy)Zω0,y;Ω1y)≥C1e(1−ε)K(p,M)Q0,y(Ω1y)≥C12e(1−ε)K(p,M),

and, hence,

 (3.2) −liminfp→0plogvannp,M≥(1−ε)limp→0pK(p,M)=2(1−ε)(eM−1).

For an upper bound we shall consider environments for which the following holds: for every there is such that for each

• and

• there is such that for all ,

 EQω0,y(τy)≤C2ye(1+ε)K(p,M),

where does not depend on . Then

 EQ0,y(τy)y≤(yZ0,y)−1(E(EQω0,y(τy)Zω0,y;Ω2y)+E(EQω0,y(τy)Zω0,y;Ω∖Ω2y))≤C2e(1+ε)K(p,M)+(yZ0,y)−1E(EQω0,y(τy)Zω0,y;Ω∖Ω2y).

By Lemma A.1 (see Appendix), . Combining this with (U1) we get

 (3.3) −limsupp→0plogvannp,M≤(1+ε)limp→0pK(p,M)=2(1+ε)(eM−1).

Since is arbitrary, relations (3.2) and (3.3) imply (1.12). Our task will be to construct , , with the desired properties. The starting point for obtaining (L2) and (U2) is Lemma A.2 which gives bounds on in terms of gaps between obstacles. The construction of , , will be carried out in Subsection 3.3 after we obtain information about a typical environment under the annealed measure . The latter is the content of the next subsection.

3.2. Environment under the annealed measure

Lemma 3.1.

Let , , , and consider an environment such that , , is the set of all occupied sites in . Denote by the probability that a random walk starting at reaches before hitting , i.e. , . Then

 (3.4) u1=e−V(0,ω)2r1=eM−V(0,ω)FM(0,r1,0); un=FM(rn−1,rn,un−1), n>1,

where ,

 (3.5) FM(ℓ,r,u)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩e−M2r(1−e−M(1−12r−1−u2ℓ))−1,if ℓ≠0;e−M2r,if ℓ=0.

The proof is given in the Appendix.

Lemma 3.2.

As

 (3.6) FM(ℓ,r,u)∼e−M2r(1−e−M),uniformly in u∈[0,1].

From now on we shall identify every environment on with the vector of successive distances between occupied sites in , where is the distance from the first positive occupied site in to the origin and is the distance from the last occupied site in to . If the interval is empty then we set and .

Corollary 3.3.

For any and with , , ,

 Q0,y (N=n,¯¯¯¯Rn=(r1,r2,…,rn)) (3.7) = Z−10,y(eM(1−p)p+1)n∏i=1(p(1−p)ri−1FM(ri−1,ri,ui−1)) = Z−10,y(1−p)y(eMϱ+1)n∏i=1(ϱFM(ri−1,ri,ui−1)), where ϱ:=p/(1−p).

Heuristic derivation of (3.1). Before we turn to rigorous analysis of (3.3) we would like to present a “back of the envelope derivation” of the gap asymptotics (3.1). When we might expect that measures converge to a limiting measure, under which the consecutive gaps are essentially i.i.d.. It is reasonable to assume that if we let or then the distances between consecutive occupied sites under this limiting measure will also go to infinity. Thus, we replace in (3.3) with its limit as given by (3.6). We get that for

 (3.8) Q0,y(N=n,¯¯¯¯Rn=(r1,r2,…,rn)) ≍ Z−10,yn∏i=1(p(1−p)ri−1e−M2ri(1−e−M)).

By [KM12, Lemma 5.5], , where is the annealed Lyapunov exponent (see (1.17)), and we replace in (3.8) with to arrive at

 Q0,y(N=n,¯¯¯¯Rn=(r1,r2,…,rn)) ≍ n∏i=1(p2(1−p)(eM−1)(eβ(1−p))riri)=n∏i=1(1K(eβ(1−p))riri),

where is the same as in (3.1). For to be a probability measure it should hold that

 (3.9) 1K∞∑r=1(eβ(1−p))rr=1.

In other words, the limiting gap size appears to have the so-called log-series distribution. Summing up the series in (3.9) we see that , i.e. , and conclude that the expected gap size is

 1K∞∑r=1(eβ(1−p))r=1K∞∑r=1(1−e−K)r=eK−1K.

Here is a layout of the rest of this subsection. We start a rigorous analysis by noticing that all information about the dependence of (3.3) on is contained in the last product. To study its behavior, we consider an auxiliary quantity, the probability that the killed random walk reaches the -th occupied site prior to the first return to if , are i.i.d. positive integer-valued random variables with probability mass function , , for some . Without loss of generality we shall assume that is occupied. Then (setting , )

 (3.10) Un(q)=∑r1,r2,…,rn∈Nn∏i=1Gq(ri)FM(ri−1,ri,ui−1).

We notice that decays exponentially fast in for each (Lemma 3.4). If we want the event that the killed random walk reaches the -th occupied site prior to the first return to to be a typical event, then we need to renormalize (3.10). In Corollary 3.5 we show that there is such that, after the renormalization, the probability of the above event is essentially equal to 1 (see (3.15)). The renormalized measures (3.18) can be effectively compared with product measures (Lemma 3.7). Such comparison allows us to use standard large deviation bounds for product measures (Corollary 3.9) and obtain sufficient control on the right-hand side of (3.3) to be able to construct and in the next subsection.

We use below simply as a shorthand for . Obviously as .

Lemma 3.4.

There is a continuous function such that for every

 (3.11) qlog(1/q)2eM(1−q)≤μ(q)≤qlog(1/q)2(eM−1)(1−q),

and for all .

Proof.

Let us fix an arbitrary and drop it from the notation. It is obvious that for . This implies that the sequence , , is superadditive, and, thus,

 (3.12) limn→∞logUnn=supnlogUnn=:logμ.

Therefore, for all .

For the lower bound, consider a killed random walk, which starts from the origin in an environment, such that all sites to the left from are empty. Let