A note on limiting behaviour of disastrous environment exponents

We consider a random walk on the $d$-dimensional lattice and investigate the asymptotic probability of the walk avoiding a "disaster" (points put down according to a regular Poisson process on space-time). We show that, given the Poisson process points, almost surely, the chance of surviving to time $t$ is like $e^{-\alpha \log (\frac1k) t } $, as $t$ tends to infinity if $k$, the jump rate of the random walk, is small.


Introduction
This note concerns a recent work of T. Shiga ([Shi]).The following model was considered: We are given a system of independent rate one Poisson processes on [0, ∞), N = {N x (t)} x∈Z d.We are also given an independent simple random walk on Z d , X(t), moving at rate k and with, say, Of course simply by integrating out over N , X we have (taking δN X(s The problem becomes non-trivial when considering It is non-trivial, but was shown in [Shi], that the random quantity p(t, N ) satisfies It was shown that as k becomes large λ tends to one in all dimensions and that in dimensions three and higher λ is equal to one for k sufficiently large.The focus of this note is on the other behaviour of λ(d, k): the behaviour as k → 0. It was shown in [Shi] that there existed two constants c 1 , c 2 ∈ (0, ∞) so that We wish to show Theorem 1.0There exists a constant α so that lim λ (d,k)   log( The paper is organized as follows: in Section One we consider a "shortest path" problem which is easily and naturally dealt with by Liggett's subadditive ergodic theorem (see [L]).This yield a constant α.In Section Two we show (Corollary 2.4) that lim inf k→0 Acknowledgement This paper was conceived during a visit to the Tokyo Institute of Technology in January of this year.It is a pleasure to thank Professor Shiga for his clear exposition of the area and for his generosity as host.
I wish also to thank the referee for a thorough reading which has resulted in a much clearer paper.

Section One
In this section we consider only the Poisson processes N .The random walk will not be directly considered at all, though sometimes it will be implicit, as in the definition of a path below: A path γ is a piecewise constant right continuous function with left limits [0, ∞) → Z d so that for all t ||γ(t) − γ(t−)|| 1 ≤ 1.The collection of paths beginning at x ∈ Z d which avoid points in N up to time t will be denoted by Γ x,t .More formally where I is the usual indicator function.In words S counts the number of jumps that γ makes in time interval [0, t].If x = 0 we suppress the suffix x.
Then the random variables satisfy the conditions for Liggett's subadditive ergodic theorem.
Given the ergodicity of our Poisson processes we conclude that the a.s.limit of 1 t α(t, N ) is non random.
We now show that the constant α of Proposition 1.1 is strcitly positive.This fact will follow from Theorem 1.0 and the results of [Shi], however we include it for completeness and because the argument given is a precursor to the block argument of Proposition 2.2.

Proposition 1.2 The constant α is strictly positive.
Fix ε > 0 small we shall give conditions on the smallness of ε as the proof progresses.Choose integer L so that L d e −L < ε.
We divide up space time into cubes We associate 0-1 random variables ψ(n, r) to these cubes by taking ψ(n, r) to be 1 if and only if We note that the ψ random variables are i.i.d. and that, by the choice of L, the probability that ψ(n, r) = 1 is < ε.
To show our result it is sufficient to show that as m tends to infinity α(mL, N ) ≥ m 2 with probability tending to one.
The trace of a path γ ∈ Γ mL is the sequence of points in The crucial observation is that for such γ, n i , Thus to show that α(mL, N ) ≥ m 2 it suffices to show that for all {n i } with it is the case that By simple large deviations arguments the probability that for any given Thus it remains only to count the number of {n} satisfying (1).We write (for positive integer g We may find K so that for all g, (2g + 3) d−1 ≤ K2 g ; we conclude that 2L is less than 2 2m .We conclude that the number of {n i } satisfying (1) is bounded by (4C) m .Thus the probability that α(mL, N ) . This tends to one as m tends to infinity provided that ε was fixed sufficiently small.

Section Two
Fix ε > 0, arbitrarily small.Given c > 0 fixed, we say that a cube Proof Given ε, c, there exists k so that for any R, we can pick points Thus we have This last term is greater than 1 − δ if R is sufficiently large. .
We have not fully specified how small we require δ to be but, conditional on this we will fix R at a level so large that the conclusions of Lemma 2.1 hold for δ and also so that Lemma 2.2 Given c and R ≥ R 0 fixed, there exists k 0 > 0 so that if 0 < k ≤ k 0 and cube [−cR, cR] d is good then for any random walk X(t) starting in the cube, the chance of survival to time R is bounded above by k R(α−2ε) .More generally given c, R ≥ R 0 we have for k ≤ k 0 that the chance that the random walk makes ≥ f αR jumps in time R is bounded above by k fR(α−ε) .
Proof Let the starting point of X be x.By definition of α(R, N ) and a cube being good we have This latter term is less than k R(α−2ε) if k is sufficiently small.
We choose c to equal 10(α + 1) and divide up the lattice into cubes good (in the old sense) after translating Poisson system (N ) spatially by 2cRn and temporally by iR.
We define random variables ψ(n, i) taking values 0 or 1 by The random variables ψ(n, i) are not independent, but it should be noted that random variables We do not require that |β j+1 − β j | 1 be less than or equal to 1.
Given ψ we associate a score to a (r − v)-chain β by Proposition 2.1 For a random walk starting at time rR in cube C(n), the chance that it survives until time vR is bounded above by where the minimum is taken over all (r-v)-chains β with β r = n.
Proof In the proof we regard v as fixed and use induction on k = v − r.The proof follows from induction on k.It is clearly true for k = 1 (or r = v − 1) and all n by Lemma 2.2.Suppose that it is true for k − 1 (and all possible n) and suppose further that X k is a random walk starting at time R(v − k) in cube C(n).We consider the random walk over time interval By the Markov property for X k and induction this summation is bounded by If R was chosen sufficiently large this is bounded by where the minimum is taken over all (r-v)-chains β with β r = n.
It remains to show that as v tends to infinity J v (β) is roughly v.It is time to properly define δ First fix K 3 d and so that for each integer f at least 1, the number of m with ||m|| ∞ = f is less than K2 f −1 /100.Lemma 2.3 Given ε > 0 there exists δ so that 0 < δ < ε/100K so that if X 1 , X 2 , • • • X N are i.i.d.Bernoulli δ) random variables for any integer N then ) N +r .

