Record statistics for random walk bridges
RRecord statistics for random walk bridges
Claude Godr`eche
Institut de Physique Th´eorique, Universit´e Paris-Saclay, CEA and CNRS, 91191Gif-sur-Yvette, France
Satya N. Majumdar
Universit´e Paris-Sud, LPTMS, CNRS (UMR 8626), 91405 Orsay Cedex, France
Gr´egory Schehr
Universit´e Paris-Sud, LPTMS, CNRS (UMR 8626), 91405 Orsay Cedex, France
Abstract.
We investigate the statistics of records in a random sequence { x B (0) =0 , x B (1) , · · · , x B ( n ) = x B (0) = 0 } of n time steps. The sequence x B ( k )’s representsthe position at step k of a random walk ‘bridge’ of n steps that starts and ends at theorigin. At each step, the increment of the position is a random jump drawn from aspecified symmetric distribution. We study the statistics of records and record agesfor such a bridge sequence, for different jump distributions. In absence of the bridgecondition, i.e., for a free random walk sequence, the statistics of the number and ages ofrecords exhibits a ‘strong’ universality for all n , i.e., they are completely independentof the jump distribution as long as the distribution is continuous. We show that thepresence of the bridge constraint destroys this strong ‘all n ’ universality. Neverthelessa ‘weaker’ universality still remains for large n , where we show that the record statisticsdepends on the jump distributions only through a single parameter 0 < µ ≤
2, knownas the L´evy index of the walk, but are insensitive to the other details of the jumpdistribution. We derive the most general results (for arbitrary jump distributions)wherever possible and also present two exactly solvable cases. We present numericalsimulations that verify our analytical results. a r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n ecord statistics for random walk bridges
1. Introduction and summary of main results
During the last few years, there has been a growing interest in the study of records.Records have not only become popular in our societies as one often hears and reads,in the media, about record breaking events, including for instance sports [1] or weatherrecords [2], but they have also found interesting applications in various areas ofsciences. Records are often useful to characterize the statistics of a (discrete) timeseries x (0) , x (1) , . . . , x ( n ) where a record happens at time k if x ( k ) is larger than theprevious values x (0) , x (1) , · · · , x ( k − n ? (b) what is the probability that a recordis broken at step n ? how long does a record survive and in particular what is the age ofthe longest lasting record? These questions are fully understood in the case where thevariables x ( i )’s are independent and identically distributed (i.i.d.) [21, 22] (see also Ref.[23] for a short review), although more refined questions about first passage propertiesof records in this i.i.d. case were only recently investigated [24, 25]. However, manyapplications of records in statistical physics have highlighted the importance of strongcorrelations between the random variables x ( i )’s, for which much less is known. Withthat perspective, it was demonstrated rather recently that one-dimensional randomwalks offer a very useful instance of strongly correlated variables where the impact ofcorrelations on records can be studied analytically [5, 6, 7, 8, 9, 10]. As we recall itbelow, a remarkable feature of the record statistics of RWs with continuous jumps isthat it is completely universal, i.e., independent of the jump distributions, even for afinite number of steps [5]. It is thus natural to ask whether this universality still holdsfor constrained random walks. This question is the main motivation for the presentwork where we consider random walk bridges, where the walker is constrained to startand end at the same point, which are relevant, for instance, to study periodic correlatedtime series.Let us consider a time series where the x ( k )’s are the positions of a random walker(RW) after step k , evolving according to the Markov rule: x (0) = 0 , x ( k ) = x ( k −
