Excited Random Walk in One Dimension
Abstract
We study the excited random walk, in which a walk that is at a site that contains cookies eats one cookie and then hops to the right with probability and to the left with probability . If the walk hops onto an empty site, there is no bias. For the 1excited walk on the halfline (one cookie initially at each site), the probability of first returning to the starting point at time scales as . Although the average return time to the origin is infinite for all , the walk eats, on average, only a finite number of cookies until this first return when . For the infinite line, the probability distribution for the 1excited walk has an unusual anomaly at the origin. The positions of the leftmost and rightmost uneaten cookies can be accurately estimated by probabilistic arguments and their corresponding distributions have powerlaw singularities near the origin. The 2excited walk on the infinite line exhibits peculiar features in the regime , where the walk is transient, including a mean displacement that grows as , with dependent on , and a breakdown of scaling for the probability distribution of the walk.
pacs:
02.50.r, 05.40.aI Introduction
In this work, we study the excited random walk in one dimension. This recentlyintroduced model PW97 ; D99 , gives rise to a stochastic process with many unusual properties P01 ; ABV03 ; BW03 ; Z04 . The definition of the excited random walk is as follows: Initially each site of a onedimensional chain contains a fixed number of cookies (with ). A random walk makes nearestneighbor hops between sites on the chain, with hopping probabilities that depend on whether or not the current site contains a cookie. When the walker hops onto a site that contains one or more cookies, the walker eats a cookie and, at the next time step, the walk hops to the right with probability and to the left with probability . If the walker hops onto an empty site, then the hopping probabilities to the left and right at the next step are both 1/2. This bias can be viewed as the foraging strategy of a primitive organism. When it encounters food, the organism consumes the food and continues to move in a preferred direction. Conversely, the organism wanders aimlessly when it does not encounter any food.
A basic feature of the excited random walk is the transitory nature of the bias; that is, when all the cookies at a given site are eaten the subsequent motion is unbiased. This nonMarkovian feature is responsible for the unusual properties of the excited random walk. Related nonMarkovian constraints arise in many important stochastic phenomena. Perhaps the best known example is the selfavoiding random walk, where returns to previouslyvisited sites are strictly forbidden SAW . This leads to an expansion, relative to the pure random walk, in which the rootmeansquare displacement grows faster than when the spatial dimension is less than 4. A related example is the “true” selfavoiding walk, where returns to previouslyvisited sites are inhibited by an energetic constraint TSAW . The opposite type of perturbation occurs in the reinforced random walk PV99 , where the walk has a preference to hop onto previouslyvisited sites. Depending of the functional form of this reinforcement, the walk may, quite surprisingly, “condense” onto a support that consists of only a finite number of sites as PV99 .
The excited random walk represents a simple and surprisingly rich example of a nonMarkovian process in which the bias at each site disappears when all the cookies on the site are eaten. Because a random walk in one dimension is recurrent, that is, the probability to eventually return to the starting point equals one, the walk returns to previouslyvisited sites infinitely often. For the 1excited walk, the reinforcement mechanism due to the presence of cookies is insufficient to counterbalance this recurrence of the random walk; thus the 1excited random walk remains recurrent for any . Nevertheless, this model exhibits many novel features. For example, the probability that a 1excited walk on an infinite line first returns to the origin at time decays as , rather than the decay for a pure random walk DA01 ; DAA03 ; BW03 .
