Maxima of Two Random Walks: Universal Statistics of Lead Changes E. Ben-Naim,1 P. L. Krapivsky,2,3 and J. Randon-Furling4 1Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2Department of Physics, Boston University, Boston, Massachusetts 02215, USA 3Institut de Physique Théorique, Université Paris-Saclay and CNRS, 91191 Gif-sur-Yvette, France 4SAMM (EA 4543), Université Paris-1 Panthéon-Sorbonne, 75013 Paris, France Weinvestigatestatisticsofleadchangesofthemaximaoftwodiscrete-timerandomwalksinone dimension. We show that the average number of lead changes grows as π−1lnt in the long-time limit. We present theoretical and numerical evidence that this asymptotic behavior is universal. 6 1 Specifically, this behavior is independent of the jump distribution: the same asymptotic underlies 0 standardBrownianmotionandsymmetricLévyflights. Wealsoshowthattheprobabilitytohaveat 2 mostnleadchangesbehavesast−1/4(lnt)n forBrownianmotionandast−β(µ)(lnt)n forsymmetric Lévyflightswithindexµ. Thedecayexponentβ ≡β(µ)variescontinuouslywiththeLévyindexµ n for 0<µ<2, while β =1/4 when µ>2. a J PACSnumbers: 05.40.Jc,05.40.Fb,02.50.Cw,02.50.Ey 8 ] h I. INTRODUCTION c e Extreme values play a crucial role in science, technol- m ogy, and engineering. They are linked to rare events, - large deviations, and optimization: minimizing a La- t a grangian, for instance, is a frequent task in physics. Ex- t t s treme values also provide an important characterization t. ofrandomprocesses,andthestudyofextremevaluesisa a significant area in statistics and probability theory [1–7]. m For a scalar random process, there are two extreme - values—the maximum and the minimum. It suffices to d n considerthemaximum. Takethemostbasicrandompro- o cess, Brownian motion [8–10]. By Brownian motion we c alwaysmeanstandardBrownianmotion,namelytheone- FIG. 1: Space-time diagram of the positions (thin lines) and [ dimensional Brownian process that starts at the origin: the maxima (thick lines) of two Brownian particles. In this 1 {B(t):t≥0} with B(0)=0. For the Brownian motion, illustration, the leadership changes twice. v the maximum process {M(t):t≥0} is defined by 1 5 M(t)= max B(s). (1) Thislesser-knownformula,whichwasdiscoveredbyLévy 0 0≤s≤t [8],provesusefulinmanysituations(see[9,12]);itcanbe 2 derivedusingthereflectionpropertyofBrownianmotion. 0 This maximum process is non-trivial as demonstrated Inthisarticle,westudytheleapfroggingofthemaxima 1. bytheabsenceofstationarity—theincrementM(t+τ)− of two Brownian motions (see Fig. 1). The probability 0 M(t) depends on both t and τ. In contrast, the posi- thatonemaximumexceedsanotherduringthetimeinter- 6 tion increment, B(t+τ)−B(t), depends only on τ; this val (0,t) was investigated only recently [13, 14]. Here we 1 stationarity is a crucial simplifying feature of Brownian explore additional features of the interplay between two : v motion. Still, the maximum process (1) has a number Brownianmaxima. Whenappropriate,weemphasizethe i of simple properties, e.g. the probability density of the universality of our results, e.g. some of our findings ap- X maximum m≡M(t) is a one-sided Gaussian ply to rather general Markov processes, e.g. to arbitrary r a symmetric Lévy flights. (cid:114) 2 (cid:18) m2(cid:19) One of our main results is that the average number of Q(m,t)= exp − , m≥0. (2) πt 2t leadchangesbetweenthetwoBrownianmaximaexhibits a universal logarithmic growth The joint probability distribution of the position and 1 maximum, B(t) = x and M(t) = m, also admits an (cid:104)n(cid:105)(cid:39) lnt (4) elegant representation [11] π in the long-time limit. This behavior holds for random (cid:114) 2 (cid:20) (2m−x)2(cid:21) walks, and more generally for identical symmetric Lévy Π(x,m,t)= (2m−x) exp − (3) πt3 2t flights. 