Transition to localization of biased walkers in a randomly absorbing environment

Physica D 168–169 (2002) 244–257

Transition to localization of biased walkers in a randomlyabsorbing environment

Vishal Mehra∗, Peter GrassbergerJohn-von-Neumann Institute for Computing, Forschungszentrum Jülich, D-52425 Jülich, Germany

Abstract

We study biased random walkers on lattices with randomly dispersed static traps in one, two and three dimensions. Asthe external bias is increased from zero the system undergoes a phase transition, most clearly manifested in the asymptoticdrift velocity of survivors which is zero below a critical bias. This transition is second order in one dimension but of firstorder in higher dimensions. The model can be mapped to a stretched polymer with attractive interaction between monomers,and this phase transition would then describe sudden unfolding of the polymer when the stretching force exceeds a criticalvalue. We also present precise simulations of the zero-bias case where we show unambiguously that the transition betweenthe Rosenstock (RS) and Donsker–Varadhan (DV) regimes is first order in dimension≥2.© 2002 Elsevier Science B.V. All rights reserved.

PACS:05.40.Fb; 02.70.−c

Keywords:Diffusion; Random traps; Localization; Biased random walks; Phase transition

1. Introduction

The behavior of a particle diffusing amidst ran-domly dispersed static traps is already interesting byitself, but also because of its connections to severalother problems. It is closely related to the Andersonmodel of delocalization and to some models of pop-ulation dynamics[1]. It can be mapped exactly ontoflux lines in a medium with parallel columnar defects[1] and to the collapse of a self-attracting polymerwithout excluded volume interaction.

The problem is usually formulated as a randomwalk on a d-dimensional lattice; a random fractionc of all lattice sites is occupied by traps which killa walker if it happens to land on them. The mainquantity of interest is the survival probability after

∗ Corresponding author.E-mail addresses:[email protected] (V. Mehra),[email protected] (P. Grassberger).

t steps,P(c, t), averaged over all walks and all trapconfigurations. For smallt and small trap concentra-tions, P(c, t) decays exponentially as predicted byRosenstock (RS)[2], but at later times it crosses overto the exact Donsker–Varadhan (DV) form[3–5]

ln P(c, t) ∼ −Adλ2/(d+2)td/(d+2), (1)

whereλ = − ln (1− c) andAd are the exactly knownnumerical constants.Eq. (1) can be understood interms of contributions from large trap-free regions.Such regions are very rare, but survival is much en-hanced in them. Thus,P(c, t) is dominated by regionswhose size grows witht asR ∼ t1/(d+2) [4].

The precise form of this crossover has been a matterof some controversy[6–9]. In particular, simulationshave been notoriously difficult ford ≥ 2. While de-viations from the RS prediction are easily seen[4], ithas proven difficult to access the validity ofEq. (1)insimulations[6–8]. Ford = 3, the situation has largely

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V. Mehra, P. Grassberger / Physica D 168–169 (2002) 244–257 245

been clarified by simulations of Anlauf[10] whichunfortunately were never published, and by analyti-cal work by Nieuwenhuizen[9]. Very recently, similarscaling was found in[11,12] to hold also ford = 2.

In this paper we first present further simulations ofthis model where we correct some previously madeclaims and verify the prediction of[9] that the RS–DVcrossover is akin to a first-order phase transition atinfinite temperature. We then show results of the modelwith a superimposed bias. In the latter, the jumpingrates in the (fixed) bias direction are enhanced by afactor b > 1 and those in the opposite direction aresuppressed by the same factor.

This external bias leads to a phase transition in theasymptotic dynamics. For small bias the long-time dy-namics is similar to the zero-bias case: the walkersstay in large trap-free regions and do not drift. In-stead, their typical positions increase slowly asx ∼R ∼ t1/(d+2). But above a critical biasbc, the surviv-ing walks drift with a finite velocity.1 These behaviorsare separated by a sharp transition which is of secondorder ind = 1 but of first order ind ≥ 2. At b ↘ bc

the drift velocity tends to zero ind = 1 but reaches anon-zero valuevc in d ≥ 2. Asymptotically the sur-vival probability decays exponentially in both phases,but the decay rate is singular at the transition point:its slope is discontinuous ind ≥ 2 while in d = 1 thesecond derivative is discontinuous.