On the other hand, the excited random walk with is more strongly influenced by the cookieinduced bias. In fact, for excited random walk becomes transient Z04 . This transition has dramatic consequences on the statistical properties of the walk as is varied. As a final introductory point, the excited random walk appears to be most interesting in one dimension because of the nontrivial competition between the reinforcement when a cookie is eaten and the fact that encounters with sites that contain cookies become progressively less frequent. In contrast, an unbiased random walk in greater than one dimension typically encounters previouslyunvisited sites F68 ; fpp . Because each such site contains a cookie, the reinforcement mechanism acts as a nonzero global bias that is superimposed on the random walk motion. Thus the asymptotic behavior of an excited walk in higher dimensions is close to that of a simple biased random walk. , the
In the next section, we begin by recounting some basic firstpassage properties for diffusion in a finite interval F68 ; fpp . We present these results separately because they underlie our analytical treatment of excited random walks. In Sec. III, we study a particularly simple version of the excited random walk – the 1excited walk with a reflecting boundary condition at the origin. We use basic tools of firstpassage processes to determine the time dependence of the size of the region in which the cookies have been eaten and thence the probability for the walk to first return to the origin. The longtime behavior of the firstpassage probability in a closely related model was obtained previously, using quite different methods, in DA01 ; DAA03 . Our approach is also complementary to the tools employed in the probabilistic studies of excited random walks given in Ref. P01 ; ABV03 ; BW03 ; Z04 . In Sec. IV, we first present numerical simulations for the unusual properties for the probability distribution and for the span of the 1excited random walk on the infinite line. We also construct a probabilistic argument to quantify the asymmetry in the number of cookies eaten from the left and right sides of the line. Finally, in Sec. V we present several intriguing properties of the 2excited walk on the infinite line, including the lack of scaling in the probability distribution, the time for the 2excited walk to traverse a finite domain, and basic features of the cookie density profile that is left behind as the walk advances.
Ii Preliminaries: First Passage in an Interval
To determine the motion of excited random walks, we require the firstpassage properties of a nearestneighbor unbiased random walk, or equivalently, continuum diffusion, in a finite interval when both boundaries are absorbing. The first such property is the eventual hitting, or exit, probability to and to for a random walk that starts at within the interval F68 ; fpp . These exit probabilities are:
(1) 
Closely related quantities are the conditional exit times to and . These are the mean times for the random walk to hit or to hit with the restriction that the other boundary is never hit. These conditional times are:
(2) 
Next, we will also need the Laplace transforms of the firstpassage probabilities for a random walk that starts at to exit at the left edge and at the right edge of the absorbing interval at time . In the continuum limit, these first passage probabilities are (Eq. (2.2.10) in Ref. fpp ):
(3) 
where we take the diffusion coefficient to be for the nearestneighbor random walk.
We will also need firstpassage properties in the interval when the boundary at is reflecting and the other boundary is absorbing. We implement the reflecting boundary condition for a discrete random walk that is at site by allowing the walk to remain at with probability 1/2 or by having hop to with probability 1/2. This definition for the reflecting boundary has the advantage that it leads to a constant probability density around this boundary for large times. With this definition, the average time to exit at the right edge of the interval as a function of is
(4) 
Finally, the Laplace transform of the firstpassage probability to exit at the right edge at time in the continuum limit is (Eq. (2.2.21) in fpp )
(5) 
We will exploit these results to now determine various basic features of the 1excited random walk in one dimension. There are two basic properties of the excited random walk that we will be able to determine analytically: (i) the location of the last (leftmost) cookie; we obtain this property in terms of the firstpassage statistics of a 1excited random walk to the right edge of the cookiefree domain, and (ii) the statistics of returns to the origin.
Iii 1Excited Random Walk on the HalfLine
Consider the 1excited random walk that is restricted to the nonnegative integers by imposing a reflecting boundary condition at the origin. Initially there is one cookie at each site and the walk is at . At the first time step, the walk either hops to with probability or it remains at site with probability (Fig. 1). At later times, a bias in the random walk motion occurs whenever the walk reaches the right edge of the cookiefree domain. The restriction of fixing one boundary condition while the other boundary condition “floats” greatly simplifies the analytical description of the 1excited walk on the halfline compared to the infinite line system.
iii.1 Time to Hit the Last Cookie
It is useful to characterize the walk by the sequence of times at which the walk hits the leftmost (last) cookie at (Fig. 1). Suppose that the walk has just reached the cookie at . Then with probability , the next step of the walk is to right, which takes one time step, while with probability , the walk steps left to and the last cookie now is at . In this latter case, the average time it takes the walk to reach site is [see Eq. (4)] plus the time for the initial step from to . Thus the mean time increment for the last cookie to move from to is
Consequently, the mean time for the walk to hit the last cookie at for the first time is
(6) 
Notice that for (), the early time behavior is governed by ballistic motion, after which there is a crossover at a time of order to the asymptotic diffusive behavior of .