2 We also study the probability f (t) to have exactly n probabilitydistributionofitspositionbecomesGaussian n lead changes during the time interval (0,t), see Fig. 1. We show that in the case of identical symmetric Lévy 1 (cid:18) x2(cid:19) P(x,t)= √ exp − , (7) flights, the probability fn(t) decays according to 2πt 2t f (t)∼t−β(µ)(lnt)n. (5) inthelong-timelimit. Themaximalpositionofthewalk n during the time interval (0,t) is The persistence exponent β(µ), that characterizes the m(t)=max{x(0),x(1),x(2),...,x(t)}. (8) probability that the lead does not change, depends only on the Lévy index µ. We evaluate numerically the quan- The probability distribution of the maximum is given by tity β(µ) for 0<µ<2. (2),andthejointposition-maximumdistributionisgiven This paper is organized as follows. In Sect. II we es- by (3). tablish the growth law (4). First, we provide a heuristic In the following, we consider two identical random derivation (Sect. IIA) relying on the probability den- walkers. We assume that both walkers start at the ori- sities (2)–(3). We also mention the generalization to gin, x (0) = x (0) = 0. Denote by x (t) and x (t) the 1 2 1 2 two random walks with different diffusion coefficients. positions of the walkers and by m (t) and m (t) the cor- 1 2 In Sect. IIB we establish a link with first-passage time responding maxima. If m (t)>m (t) the first walker is 1 2 densities and the arcsine law, and we employ it to pro- considered to be the leader. If m (t)=m (t) the notion 1 2 vide a more rigorous and more general derivation of (4). of the leader is ambiguous. This happens at t = 0, and In Sect. III we study the statistics of lead changes for we postulate that the first walker is the original leader. symmetric Lévy processes, particularly symmetric Lévy There are no lead changes as long as both walkers re- flights with index 0 < µ < 2. The average number of maininthex<0half-line. Oncethewalkersspendtime leadchangesisshowntoexhibitthesameuniversallead- in the x > 0 half-line, the identity of the leader can be ing asymptotic behavior (4) independent of the Lévy in- ambiguous if the probability density is discrete, e.g., dex. In Sect. IV we show that for identical symmetric random walks the probability to observe exactly n lead P(∆)= 1δ(∆−1)+ 1δ(∆+1) (9) changesbetweenthemaximabehavesast−1/4(lnt)n. We 2 2 conclude with a discussion (Sect. V). In the following, we tacitly assume that the probability densitydoesnotcontaindeltafunctions, soonce thetwo maxima become positive they remain distinct. Eventually the maximum of the second walker will II. THE AVERAGE NUMBER OF LEAD CHANGES overtake that of the first and the second walker turns into the leader. Yet, at some later moment, the leader- shipwillchangeagain. Thisleapfroggingproceedsindef- The question about the distribution of the number of initely. How does the average number of lead changes lead changes of Brownian particles is ill-defined if one vary with time? What is the distribution of the number considers the positions—there is either no changes or ofleadchanges? Inthis sectionweanswerthefirstques- infinitely many changes. The situation is different for tion. First, we provide a derivation of the asymptotic the maxima: the probability density f (t) to have ex- n growth law (4) using heuristic arguments. actly n changes during the time interval (0,t) is a well- definedquantityiftheinitialpositionsdiffer. Discretiza- tion is still necessary in simulations. The discrete-time A. Heuristic arguments framework also simplifies the heuristic reasoning which we employ below. The corresponding results for the Wenowarguethattheaveragenumberofleadchanges continuous-time case follows from the central-limit theo- exhibits the logarithmic growth (4). Let m (t) < m (t) rem: arandomwalkwithasymmetricjumpdistribution 2 1 and the lead changes soon after that. We use the short- that has a finite variance converges to a Brownian mo- hand notation m ≡ m (t). The probability density for tion. 1 this quantity is given by (2). We are interested in the Thus, we consider two discrete time random walks on asymptotic behavior (t (cid:29) 1) and in this regime, the aone-dimensionalline. Thepositionofeachwalkevolves random walker is generally far behind the maximum. according to This behavior is intuitively obvious, and it can be made quantitative—using (3) we compute the average size of x(t+1)=x(t)+∆(t). (6) the gap between the maximum and the position The displacements ∆(t) are drawn independently from (cid:90) ∞ (cid:90) m the same probability density P(∆) which is assumed to (cid:104)m−x(cid:105) = dm dx(m−x)Π(x,m,t) be symmetric, and hence (cid:104)∆(cid:105) = 0. We set the variance 0 −∞ to unity, (cid:104)∆2(cid:105) = (cid:82) d∆∆2P(∆) = 1. Further, we