The averaging over disorder can easily be done ex-actly and leads to

P(c, t) = e−γ t (2d)−tZt , (2)

with

Zt =∑walks

(1 − c)s bx. (3)

Here,

γ = ln

(b + b−1 − 2

2d+ 1

), (4)

and s is the number ofdistinct sites visited. Fi-nally, Zt can be understood as the partition sum of a

1 Actually, this “velocity” is defined as the change d〈x〉/dt ,where〈x〉 is the average position of the survivors. Since walkersget absorbed, this is not equal to the average velocity of individualwalkers.

self-attractingpolymer (without excluded volume in-teraction; we assumekBT = 1) of lengtht , stretchedby applying opposite forces±ln b to its two ends.The self-attraction comes about by the fact that eachmonomer which is placed on a new site is punishedby a Boltzmann factor 1− c, while no such factor isapplied when a monomer is placed at a site which isalready occupied by another monomer.

Although this is rather artificial as a polymer model,the mapping is very useful for suggesting efficient sim-ulation methods and for interpreting the results.

Our simulations are performed using the Pruned-Enriched Rosenbluth Method (PERM)[13] which isa growth scheme to build weighted walks. Walks withtoo small weights are pruned, promising configura-tions are further encouraged by cloning them if theirweight exceeds a (length-dependent) given value.PERM has been used to simulate efficiently a largenumber of polymer problems includingΘ-collapse,DNA denaturation, protein folding, and critical un-mixing [14]. In addition to the “population control”provided by cloning and pruning, walks are alsoguidedby assigning different a priori probabilities todifferent jumps. In particular, we chose these suchthat previously unvisited sites are unfavored againstpreviously visited ones,punvisited = (1 − c)pvisited,and jumps in the bias direction are more likely thanothers,p+x :p−x :p⊥ = √

b:√

1/b:1. These a priori bi-ases are of course compensated by appropriate weightfactors∝ 1/p.

2. Unbiased walkers

For b = 1 walks are asymptotically subdiffusive orlocalized; the DV theory predicts the mean squareddisplacement to be〈R2〉 ∼ t2/(d+2). However, forsmall trap concentrations walks are Brownian at smalltimes (〈R2〉 ∼ t) reverting to DV scaling asymptoti-cally from above[10]. This curious non-monotonicity(also observed in other models of polymer collapse[13,15,16]results because the most successful walks atshorter times donot survive for longer periods whereDV scaling holds (Fig. 1a). The same is seen, albeitmuch less dramatically, ind = 2 (Fig. 1b). It is not

246 V. Mehra, P. Grassberger / Physica D 168–169 (2002) 244–257

Fig. 1. (a)〈R2〉 vs. t in d = 3 for unbiased walks, showing the sudden localization indicated by the drop in〈R2〉. The time at which thislocalization occurs decreases with increasing trap concentration. The transition proceeds more gradually for largerc, and for c ≥ 0.7 nomaximum is obtained. (b) Similar data ford = 2. The collapse there is much less pronounced.


seen ind = 1 where walks remain localized for alltimes and〈R2〉 is monotonic.

The sharp collapse ind = 3 suggests that theRS–DV crossover is indeed a first-order phase transi-tion in the limitst → ∞ and zero trap concentration,as predicted in[9]. In the polymer model, it would bea first-order transition at infinite temperature. A moredirect support of this is provided by histograms of thedistribution of visited sites. For a first-order transitionwe expect two peaks for larget , the positions of whichare separated by a distance∝ t . This is indeed seen ind = 3 (Fig. 2), but not ford = 2. The latter is not sur-prising in view of the much softer collapse ind = 2.It could just mean that the two-peak structure devel-ops only fort-values beyond those obtainable by oursimulations (t ≈ 105).

Combining the RS and DV limits into a single scal-ing ansatz, Anlauf[10] suggested ford = 3 that theaverage number of visited sites scales as

t−1〈s〉 = g3(λt2/3). (5)

A similar ansatz was made ford = 2 in [11], but asshown in[12] the proper ansatz ford = 2 must involve

Fig. 2. Histogram of visited sites for unbiased walks ind = 3. The two peaks correspond to the collapsed (DV, left) and extended(RS, right) phases. Trap concentrations (resp. monomer interaction strengths) have been chosen such that both peaks have equal heights:c = 0.01993(t = 8192); c = 0.01283(t = 16,384); c = 0.00823(t = 32,768); c = 0.00527(t = 65,536).