Proposition 2.2 With probability one for all v sufficiently large
Proof We simply count.Given our definition of J(β) we need only consider those For fixed code m 0 , m 1 • • • m v−2 with m j ≤ v/9 there are (by our choice of K) less than or equal to K v−1 j=v−2 j=0 2 m i −1 possible v-chains.For any such β, J v (β) = 9 m j + ψ(β j , j) and so ) 9 m j .
So the probability that for some β with code m ) But the number of codes which sum to less than v/9 is (assuming w.l.o.g that v/9 is an integer) exactly The proposition now follows from the Borel Cantelli Lemma.
Proof By Proposition 2.1 we have that for By Proposition 2.2 we have therefore that for large enough v Since ε is arbitrarily small the Corollary follows.

Section Three
In this section we will use block/percolation arguments that since [BG] may be regarded as standard.Simply to avoid notational encumbrance we will write out the proof for the case d = 1 but the argument easily extends to all dimensions.
Fix ε > 0. By Proposition 1.1 we have that for R sufficiently large Now note that, by our definition of α, the event {α(R, N ) ≤ R(α + ε)} is the same as the event {∃γ ∈ Γ R with S(γ, R) ≤ R(α + ε) and |γ(R)| ≤ R(α + ε)}.Thus for R sufficiently large has probability less than ε 6 .These two events are increasing functions of the Poisson processes and, by symmetry, have equal probabilities, so by the FKG inequalities (as in [BG]) we have , that is, and, by symmetry, We remark that such paths must be contained in space time rectangle Thus outside probability strictly less than . Now provided that δ is chosen sufficiently small we have also that with probability > 1 − ε 2 we have γ satisfying in addition to(i) and (ii) above (iii) No two jump times of γ are within 2δ of each other or of time 0 or time R ε .Also the path γ is at all times at least 2δ away from points of N (considered now as a random subset of space time).
We define a 2-dependent oriented percolation scheme on {(m, n) : n ≥ 0, m + n ≡ 0(mod(2))} as follows: We say that the bond from (m, n) to (m ') No two jump times of γ are within 2δ of each other or of time n R ε or time (n + 1) R ε .Also the path γ is at all times at least 2δ away from points of N Then we have that (provided ε was chosen sufficiently small) the percolation system is supercritical (see the appendix of [D2], which while formally treating oriented bond percolation, is valid for our bond percolation).That is with probability one there is a point (0, n) with infinitely many "descendents".

Lemma 3.1
If k is sufficiently small then for all (m, n) if the percolation bond (m, n) → (m ± 1, n + 1) is open then with probability at least k R ε (α+ε)(1+3ε) a random walk started at mR(α + ε) at time n R ε will survive until time (n + 1) R ε and will be in position (m ± 1)R(α + ε) at this time.
Proof Let a path satisfying (i),(ii') and (iii') be γ.Let its jumps be at times 0 < t 1 , t 2 , • • • t r r ≤ R(α + ε)(1 + 2ε)/ε.We consider the event that our random walk makes precisely r jumps in the time interval , these jumps occurring within the intervals (t i − δ/3, t i + δ/3) (one jump in each interval) and the jumps are equal to the corresponding jumps of γ.This event is contained in the event of interest and has probability at least Proof Given our percolation scheme we have (provided ε was chosen sufficiently small) that there exists n 0 so that (0, n 0 ) is a point of percolation.That is to say there exists 0 = m 0 , m 1 , • • • m j • • • so that ∀j ≥ 1, the bond between (m j−1 , n 0 + j − 1) and (m j , n 0 + j) is open.
It follows from induction and Lemma 3.1 that a random walk starting at site 0 at time n 0 has chance at least k R ε (α+ε)(1+3ε)j of surviving until time (n 0 + j) R ε and being at m j at this time.The chance that a random walk starting at site 0 at time 0 reaches site 0 at time n 0 R ε is strictly positive (N a.s.).So we have for some c k (ω) > 0 that for k ≤ k 0 .Thus λ(k, d) ≤ ln( 1 k )(α + ε)(1 + 3ε).The corollary follows from the arbitrariness of ε.

1 k
and in Section Three we show (Corollary 3.1) lim sup k→0 λ(d,k) log( ) ≤ α, thus completing the proof of Theorem 1.0.Both of the last two sections rely heavily on block arguments as popularized in [D], [D1].