always (cid:114)2t = (10) assume that the walk starts at the origin x(0) = 0. The π 3 The second maximum can overtake the first if x (t) is 2 close to m. Since x ≤ m < m, both x and m are m m 2 2 2 2 close to m and the probability of this event is 0 0 (cid:114) 2 (cid:18) m2(cid:19) Π(m,m,t)= m exp − . (11) πt3 2t 0 t 0 τ t * Once the second particle reaches the leading maximum it will overtake it given that the first particle is typically m far behind, see (10). Therefore, the lead changes with n o rate siti 0 2m (cid:18) m2(cid:19) po Π(m,m,t)Q(m,t)= exp − πt2 t 0 t t+τ 2t * Integrating over all possible m yields time d(cid:104)n(cid:105) (cid:90) ∞ 2m (cid:18) m2(cid:19) 1 = dm exp − = (12) FIG. 2: Path transformation leading to (16): split the sec- dt πt2 t πt 0 ond Brownian path (top right) into its pre-maximum part which gives the announced growth law (4). anditspost-maximumpart(thesetwopartsareindependent The average (4) already demonstrates that the lead by property of the time at which a Markov process attains its maximum); reverse time in the pre-maximum part (in changing process is not Poissonian. For a Poisson pro- green)andshiftdownwardby−mthepost-maximumpart(in cess, the probability U to have exactly n lead changes n blue); concatenating this with the other, independent Brow- is fully characterized by the average ν(t)≡(cid:104)n(t)(cid:105): nian path (top left), one obtains a new Brownian path of νn double duration (bottom). Note that its final value will be Un = n!e−ν. (13) non-positive, and its maximum, also equal to m, will be at- tained at time t. This distribution would imply that the probability of having no lead change decays as exp[−(lnt)/π]∼t−1/π, that is, slower than the f0 ∼ t−1/4 behavior which has Themultiplicativefactor2ontheright-handsideof (15) been established analytically [13, 14]. takesintoaccountthatm (t)couldhavebeentheleader. 2 It is straightforward to generalize equation (4) to the InthecaseofBrownianmotion,thefirst-passagedensity situation where the two random walks have different dif- Φ(m,t) is known [15], e.g. it can be derived using the fusion coefficients, denoted by D1 and D2. By repeating method of images. This allows one to compute the inte- the steps above, we get gral in (15). √ 2 D D Interestingly, one can evaluate the integral in Eq. (15) (cid:104)n(cid:105)(cid:39) 1 2 lnt. (14) without reference to the explicit forms of Q and Φ. This π D +D 1 2 evaluation uses path transformations in the same spirit In particular, in the limit when one of the particles dif- asthewell-knownVerwaatconstructionforBrownianex- fuses much more√slowly than the other, (cid:15)=D1/D2 →0, cursions[16]. Onenoticesthattheintegrandin(15)cor- we have (cid:104)n(cid:105)(cid:39)(2 (cid:15)/π)lnt. responds to the product of densities associated with two independent paths: (i) a path first hitting m at time t, and(ii)apathhavingon(0,t)amaximumequaltomat- B. First-passage analysis tained at some time τ . One then establishes a bijection ∗ between pairs of such paths and paths of duration 2t, as Here, we present an alternative approach which com- illustratedinFig.2. Thispathtransformationprocedure plementstheheuristicargumentsgiveninSect.IIA.This produces paths of duration 2t attaining their maximum approach is formulated in continuous time, and it ac- at time t, and having a non-positive final value. countsfortheuniversalityobservedinthenumericalsim- When integrating over m the product Φ(m,t)Q(m,t), ulations (see Fig. 3). one therefore obtains one half of the probability density Ifm1(t)=m>0,theprobabilitythatm2 becomesthe associated with Brownian paths of duration 2t attaining leaderintheinterval(t,t+∆t)isgivenbytheprobability their maximum at time t. This density is easily derived thatthefirst-passagetimeofx2atlevelmisin(t,t+∆t). fromthefamousarcsinelaw[8,17],whichstatesthatthe Let us write Φ(m,t) for the first-passage density, and probability density for the time τ at which a Brownian recall that the density of the maximum, Q(m,t) is given motion attains its maximum over a fixed time interval by(2). Byintegratingoverm,weobtaintheaveragerate (0,T) is given by 1/[π(cid:112)τ(T −τ)]. One thus obtains of lead changes as a first-passage equation d(cid:104)n(cid:105) (cid:90) ∞ (cid:90) ∞ 1 1 