logarithmic corrections,

t−1〈s〉 ln t = g2

(λt1/2

ln t

). (6)

As seen fromFig. 3a, Eq. (5)does not hold with thesimple scaling function proposed in[10]. Anlauf’sdata had suggested a piecewise linearg3(x) on alog–log plot, with g3(x) = const. for x < x∗ andg3(x) = const./x2/5 for x > x∗. We see instead fromFig. 3a thatg3(x) develops a kink with infinite slopeatx = x∗. This is not too surprising. While a behavioras claimed in[10] would be natural for a second-ordertransition, a more singular behavior is expected for afirst-order transition. Ford = 2 the scaling functionis much smoother (seeFig. 3b), although this mightbe a transient effect only. We finally point out that werequire the precise asymptotics for two-dimensionalrandom walks as derived in[17], with ln t replacedby ln (8t), to obtain a good data collapse.

UsingEq. (5), Anlauf had proposed a similar scalingfor P(c, t):

t−1/3 ln P(c, t) = f3(λt2/3). (7)


Fig. 3. (a) 〈s〉/t vs. λt2/3 in d = 3 for unbiased walks. For larget the data collapse onto a scaling function which is flat for smallarguments∝ (λt2/3)−2/5 for large arguments, and has a cusp with infinite right slope in between. It is not clear from the data whether thescaling function is continuous or discontinuous at the cusp. (b) Similar ford = 2. Notice that we replaced lnt in Eq. (6) by ln (8t) inorder to improve the data collapse. The straight line is the asymptotic DV prediction.


Using his simpleg3(x) he derived, by thermody-namic integration, anf3(x) which gave a very slowconvergence to the asymptotic DV scaling after hav-ing deviated sharply and suddenly from RS behavior.Although we disagree with his scaling function, our

Fig. 4. (a) Scaling plot forP(c, t) for unbiased walks ind = 3. The straight lines are the RS and DV predictions. While the RS predictionis realized numerically (for small concentrationsc) up to the transition point, the true DV scaling is reached only very slowly. (b) Similarfor d = 2.

results forP(c, t) (Fig. 4a) fully confirm with hispredictions within the numerical errors. Notice thatthis disagrees slightly with the results found in[12].Again the analogous ansatz ford = 2 should involvelogarithms[12], and for a faster convergence we must


replace lnt by ln (8t),

ln P(c, t)

ln (8t)= f2

(λt1/2

ln (8t)

). (8)

Agreement with simulations is very good, and againwe see very slow convergence to the DV asymptotics,in contrast to[12].

3. Non-zero bias

For any non-zero bias the long-time probability de-cay is exponential[18,19]. This is most easily seen asfollows. Let φ(x, x⊥, t; b; C) be the probability den-sity, for any fixed trap configurationC and biasb, thata walker who started att = 0 in the originx = 0 is attime t at the positionx = (x, x⊥). One verifies easilythat this is related to the analogous distribution of thebias-free case by (see[20] for the continuum case)

φ(x, x⊥, t; b; C) = e−γ tbxφ(x, x⊥, t; 1; C). (9)

This ‘gauge transformation’ means in particular thatthe return probability

Preturn(t) = 〈φ(0, 0, t; b)〉c (10)

decays exactly as

Preturn(t) = e−γ t P(b=1)return (t), (11)

i.e. as an exponential multiplied by the DV stretchedexponential sinceP (b=1)

return (t) also satisfies the DV esti-mate up to algebraic factors.

For the survival probability we obtain

ln P(c, t) = ln P (b=1)(c, t) − γ t

+ ln 〈 cosh(x ln b)〉walks,C. (12)

Since the last term is positive definite, we have the ex-act inequality lnP(c, t) ≥ ln P (b=1)(c, t)−γ t . More-over, if the walkers stay localized in spite of the bias,x increases sublinearly witht and thus the asymptoticdecay rate is

α ≡ − limt→∞

1

tln P(c, t) = γ (localized). (13)

Notice that this is independent of the trap concentra-tion. A smaller decay rate can be obtained only when

the walkers drift, but then the chance to hit a trap isfinite at any time step, andP(c, t) decays again ex-ponentially. Indeed, in the drifting regime any corre-lations should be short, and thus both the decay rateand the drift velocity are attained already after a shorttime.