We now extend this analysis to obtain the distribution of firstpassage times for the walk to hit the last cookie at at time . Let be the probability that the time interval between the walk first hitting site and first hitting site equals . Similarly, let be the firstpassage probability that the walk hits site for the first time at time . Then
with . Because this relation is a convolution, the Laplace transform obeys
(7) 
Now the Laplace transform of is, using Eq. (5),
(8) 
The first term accounts for the direct step from to . The contribution of this direct process to the discrete generating function would be . In the continuum limit, the factor maps to the Laplace variable . The second term accounts for a single step to the left (factor ) and the subsequent firstpassage from to in the presence of a reflecting boundary at the origin. The first step of the walk is accounted for by defining .
We anticipate that the dominant contribution to in the longtime limit comes from terms where AHZ . In this limit, Eq. (8) reduces to
by expanding of Eq. (5) for large . Using this approximation for and taking the limit , corresponding to longtime behavior, can be written as
(9)  
where we use by the continuous variable in the last two lines. and we replace the discrete variable
From this formal solution, the cumulants of the firstpassage time may be obtained as the coefficients of in the power series of . For this we use the product representation of the function, and the series expansion of to give
Recognizing that the second sum can be expressed in terms of the Riemann zeta function, the cumulants are
(10) 
where
which gives , etc. This result agrees with the direct solution given in Eq. (6) for the average firstpassage time .
Finally, the time dependence of the firstpassage probability for the last cookie – the probability that the cookie at is first hit at time – is obtained by inverting the Laplace transform in Eq. (9). To simplify this calculation, we introduce the scaling variables
to rewrite the firstpassage probability in the scaling form
(11) 
For the case of no bias (), the integrand has simple poles on the negative real axis at
By evaluating the corresponding residues at these poles, we find the scaling function
(12) 
Because of the presence of an exponential cutoff, this series is easy to compute numerically and the result is displayed in Fig. 2, together with the small limiting behavior (corresponding to small for fixed or large for fixed ) that stems from the first term of Eq. (12):
For general the poles in Eq. (11) are branch points and we therefore evaluate this integral in a simpler manner by first using the binomial theorem to write Eq. (9) as
(13) 
Each term in this series can be easily back transformed term by term (see 29.3.82 in Ref. AS ). The scaling function thus has the form
(14) 
For , this function is identical to in Eq. (12) in spite of the apparent visual differences in these two formulae, since both functions originate from the same Laplace transform. Eq. (14) can be viewed as the large expansion of . As shown in Fig. 2, the term in this series representation
(15) 
together with the term in the small series for Eq. (12) give a fairly good approximation for over the entire range of for . Fig. 3 shows the solution Eq. (14), together with the exact numerical propagation of the probability distribution at different times (see Appendix).
iii.2 Location of the Last Cookie
Another important characterization of the excited random walk is the probability distribution for the location of the last cookie. Now the last cookie either remains stationary or it moves only to the right during each step of the underlying excited random walk. Consequently, we can write a simple relation between the firstpassage probability for hitting the last cookie, derived in the previous section, and the probability distribution of this last cookie.
The probability that the last cookie is located at site at time equals the joint probability that a walk first hopped to site before time , and that this walk then first hopped to site after time . That is,
(16) 
In the continuum limit, the Laplace transform of this equation is
where we used Eq. (7) to simplify the last factor. In the continuous space limit, the relation between and becomes
which is equivalent to
(17) 
Substituting the scaling form for into Eq. (17) and performing the derivative, we express the integrand in scaled units, after which the integral can be evaluated easily.
These steps lead to the following simple connection between the firstpassage and occupation probabilities of the last cookie:
(18) 
with the same scaling function for both quantities. We exploit this connection between and to obtain the finite time data in Fig. 3 by first calculating the position distribution of the last cookie from the numerical evaluation of the Master equation (see Appendix), and then using Eq. (18) to arrive to the scaling function .