1 =2 dmΦ(m,t)Q(m,t). (15) dmQ(m,t)Φ(m,t)= (cid:112) = . (16) dt 0 0 2 π t(2t−t) 2πt 4 By combining (16) with (15), one recovers the heuris- 6 tic prediction (12) (cid:104)n(cid:105) (cid:39) Alnt and confirms A = 1/π. 0.34 Moreover, the derivation is now sufficiently general as it 5 0.32 applies to all random walks with symmetric jump distri- 0.30 butions. Hencetheleadingasymptoticbehavior(4)does 4 0.28 ntroitbudteiponen,dinofnultlhaegdreeetmaielsntofwtihthe (tshyemumnievterrics)aljiutymopfdtihse- 3 0.26101 102 103 10t4 105 106 107 amplitude observed in numerical simulations. The first- µ=1/2 2 µ=1 passageequationandpathtransformationprocedurecan µ=3/2 be applied to random walks that do not converge to 1 random walk Brownian motion, as discussed in the next section. (ln t)/π 0 0 1 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 10 III. LE´VY FLIGHTS t FIG. 3: The average number of lead changes (cid:104)n(cid:105) versus time Brownian motion belongs to a family of (symmetric) t. Shown (top to bottom) are simulation results for identical Lévyprocesses[8,18–20]. Theseprocessesarestationary, Lévy flights with index µ=1/2,µ=1,µ=3/2 and for ran- homogeneous and stable. In a discrete time realization, domwalks(µ>2). Theinsetshowsdln(cid:104)n(cid:105)/dlntversustime a Lévy process becomes a Lévy flight. Lévy flights are t,demonstratingthattheamplitudeA(µ)in(20)isuniversal ubiquitousinNature,see[21–24]. Here,weexaminelead and equal to 1/π. changes of the maxima of two identical Lévy flights. For Lévyflights,thejumpdistributionP(∆)hasabroadtail, 100 andtheLévyindexµquantifiesthistail,P(∆)∼|∆|−µ−1 µ=3/2 as |∆| → ∞. For simplicity, we choose the purely alge- µ=1 braic jump distribution, µ=1/2 10-1 (cid:40) P(∆)= 0µ2|∆|−µ−1 ||∆∆||<>11., (17) (t,µ) 0 f 10-2 When µ > 2, Lévy flights are equivalent to an ordinary random walk. For true Lévy flights 0 < µ ≤ 2; in this rangethevariance(cid:104)∆2(cid:105)diverges. For0<µ≤1eventhe averagejumplength(cid:104)|∆|(cid:105)becomesinfinite. Generallyfor 10-3 µ<2 the displacement x(t) scales as t1/µ. 100 101 102 103 104 105 106 107 108 t Wenowshowthattheaveragenumberofleadchanges exhibits logarithmic growth. Denote by Q (m,t) the µ FIG.4: Thesurvivalprobabilityf (t,µ)versustimetforLévy 0 probability density of the maximum process and by flights. Shownare(fromtoptobottom)simulationresultsfor Φµ(m,t) the density of the first passage time at level m. the values µ=3/2, µ=1, and µ=1/2. Following the same reasoning as in Sect. IIB, we write for the rate at which lead changes take place: Consequently, the average number of lead changes grows d(cid:104)n(cid:105) (cid:90) ∞ logarithmically with time, =2 dmΦ (m,t)Q (m,t). (18) dt µ µ 0 (cid:104)n(cid:105)(cid:39)A(µ) lnt. (20) We now invoke the scaling properties of Lévy flights Our numerical simulations confirm that for identical Φ (m,t)(cid:39)m−µΦ (1,m−µt), random walks on a line, and even for random walks on µ µ (19) the lattice with the discrete jump distribution (9), the Q (m,t)(cid:39)t−1/µQ (t−1/µm,1). µ µ amplitudeA(µ)isindependentofthedetailsofthejump distribution. Furthermore, the universality continues to These scaling forms reflect that the maximum process hold for the Lévy flights, so A=1/π independent of the is (µ)-stable and the first-passage time process is (1/µ)- Lévyindexµ. Thisisconfirmedbynumericalsimulations stable. Combining (18) and (19) we get (see Fig. 3). Theuniversalitycanbeunderstoodbynotingthatthe d(cid:104)n(cid:105) (cid:90) ∞ = 2 dmm−µΦ (1,m−µt)t−1/µQ (t−1/µm,1) path transformation procedure described at the end of dt 0 µ µ the previous section, as well as the arcsine law, are valid 2(cid:90) ∞ dz A(µ) for Lévy flights. The key requirements are to have cyclic (cid:39) Φ (1,z−µ)Q (z,1)= . t zµ µ µ t exchangeability of the jumps (which is guaranteed when 0 5 0.36 4 0.35 0.34 3.5 t1/4 F 1 0.33 1/2 (t1/4 F ) 0.32 3 2 0.31 β 0.3 2.5 0.29 0.28 2 0.27 1.5 0.26 0.25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 µ 0 1 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 10 t FIG. 5: The persistence exponent β versus the index µ char- acterizing (symmetric) Lévy flights. FIG.6: Thequantities(Fnt1/4)1/nversustimetforn=1and n = 2. The data corresponds to simulations of two ordinary random walks in one