3.1. d = 1

In one dimension it is possible to derive analytic es-timates for the survival probability and the drift veloc-ity of delocalized walks. An exact lower bound on thesurvival probability in terms of probabilities of largetrap-free regions is obtained in a way similar to[4],by estimating the contributions of trap-free regionsof length l downstream of the starting point of thewalker. Take some positive integerl and consider con-figurations with no trap in the interval [0, l]. We thenhave

P(c, t) ≥ (1 − c)lPl(t), (14)

where the first factor on the RHS is the probability tofind such a gap in the trap configuration, andPl(t) isthe probability to survive in it. The latter can be esti-mated usingEq. (9), the method of images to incorpo-rate the boundary conditions atx = −1 andx = l+1,and Stirling’s formula. We obtain, up to a constantdependent only onb:

ln Pl(t) ≥ l ln b − γ t −[t + l

2ln

(1 + l

t

)

+ t − l

2ln

(1 − l

t

)]. (15)

The RHS of Eq. (14) is maximal for l = l∗ ≡t tanh [ ln((1 − c)b)] when b > bc ≡ 1/(1 − c),and for l = o(t) when b < bc. Thus the decay rateis bounded byα ≤ γ − ln [ cosh( ln (1 − c)b))]for b > bc. Actually we claim that this bound issaturated, since it can also be understood as theresult of a (non-rigorous) saddle point approxima-tion. Summarizing, we have thus found that the de-cay rate is equal toγ for b < bc and is given byα = γ − ln [ cosh( ln (1 − c)b)] for b > bc. Sim-ilarly, the drift velocity of the survivors, defined asv = lim t→∞〈x〉/t , vanishes forb < bc and is given


Fig. 5. lnP(c, t) + γ t for d = 1 and trap concentrationc = 0.1. Statistical errors are smaller than the thickness of the curves.

by v = tanh [ ln((1 − c)b)] for b > bc. In the con-tinuum limit (c → 0, (1 − c)b = finite) this agreeswith [20–22], but for finite c it disagrees with[21].2

Thus there is a phase transition atb = bc. It issecond order in the sense thatv, α, and dα/db arecontinuous atb = bc, while dv/db and d2α/db2 arediscontinuous.

These predictions are fully confirmed by simula-tions, as seen fromFigs. 5 and 6. In Fig. 5 we seeln P(c, t) + γ t versust for c = 0.1 and for severalvalues ofb. First of all we see that all curves areabove that forb = 1, as predicted theoretically. Sec-ondly, we see that all curves forb ≤ 1.11 becomehorizontal fort → ∞, in agreement with the predic-tion bc = 1/(1 − c) = 1.111. . . The decay rates ob-tained fromFig. 5 for b > bc are shown inFig. 6a,together with the theoretical predictions. We see thatthe transition is indeed smooth, as obtained theoreti-cally. Finally, the drift velocities forc = 0.1 are shownin Fig. 6a. Again agreement with theory is perfect.We should add that similar results were also found forother values ofc, and that〈x〉 increased always mono-

2 This seems to be due to an unjustified Gaussian approximationto Eq. (15)made in[21], lnPl(t) ≈ l ln b − γ t − l2/(2t).

tonically with t , in contrast to what is found ford > 1(seeSection 3.2).

3.2. d ≥ 2

In higher dimensions we do not have similar ana-lytic results. Moreover, direct simulations of the tran-sition region are very difficult, even with our efficientalgorithm. ButEq. (13)together with the expected fastconvergence of the decay rate and drift velocity in thedelocalized phase allow us, in spite of this difficulty,to computev, α, andbc with very high precision.

In Fig. 7 we show the average displacements〈x〉in d = 2 againstt , for c = 0.125 and for severalvalues ofb. Very similar data are obtained ford = 3and for other values ofc. Obviouslybc > 1.65, sincefor all b ≤ 1.65 the curves decrease after an ini-tial rise 〈x〉 ∝ t . The reason for this non-monotonicbehavior is the same as inFig. 1. Walkers whoventure far out do better initially, but finally onlythose win who started in a large trap-free regionand stayed in it. FromFig. 7 one might guess that1.65 < bc < 1.7, but this would be wrong. Alsofor b = 1.7 the curve should bend down ultimately,but this will happen very late (see below) and our


Fig. 6. (a) Decay rateα as a function of biasb, for c = 0.1 (d = 1). For b ≥ 1.2 the statistical errors are much smaller than the size ofthe symbols, for smallerb there is an increasing systematic uncertainty due to the non-trivial extrapolation fort → ∞. (b) Analogousresults for the drift velocity. Dashed curves are analytic predictions.

algorithm might easily miss it (in the transition regionit generates many drifting walks with low weight andonly very few compact walks, but these with highweight).