Finally, we use Eqs. (14) and (18) to obtain the exact expression for the average position of the last cookie:
(19) 
where the proportionality constant
is obtained by first integrating (14) term by term and then using the identity knuth
The surprising point of this calculation is that as , while the mean firstpassage time for the walk to reach a distance scales as . More precisely,
(20) 
while naive dimensional scaling would suggest that implies . The fact that this naive scaling does not hold is a result of the broadness of the distribution of firstpassage times for the last cookie.
iii.3 First Passage to the Origin
As a complement to the statistical properties of the last cookie, we now study first passage to the origin, namely, the probability for the 1excited walk to reach for the first time at time . We first provide a heuristic argument (previously given in Ref. BW03 ) that the 1excited walk is recurrent and then we extend this line of reasoning to argue that the firstpassage probability asymptotically scales at .
If the walk is at a point for the first time, then the probability for the walk to subsequently reach without hitting the origin is accounts for the event where the walk immediately eats a cookie by making a step to the right, and the factor [see Eq. (1)] accounts for the probability that the first step is to the left and that the walk eventually eats another cookie on the right without hitting the origin. Thus, if the walk starts at site with the system initially full of cookies, the probability for the walk to reach (at least) an arbitrary point without hitting the origin is . Here the factor
(21)  
Since as , the random walk must necessarily return to the origin; that is, the 1excited random walk is recurrent, unless .
Since is the probability that the walk eats at least cookies without hitting the origin, is the probability that the walk eats exactly cookies. The average number cookies eaten by a walk that does not hit the origin is therefore . Thus for ) a walk that manages to avoid the origin typically wanders only a finite distance away (for a more rigorous argument see Sec. IV.3).
If the walk is at site for the first time, the conditional average time to reach site without hitting the origin is one with probability , and with probability (see Eq. (2)), that is . Thus the average time to reach site , without hitting the origin, is for large . From this relation between and , we infer that the survival probability, namely, the probability that the walk does not reach the origin by time , should vary as . Consequently, the firstpassage probability should asymptotically decay as
(22) 
For , the average time to reach the origin is infinite. Quite strangely, although it takes infinite time for the walk to hit the origin for any , the walk eats only a finite number of cookies, i.e., wanders only a finite distance from the origin, for . This result remains valid even if there is an initial cookiefree region at the origin, as this initial condition would affect the preasymptotic factors in Eq. (21).
To compute the firstpassage probability to the origin more precisely, it is useful to define two auxiliary quantities, and , as follows:

the probability of first hitting without hitting a time after the walk first hits .

the probability of first hitting without hitting a time after the walk first hits .
In the Laplace domain, these firstpassage probabilities are, using Eq. (II),
For example, for , the first step must be to the left (factor ) and then the factor is the firstpassage probability to the origin, without hitting , for a walk that starts at . Similarly, for , the term accounts for the transition in a single step. The factor in the second term accounts for a single step to the left after which the walk is at while the last cookie is at . Then the factor gives the firstpassage probability to without hitting the origin, when starting from .
Finally, we obtain the Laplace transform of the firstpassage probability to the origin for a walk that begins at with the system initially full of cookies by summing over all paths that contain first passages to the last cookie before the origin is reached. This gives
(23)  
In the limit , this Laplace transform is just the integral of the firstpassage probability over all time or, equivalently, the probability of eventual return to the origin. It is easy to verify that this return probability equals one. In the limit, the auxiliary probabilities and are
(24) 
Then
(25)  
This expression obviously equals one for all , while for extreme case of , . Thus eventual return to the origin is certain for any bias .
More formally, we denote the sum of the first terms of Eq. (23) as
(26) 
For finite , we use Eq. (24) to give, in the limit ,
(27) 
which goes to one as concmat . The use of the forms for and from (24) remains valid for of the order of or less. We now assume that the main contribution to comes from terms with in Eq. (26). That is,
(28) 
This finite sum can be written as the difference of the infinite sum (which equals 1) and the sum from to . In this second sum, for all terms as . Therefore we can replace the ratio of gamma functions by and the sum by an integral. These steps lead to
(29) 
This result for implies that for large times the firstpassage probability to the origin indeed behaves as .
