Maxima of Two Random Walks: Universal Statistics of Lead Changes

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by  E. Ben-Naim

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 Th´eorique, Universit´e Paris-Saclay and CNRS, 91191 Gif-sur-Yvette, France
4SAMM (EA 4543), Universit´e Paris-1 Panth´eon-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 standardBrownianmotionandsymmetricL´evyflights. Wealsoshowthattheprobabilitytohaveat
2 mostnleadchangesbehavesast−1/4(lnt)n forBrownianmotionandast−β(µ)(lnt)n forsymmetric
L´evyflightswithindexµ. Thedecayexponentβ ≡β(µ)variescontinuouslywiththeL´evyindexµ
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,whichwasdiscoveredbyL´evy
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 L´evy 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 L´evy
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 L´evy 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 L´evy 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 L´evy processes, particularly symmetric L´evy
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 L´evy 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- <n>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
L´evyprocesses[8,18–20]. Theseprocessesarestationary,
L´evy 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 L´evy process becomes a L´evy flight. L´evy 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 L´evy flights. For
L´evyflights,thejumpdistributionP(∆)hasabroadtail, 100
andtheL´evyindexµquantifiesthistail,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, L´evy flights are equivalent to an ordinary
random walk. For true L´evy 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)versustimetforL´evy
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 L´evy 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 L´evy flights, so A=1/π independent of the
is (µ)-stable and the first-passage time process is (1/µ)-
L´evyindexµ. 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 L´evy 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) L´evy 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 L´evy processes, and in particular,
symmetric L´evy 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, L´evy 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 1<s<t−1], we find
4
In the marginal case µ=2, the mean-square displace- (cid:90) t−1
ment hasalogarithmic enhancementoverthe classicdif- f (t)∼ dss−1t−1/4 ∼t−1/4(lnt).
1
fusive growth, (cid:104)x2(cid:105) ∼ tlnt. Consequently, the conver- 1
gence toward the ultimate asymptotic behavior is very
Hence,thereisalogarithmicenhancementoftheprob-
slownearµ≈2. Ournumericalsimulationssuggestthat
ability to have one lead change compared with having
in the marginal case µ = 2, there may be a logarithmic
none. The above argument can be generalized to arbi-
correction to the algebraic decay (21).
trary n to yield
f ∼t−1/4(lnt)n. (23)
n
IV. DISTRIBUTION OF THE NUMBER OF One can establish this general behavior by induction:
LEAD CHANGES
(cid:90) t−1 (cid:20) df (s)(cid:21)(cid:90) ∞
f (t) ∼ ds − n dτρ (τ)
We now focus on the probability fn(t) to have exactly n+1 1 ds t s
n lead changes until time t. For two identical random (cid:90) t−1
walks, the probability that there are no lead changes de- ∼ ds(lns)ns−5/4s1/4t−1/4
cays as f ∼ t−1/4 in the long-time limit [13, 14]. Since 1
0 (cid:90) t−1
fn(t) is the probability that the (n+1)-st lead change ∼ t−1/4 ds(lns)ns−1
takes place after time t, the probability density for the
1
(n + 1)-st lead change to occur at time t is given by ∼ t−1/4(lnt)n+1.
−df /dt. In particular, from f , one can write down the
n 0
conditionalprobabilitydensityρ ofthetimeofthenext Thus, there are logarithmic corrections for all n.
s
6
Weprobedthebehaviorofthecumulativedistribution One natural generalization of the competing maxima
(cid:80)
F (t)= f (t) using numerical simulations. The problem is to an arbitrary number k of identical random
n 0≤k≤n k
quantityF (t)istheprobabilitythatthenumberoflead walksorL´evyflights. Weexpectthattheaveragenumber
n
changes in time interval (0,t) does not exceed n. The ofleadchangesamongthemaximaofk identicalrandom
dominantcontributiontoF isprovidedbyf andhence walkers still exhibits a universal logarithmic growth, al-
n n
beit with a k-dependent prefactor. Another quantity of
F (t)∼t−1/4(lnt)n. (24) interest is the survival probability S (t,µ) that k L´evy
n k
maxima remain ordered, m (τ)>m (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
ForL´evyflights,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
L´evy 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
universalityalsoholdsforsymmetricL´evyflights, andin
changes [39]. For L´evy 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” be-
tween maxima of two identical random walks. In partic-
ular, we examined the average number of lead changes
and the probability f (t) to have exactly n changes. We Acknowledgments
n
found that the average number of lead changes exhibits
a universal logarithmic growth (4). This asymptotic be- We benefited from discussions with S. N. Majumdar.
havior also holds for symmetric L´evy flights with arbi- Two of us (PLK and JRF) thank the Galileo Galilei
trary index. In contrast, the probability distribution Institute for Theoretical Physics for hospitality during
f (t) is not universal: the persistence exponent β(µ) in theprogramon“StatisticalMechanics, Integrabilityand
n
(25) depends on the L´evy index µ. The most interesting Combinatorics” and the INFN for partial support. The
challengeforfutureworkistodetermineanalyticallythe workofEBNwassupportedthroughUS-DOEgrantDE-
continuously varying exponent β(µ) in the range µ<2. AC52-06NA25396.
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