The true transition point can be estimated from thesurvival probabilities, shown inFig. 8 for d = 2. Ac-tually, in view of Eq. (12), we plot lnP(c, t) + γ t .For smallb we see curves which follow closely the

curve forb = 1, but are slightly above it. These curvescontain a linear part at smallt whose slope decreasesand whose length increases withb. The crossover fromthis linear part to the part following the DV curve israther sharp and stays so with increasingb. Since theDV curve is asymptotically flat, this scenario shouldhold up to the value ofb where the “initial” part be-comes horizontal. Beyond that value we are in the


Fig. 7. Average displacement in the bias direction forc = 0.125 in d = 2.

drifting phase. Obviously the initial straight parts inFig. 8 correspond precisely to the (transient) driftingphase seen already inFig. 7. Indeed, the crossoversbetween the two regimes are at the same positions inFigs. 7 and 8.

This means thatbc is implicitly given by the con-dition αdrifting = γ , whereαdrifting is the decay ratein the drifting phase. While this phase is asymptoti-cally stable forαdrifting < γ , it is only transient forαdrifting > γ . Even in the transient regime it can bemeasured precisely (together with the drift velocity),if this regime lasts long enough. Moreover, the drift-ing phase exists also forb < bc and larget , but is onlymetastable in this regime. This is clearly seen from his-tograms of thex-distribution. InFig. 9 (which refersto d = 2 andc = 0.125) we see clearly two peaks inP(x), for t ≥ 900. The left peak is atx/t ≈ 0. Indeed,it is slightly to the right ofx/t = 0, but it moves tox/t = 0 for t → ∞. It of course corresponds to thelocalized phase. The right peak is the drifting phase.The fact that these peaks become sharper and move

apart with increasingt shows that the transition isfirstorder. For d > 2 we expect the transition then to beof first order a fortiori.

Since γ is known analytically, we can locatebc

by simulations in the delocalized phase alone. Thisprocedure is indicated inFig. 10 where the dashedcurve isγ (b) and the various continuous curves arefinite-timedecay rates obtained fromshort-timerunswith different trap concentrations. The error bars of thelatter are much smaller than the size of the symbols.The intersection points givebc andαc in terms of thetrap concentrationc.

The scaling behavior forc → 0 is easily obtainedby observing thatγ ≈ (b − 1)2/(2d) for b ≈ 1, whileα ∼ c in the drifting phase (seeFig. 11). The latter istrue for smallc since the RS approximation becomesexact in this limit. Therefore we predict thatbc − 1 ∼√

c. This is verified inFig. 12for d = 2 and 3. Finally,from Fig. 13we see that alsovc ∼ √

c, i.e. vc ∝ bc.The latter is very natural, but we must remember thatit is not true ind = 1.


Fig. 8. lnP(c, t) + γ t for d = 2 and trap concentrationc = 0.125. Since these curves must be increasing withb, the fact that the curvefor b = 1.65 crosses fort > 6000 below those forb = 1.555–1.635 is due to numerical inaccuracy.

4. Alternative interpretations

4.1. Population dynamics

Up to now, we dealt with independent walkers. De-tailed studies of this model were often criticized on thegrounds that the tiny probabilities that obtain in the DVregime render the results physically irrelevant (in themodel with bias, probabilities are even smaller). Thiscriticism can be countered by adding autocatalytic par-ticle production (local production of offsprings)[1].

Thus we consider a model where particles can hop,get absorbed when hitting a trap, and can reproducewith some fixed rateσ . For simplicity we do not in-clude any interaction between particles, in particularwe do not take into account any excluded volume ef-fect or any competition for food or other resources.The only effect of reproduction is then that the averagenumber of survivors at generationt is P(c, t)(1+σ)t .

In the bias-free case any non-zeroσ will thus leadto an explosion of the population, indicating that themodel is basically sick. But in the case with drift a suit-ably chosen rateσ will lead to either a much reduceddecay of the particle number, or even to a stationarypopulation size. In this case the neglect of excludedvolume and saturation effects will be much more be-nign, and the effects studied in this paper might havea bigger chance to be observable in real experiments.