Iv 1Excited Walk on the Infinite Line
iv.1 FirstPassage Probabilities
Using the same firstpassage analysis as that given for the 1excited walk on the halfline, we can immediately deduce the firstpassage probabilities for the 1excited walk on the infinite line to hit the left side and the right side of a cookiefree region. Thus the firstpassage probability for the walk to hit the left side of this gap is the same as the firstpassage probability to the origin for the halfline system. From Eq (22), , where we now write the subscript to denote hitting the left side of the gap. The firstpassage probability to the right side of the gap can immediately be obtained merely by interchanging and ; thus . Since the exponents in both of these firstpassage probabilities are smaller than 2, the mean time to hit either edge of the cookiefree domain is infinite.
iv.2 The Probability Distribution
Another basic characteristic of the 1excited walk is the probability distribution of displacements. An analytical treatment of this quantity is more involved than the semiinfinite system because of the existence of two moving boundaries. We thus turn to numerical simulations to provide a qualitative picture of this distribution.
Fig. 4 shows the scaled probability distribution for , 0.8, and 0.9. For , the probability distribution has a nearly Gaussian, but slightly asymmetric, appearance. As increases toward 1, this asymmetry becomes more pronounced, as shown in the figure. In addition, an apparent singularity in the distribution develops near the origin as . By running simulations for and to steps, we observe quite accurate data collapse for . However for , there is a small but persistent departure from data collapse for (inset to Fig. 4. We do not understand whether this anomaly arises from a slow correction to scaling term in the probability distribution or whether there exists an effect that lies outside of scaling.
For all values of , both and the variance grow as . The time dependence for the displacement is easy to understand on heuristic grounds. Once a cookiefree domain of linear dimension has been created, the domain then grows only when the walk reaches either edge of the domain. Since the motion of the walk is purely random within the domain, the probability that the walk hits either edge is proportional to . If the walk hits the right edge of the domain, then the walk will move ballistically for typically steps as . Thus the equation of motion for the domain length will be as . The solution to this equation gives for any value of . However, it bears mention that this naive approach gives the wrong value for the amplitude – – rather than that was derived in Eq. (20).
iv.3 Asymmetry
A useful geometric characteristic of the 1excited walk is the asymmetry in the span of the walk. Here the span is the length of the cookiefree region. For an unbiased random walk, the growth of the span with time and its distribution are well understood W , and we now investigate how the bias inherent in the 1excited walk influences the span. It is useful to characterize the span by the positions of the last cookie on the right and the last cookie on the left of the cookiefree region. We can understand the motion of these extreme cookies by the following heuristic argument. Consider the situation where the cookiefree gap has length and where the walk has just stepped to the extreme cookie on the right. As shown at the beginning of Sec. III.3, the probability that the walk eats another right cookie without eating a left cookie is .
With this basic result, the probability that the walk eats precisely consecutive cookies (we term this event a single “meal”) from the right edge of the cookiefree region is
(30) 
where we use the convention for . The probability that the walk eats precisely one cookie (the one at the starting position) is then .
When the initial cookiefree gap is large, this probability simplifies to
The dependence can be scaled out using the dimensionless variable and the rescaled probability
(31) 
Note that this probability is normalized, =1. Using this probability, the average relative number of consecutive cookies eaten from the right side of the gap, , is
(32) 
This result is paradoxical at first sight because, as we showed in Sec. III.3, the average time to return to the left side of the gap is infinite. However, the walk eats, on average, only a finite number of cookies if , even though the time taken for its meal on the right side of the gap is infinite.
The average relative number of eaten cookies from the left is also given by Eq. (32) after interchanging and . Although the 1excited walk eats, on average, infinitely many consecutive cookies from a single side of the gap during a single meal, the average ratio turns out to be finite for any value of . To show this we compute the joint probability of eating consecutive cookies on the right and then consecutive cookies on the left before the next right cookie is eaten, under the initial condition that the walk starts at the position of the right extreme cookie. This joint probability can be derived analogously to Eq. (30)
(33) 
It is again useful to define the rescaled variables and , as well as the rescaled probability
(34) 
where is the total number of cookies eaten in one rightleft cycle. One may check that this probability is normalized, .