dimension. theseareindependentandidenticallydistributed),acon- tinuous and symmetric jump distribution, and a certain lead change, given that it takes place after time s: type of regularity for the supremum, see [18, 20, 25, 26]. (cid:20) (cid:21) The validity of the arcsine law implies that we can use ρ (τ)= 1 −df0(τ) ∼s1/4τ−5/4. (22) (16). Thus the leading asymptotic (4) holds for a wide s f (s) dτ 0 class of symmetric Lévy processes, and in particular, symmetric Lévy flights. Furthermore,thequantityf1(t)istheprobabilitythat the first lead change occurs at some time s≤t and that Next, we examine the probability f (t,µ) that there 0 the next one occurs at some time τ ≥t. The probability are no lead changes till time t. Our simulations show density associated with such a configuration is simply that this probability decays algebraically (see Fig. 4): proportional to f (t)∼t−β(µ). (21) (cid:20) df (cid:21) (cid:90) ∞ 0 − 0 × dτρ (τ)∼s−5/4×s1/4t−1/4 ∼s−1t−1/4. ds s t The persistence exponent β(µ) varies continuously with µ, namelyβ(µ)isamonotonicallydecreasingfunctionof Integrating over s [we could introduce a cut-off ∆s, but µ, see Fig. 5. For µ > 2, Lévy flights are equivalent to this is akin to a discretization of time, so in the large t ordinary random walks, so β(µ)= 1 for µ>2. limit we might as well set 1m (τ)>...>m (τ), 1 2 k We verified Eq. (24) numerically for n = 1 and n = 2, for τ =1,2,...,t. Based on (21), we anticipate that the see Fig. 6. probability S (t,µ) decays algebraically, k ForLévyflights,thedistributionofthenumberoflead changes can be established in a similar manner. Using Sk(t,µ)∼t−βk(µ). (26) (21) and following the derivation of (23) we get Similaralgebraicdecayswithk-dependentexponentsde- f (t)∼t−β(µ)(lnt)n, (25) n scribetheprobabilitythatthepositionsofrandomwalks remain perfectly ordered, the probability that one ran- where β(µ) is the aforementioned persistence exponent. dom walk remains the leader, etc. [32–37]. Using numerical simulations, we verified that (25) holds For k = 2, the exponent appearing in (26) is our ba- forn=1andn=2forafewrepresentativevaluesofthe sic persistence exponent: β (µ) ≡ β(µ). The exponents Lévy index in the µ<2 range. 2 β (µ) do not depend on µ when µ > 2, i.e., for ran- k dom walks. For ordinary random walks, the exponents β were studied numerically in [13], while approximate V. DISCUSSION k values for these exponents were computed in [38]. Generallyonewouldliketoexplorethestatisticsofor- The simplest model of correlated random variables is dering and lead changes for a collection of random vari- a one-dimensional discrete-time random walk. Its maxi- ables. Even the case of Markovian random variables is mumevolvesbyarandomprocessthatexhibitsanumber far from understood. Non-Markovian random variables ofremarkablefeatures. Someofthesepropertiesareclas- appearintractable,butsometimestheprogressisfeasible sical [8–10], while others were discovered only recently duetohiddenconnectiontoMarkovianprocesses. Forin- [12, 27–29]. For instance, the probability distribution stance, inthecaseofBrownianmaximathefirst-passage forthetotalnumberofdistinctmaximalvalues(records) processes are Markovian. This allows one to use results achieved by the walk is universal [27], i.e., independent fromclassicalfluctuationtheorytoderivethepersistence of the details of the jump distribution, as long as the exponentβ =1/4,see[14],andyieldsanotherderivation jump distribution is symmetric and not discrete. This of the logarithmic growth of the average number of lead universalityalsoholdsforsymmetricLévyflights, andin changes [39]. For Lévy flights, however, the existence of fact, it is rooted in the Sparre Andersen theorem [30]. leapovers at the first passage time of certain points [40] In the case of multiple identical random walks, the total leads to a breakdown of this approach. New results in number of records has been investigated in [31]. this direction represent challenges for future work. In this article, we studied a sort of “competition” between maxima of two identical random walks. 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Maxima of Two Random Walks: Universal Statistics of Lead Changes

Similar Maxima of Two Random Walks: Universal Statistics of Lead Changes