Finally, including excluded volume effects wouldlead to directed percolation (contact process) in a ran-dom medium[23].

4.2. Stretched collapsed polymers

We have already pointed out thatZt (Eq. (3)) canbe understood as the partition sum of a polymer withself-attraction of strengthkBT λ, and stretched bya force kBT ln b. The decay rate is then essentially


Fig. 9. Histograms of the rescaled end point distanceP(x) vs. x/t for d = 2 and c = 0.125. Biases were adjusted so that both peakshave equal height:b = 1.510(t = 900); b = 1.577(t = 1800); b = 1.608(t = 2700); b = 1.628(t = 3600). Normalization is arbitrary.The peak atx/t ≈ 0 is due to the localized phase, the other corresponds to the drifting phase.

Fig. 10. Decay ratesα measured in the drifting (delocalized)phase,d = 3. The dashed curve is the predictionα = γ (b) forthe localized phase. To the left of this curve, the drifting phaseis only transient. In spite of that,α is easily measured with highprecision, thus the statistical errors are≤ 10−4, much smaller thanthe symbols.

Fig. 11. Log–log plot of the critical decay rate vs.c. For smallc,both curves scale∝ c.


Fig. 12. Log–log plot of the logarithm of the critical bias vs.c.For smallc, both curves scale as

√c.

the free energy per monomer in the thermodynamiclimit.

The fact that unfolding of a collapsed polymer is afirst-order transition is well known[24–26]. The co-existence of stretched and coiled phases corresponds

Fig. 13. Log–log plot of the critical drift velocity vs.c.

to a “tadpole” configuration[24,27] where part ofthe chain is collapsed and the rest sticks out of thecoil, so that the total lengthx is equal to some pre-scribed value. Our finding that the critical bias is∝ √

c means then that the string tension in a tad-pole configuration scales, in the limit of a very longchain and close to the collapse point, as the squareroot of the monomer–monomer attraction strength.This seems to be a new result, and it remains tosee whether it holds also for more realistic polymermodels.

The main difference to standard polymer models isthe fact that we have neglected here excluded volumeeffects (they have no counterpart in the trap prob-lem). Thus the collapse of the unstretched polymerhappens at infinite temperature in the present model.In the standard polymer models with next-nearestneighbor attraction or with three body repulsion theΘ-collapse is second order, in contrast to the presentmodel. But the dependence of〈R2〉 on chain length isalso non-monotonic ind ≥ 3 [13,15,16]. In d = 4 thecollapse is indeed “quasi-first order”[16], and specialcare is needed to realize that it actually is continuous.We believe that in the present model the transition istruly of first order.

4.3. Magnetic flux lines

A last interpretation is in terms of magnetic fluxlines in a type II superconductor with parallel colum-nar defects[1]. These defects are quenched and ran-domly distributed. They tend to pin the flux lines, thuspreserving superconductivity in the sample. In the un-biased case, the mean orientation of the flux lines isparallel to the defects, caused by a magnetic field inthe same direction. The biased case corresponds thento a non-zero angle between the field and the defects.For small magnetic field the flux lines will essentiallyfollow the defects. But above a critical field strength(resp. tilting angle) the average angle between fluxlines and defects is non-zero. Our main result is thatthis transition is first order in the sense that the av-erage angle jumps by a finite amount when passingthrough the critical field strength. This jump scales asthe square root of the defect density.


5. Conclusion

Using extensive Monte Carlo simulations by meansof the PERM algorithm, we have studied the problemof particles performing unbiased and biased randomwalks in media with randomly located traps. In thebias-free case we find the first clear numeric indicationfor the first order nature of the RS–DV transition inthree dimensions. Ind = 2 the situation is less clear,but we showed that also there the transition is rathersharp. When the diffusion is biased, we find a tran-sition from localized to drifting walkers at a criticalbias which depends on the trap density. This transi-tion is second order ind = 1 but first order ind ≥ 2.Combining the simulations with exact analytic esti-mates we were able to draw the phase diagram withhigh accuracy.

Acknowledgements

We are indebted to Walter Nadler for numerous dis-cussions and for critically reading the manuscript, andto Gerard Barkema for correspondence.

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Transition to localization of biased walkers in a randomly absorbing environment

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