Finally, the average ratio for the number of right cookies to total cookies eaten in a single cycle is
This integral can be performed with the help of 3.235 in Ref. GR (both for and ), and also using some basic properties of the Euler function AS to give
(36) 
This gives the asymptotically exact ratio for the number of right cookies to total cookies eaten in one cycle, under the assumption that the initial cookiefree gap is large. Although or is infinite, the dependence of the ratio is wellbehaved (Fig. 5).
Despite the fact that the average number of consecutive cookies eaten from the left or the right side of the gap in one cycle is infinite, our data clearly show that the positions of the two extreme cookies both grow as , but with very different amplitudes. To obtain an approximation for these amplitudes we now calculate the typical number and of eaten consecutive cookies from each side.
We estimate the value of by setting the probability of eating at least consecutive cookies on the right equal the probability of going to the left before right cookies are eaten. That is,
(37) 
This criterion gives a crude estimate for the number of right cookies eaten during a single consecutive string. For , Eq. (37) becomes
which leads to the estimate that the typical length of the string of eaten right cookies is . Applying this same argument to the left side of the cookiefree region, a string of consecutive left cookies are typically eaten. Their ratio thus equals the constant value
(38) 
As a function of time, clearly approaches a constant value as and it is easy to estimate its asymptotic value which is shown in Fig. 6, together with the prediction of Eq. (38).
To conclude this section, we point out another intriguing feature of the extreme cookies; namely the probability distribution in their positions. Quite strikingly, these distributions have shortdistance powerlaw tails, both for the left and for the right extreme cookies (Fig. 7) for . In contrast, for the pure random walk case of , the extreme cookie distribution is essentially the same as the span distribution, which is known to be a Gaussianlike function W . The origin of the shortdistance powerlaw tails in the probability distributions of the last cookie seem to be connected with the anomalous scaling of the firstpassage probability, but we do not have a definitive explanation for this phenomenon.
V 2Excited Random Walk
In the 2excited random walk, each lattice site initially has two cookies and the bias occurs as long as the walk lands on a site that contains at least one cookie. This 2excited random walk undergoes a transition between recurrence and transience as is increases beyond a critical value Z04 . Indeed, the excited walk for any appears to have qualitatively similar behavior as the 2excited walk. For simplicity, we thus focus on the 2excited random walk.
We first recount the qualitative argument for the transition between recurrence and transience of the excited walk that was given in Ref. Z04 . Whenever the walk is in a region that contains cookies, there is ostensibly a bias velocity . Thus the time that the walk takes to travel a distance should be . If this time is less than the total number of cookies in this region, then the walk does not have sufficient time to consume all the cookies as it moves. Consequently, the bias will persist forever and the walk will be transient. The condition for transience is therefore , or
(39) 
As we shall see, the behavior of the 2excited walk is qualitatively different in the regimes and .
v.1 Probability distributions
Fig. 8 shows scaled displacement probability distributions for the 2excited random walk for representative values of . As passes through , there is a sudden drop in the integrated probability that the walk lies in the range . Another basic feature of the distribution is the departure from scaling, as shown in Fig. 9. For , data collapse has not yet set in by time steps. If scaling were to hold, it would only arise after an extraordinarily long time. In the absence of such a clearly identifiable long time scale, it seems that the asymptotic distribution will not obey singleparameter scaling.
The transition in the behavior of the probability distribution is mirrored in the spatial profile of the uneaten cookies. For , the 2excited random walk is recurrent and thus the region close to the origin should ultimate be devoid of cookies. This behavior is illustrated in Fig. 10, where the scaled density of uneaten cookies is shown. For this density profile follows singleparameter scaling and the also decays to zero near the origin because of the recurrence of the walk for . On the other hand, for , the cookie density shows a very small departure from scaling and the density goes to a nonzero limit near the origin as a result of the transient nature of the walk. We can understand this limiting cookie density near the origin by the same argument that led to the criterion for in Eq. (39). For general , a walk typically resides for a time at each site and should eat cookies, thus leaving a residue of uneaten cookies per site. For the case shown in Fig. 10, we therefore expect 3/4 of a cookie per site near the origin.
A more detailed characterization of the cookie profile is provided by the size distributions of domains that contain one cookie and contain zero cookies. The distribution of onecookie domains is an exponentially decaying function of domain length (Fig. 11). This behavior is simple to understand. The process that gives rise to a onecookie domain as is a walk that makes a single step to the left after a run of consecutive steps to the right. During this run of rightward steps into previouslyunvisited territory (each occurring with probability ), one cookie is left behind at each site. A step to the left (which occurs with probability ) terminates this onecookie domain by a cookiefree gap. Thus the probability of a onecookie domain of length is simply . From linear leastsquares fits to the data in Fig. 11, the slopes for , and 0.9 are, respectively, , , and . On the other hand the corresponding slopes from by the distribution are , , and , in excellent agreement with the data, especially as .
The length distribution of zerocookie domains is more intriguing. Such domains arise both by the growth of isolated zerocookie domain and by the merging of two neighboring zerocookie domains when an intervening onecookie domain disappears. The zerocookie domains also have qualitatively different behavior for and . In the former case, the distribution has a secondary peak close to the maximumlength domains, while the distribution appears to have a pure powerlaw form for .
v.2 Transit time
Our numerical simulations indicate that the mean displacement of the 2excited walk grows with time as , with apparently dependent on . For the 3 cases of , 0.8 and 0.9, leastsquares fits to the data in the range give , and 0.85, respectively. Naively, one might anticipate that the transience of the walk for would lead to an average displacement that grows linearly with time. Indeed, the argument that gave the threshold bias was based on the walk possessing a nonzero bias velocity proportional to for . On the other hand, it has been proven that the mean velocity of the 2excited walk, defined as equals zero for all Z04 .
While we do not know how to determine the time dependence of the displacement, we can address the complementary problem of estimating how long it takes a 2excited walk to traverse a finite interval. We already know, from the 1excited walk, that the mean time to traverse any cookiefree interval within an infinite system is divergent. Thus we now consider the time for a 2excited walk to traverse a finite cookiefree interval. As we shall show, the transit time grows faster than linearly in the interval length when is close to but strictly less than 1.
The 2excited random walk starts at and we compute the time to reach , where . This inequality ensures that a “defect”, namely, a region where the walk takes a small number of consecutive steps to the left after a long string of rightward steps, will certainly occur in the interval. This defect represents the lowestorder perturbation to the purely ballistic motion for the 2excited walk when . After the defect has been completed and a step to the right has eventually been made, the walk finds itself at the point in a cookiefree region . We ask: what is the mean time for the walk to reach the right edge of this interval? As illustrated in Fig. 13, this system is a 1excited random walk in disguise.
We denote by the mean time for the 2excited walk to reach the point in this geometry. To simplify the calculation, we assume that whenever the left edge of the domain is reached and the last cookie is eaten, the next step of the walk is always to the right. Because this next step could be to the left, our approach represents a lower bound to the transit time that we denote by .
As defined in Eq. (1) and (2), with probability , the walk reaches without touching and the time for this event is . On the other hand, with probability the walk hits the left edge and then, according to our approximation, hops one step to the right. Since this second argument always equals 2 in all formulae involving and , we henceforth drop this argument for simplicity. After these two events, the walk is a distance 2 from the left edge of a cookiefree domain of length and the elapsed time is . Thus obeys the recursion
(40) 
If we allowed the walk to hop to the left (with probability ) when the leftmost cookie is reached, then Eq. (40) would be replaced by the exact equation
The additional terms represent the contribution of paths in which the random walk makes 1, 2, 3, etc. consecutive steps to the left upon hitting the last cookie before a step to the right is made. This exact equation is not recursive and for simplicity we therefore consider the approximation recursion for the lower bound in what follows.
By iterating (40), we express compactly as
(41)  
where is the unconditional exit time, namely, the time for the random walk to exit either side of the interval when the starting point is . Using this result for , as well as that for in Eq. (41), we obtain after straightforward algebra:
(42) 
The upper limit in the sum accounts for case where a cookiefree defect of length occurs close to the right edge of the domain so that terms or order are needed to account for the growth of the cookiefree defect. Thus we infer that mean time for a 2excited walk to traverse a finite system of length scales at least as fast as . This result is then consistent with growing slower than linearly with .
Vi Summary
The excited random walk is a very rich stochastic process in which the random walk motion is influenced by a transitory bias. Each site of a lattice is initially populated with a finite number of cookies. When a walk reaches a site that contains cookies, the walk eats the cookie and there is a right/left bias at the next step. On the other hand, the walk undergoes unbiased motion if the visited site does not contain cookies. Thus as the cookies are eaten, the bias gradually disappears. If a walk is recurrent, then the interplay between the recurrence and the bias is particular delicate and the excited random walk will have very different behavior from pure diffusion or from diffusion with a constant bias. For this reason, the excited random walk appears to be most interesting in one dimension.
In the simplest case of the 1excited walk on the positive halfline, we employed a physical approach, complementary to that of Dickman et al. DA01 ; DAA03 , to show that the firstpassage probability for the walk to hit the origin at time decays as for any . Thus the 1excited walk is always recurrent, except for the trivial case of perfect bias . There is especially paradoxical behavior in the case of , where the cookieinduced bias is directed toward the origin. Here the walk exhibits two seemingly incompatible features: the mean time for the walk to hit the origin is infinite, while the walk strays only a finite distance from the origin during this infinite excursion.
On the infinite line, the 1excited walk has the same firstpassage characteristics as on the semiinfinite line. However, the probability distribution of the walk and the probability distributions for the positions of the last cookie exhibit unusual features. For the probability distribution of the walk, there appears to be a singularity at the origin that does not obey scaling while the rest of the distribution does scale. There is also an asymmetry in the span of the walk that we were able to characterize by a probabilistic argument. The probability distributions for the positions of the leftmost and rightmost cookies have power law tails that we have been unable, thus far, to explain. These tails seem to be connected with the powerlaw behavior for the firstpassage probability itself.
The 2excited walk shows a wide range of strange phenomena. A transition between (conventional) recurrence and transience occurs as passes through a threshold value . For , the 2excited walk appears to qualitatively resemble the 1excited walk. On the other hand, for , the 2excited walk is transient and this attribute appears responsible for several unexpected features, such as the lack of scaling in the probability distribution of the walk and in the spatial distribution of cookies, as well as the existence of very different distributions of 1cookie and 0cookie domains. Most of these features do not yet have clean explanations and the model is ripe for further exploration.
Acknowledgements.
We thank E. BenNaim and P. Krapivsky for useful discussions. TA gratefully acknowledges financial support from the Swiss National Science Foundation under the fellowship 8220067591. SR acknowledges financial support from NSF grants DMR0227670 (at BU) and DOE grant W7405ENG36 (at LANL). *Appendix A Master equation
In this appendix, we outline a Master equation approach to determine the survival probability of the 1excited random walk on the infinite halfline. We merely follow the method introduced by Dickman et al. DA01 ; DAA03 to describe an unbiased walk between a fixed absorbing boundary at and a moving reflecting boundary that moves to the right with a specified probability each time it is hit by the walk. This perturbed random walk motion is essentially the same mechanism as that of the excited random walk. While the excited walk is ostensibly not Markovian, the process can be made Markovian by considering an extended state space that is defined by the position of the walk and also the positions of the leftmost and rightmost cookies, and . The exact master equation for the joint probability distribution of the position of the walk and the positions of the last cookies can be obtained in terms of a Markov process.
We consider the simpler problem of the 1excited random walk with an absorbing boundary at the origin. The system is now described by the joint probability for the cookie to be at and the leftmost cookie to be at , . This distribution obeys the recursion formula
(43) 
for . At the origin there is an absorbing boundary condition
(44) 
while the effect of the cookies is described by slightly modified equations for the probability when the walk is sufficiently close to the position of the last cookie:
These equations are defined over the wedge domain . We choose the initial conditions that the walk starts at and that the last cookie is also at ; thus . These equations can be iterated then numerically and this is the method that we used for the simulations presented in Sec. II.
From these Master equations, we compute the probability for the excited random walk to reach the origin for the first time at time , from which the survival probability of the walk follows. We merely outline this derivation as it very similar to that given by Dickman et al. DA01 ; DAA03 . We first rewrite the Master equations in terms of the generating function . In this representation, the solution can be written in the product form
(46) 
where