1) + η ( k ) , (1.1)where η ( k )’s are i.i.d. random variables. We denote by p ( η ) their symmetric probabilitydistribution function (PDF) [with p ( η ) = p ( − η )]. In the following, we will focus onthe positions of RW bridges { x B ( k ) } , 0 ≤ k ≤ n which are RWs as defined in (1.1)with the additional constraint that they return back to the origin after n time steps, x B ( n ) = x B (0) = 0 (see Fig. 1).For the purpose of our study, it is important to distinguish between jumpdistributions p ( η ) that are respectively discrete and continuous. A representative of ecord statistics for random walk bridges p d ( η ) = 12 δ ( η + 1) + 12 δ ( η − , (1.2)where the superscript d refers to this particular discrete case. In this case, the walkermoves on a lattice of the integers. In contrast, when the jump distribution is continuous,we will denote it generally by a superscript c , i.e., p c ( η ). Let ˆ p c ( k ) = (cid:82) ∞−∞ d η p c ( η ) e i k η denote the Fourier transform of the jump distribution. It turns out that for manyproperties associated with the record statistics, only the small k -behavior of the jumpdistribution matters. Generically, for small k , one typically hasˆ p c ( k ) = 1 − | a k | µ + o ( | k | µ ) , (1.3)where 0 < µ ≤ a > µ = 2, the variance σ = (cid:104) η (cid:105) is finite. Consequently, a = σ / n , to the Brownian motion. In contrast,for µ <
2, the jump distribution has a heavy tail, p c ( η ) ∼ | η | − − µ for large η , leadingto a diverging second moment. A random walk with µ < µ . Among the class of continuous jump distributions with a finite σ , a specialrole is played by the exponential distribution p e ( η ) = 12 b e −| η | /b (1.4)where b > µ = 2 and a = b .We will see later that this exponential distribution belonging to the continuous family,and the lattice RW in Eq. (1.2) belonging to the discrete family, represent two rarecases where the record statistics for the bridge sequence is exactly solvable.When studying the sequence of records of a time series x (0) , x (1) , · · · , x ( n ), a recordhappens at step k if it is strictly larger than the previous entries, i.e., x ( k ) > max( x (0) , x (1) , · · · , x ( k − x (0) is a record. We emphasize that the strict inequality in Eq.(1.5) is of special importance in the case of discrete random walks (1.2). We denoteby R ( n ) ≥ n steps. Other important observables for ourstudy are the ages of these records (see Fig. 1). We define τ k as the number of stepsbetween the k -th and ( k + 1)-th records: this is the age of the k -th record, i.e. thetime up to which the k -th record survives. Note that the last record occurring before n is still a record at step n and we denote by A n the current age of this record at step n (see Fig. 1). The typical fluctuations of these sequences of ages is rather simple tostudy. For instance, the typical age of a record is simply related to the average numberof records (cid:104) R ( n ) (cid:105) via (cid:96) typ ∼ n/ (cid:104) R ( n ) (cid:105) . However, it has been shown that the statisticsof these sequences is dominated, for RWs, by rare events [5, 10]. To characterize the ecord statistics for random walk bridges τ A n τ τ ρ timesteps nx max ,B ( n ) x B ( k ) τ ρ ρ ρ Figure 1.
A realization of a random walk bridge x B ( k ) with an exponential jumpdistribution ( α = e) with n = 20 steps. Here the number of records is R e B ( n ) = 5. The τ i ’s denote the ages of the records and the ρ i ’s are the increments between successiverecords. Finally, A n denotes the age of the last record. The joint PDF of the τ i ’s, ρ i ’s, A n and R e B ( n ) is given in Eq. (3.21) and constitutes one of the main result of thispaper. fluctuations of these rare events, it is useful to consider the probability Q ( n ) that thelast time interval A n is the longest one, which we call in the following the probabilityof record breaking. For a RW with a given number of records R ( n ) = m , the sequenceof the ages of the records is τ , . . . , τ m − and A n . It is then useful to introduce the jointprobability Q ( m, n ) defined as Q ( m, n ) = Pr( A n ≥ max( τ , . . . , τ m − ) , R ( n ) = m ) . (1.6)The probability of record breaking Q ( n ) is then obtained by summing Q ( m, n ) over allpossible number of records in the time series, i.e., Q ( n ) = ∞ (cid:88) m =1 Q ( m, n ) . (1.7)Another important observable to characterize the rare and extreme fluctuations of thesequences of ages is the age of the longest lasting record (cid:96) max ( n ) which is defined as (cid:96) max ( n ) = max( τ , τ , . . . , τ m − , A n ) , (1.8)whose fluctuations, for RWs, were recently demonstrated to be very sensitive to the lastrecord [10]. Notations:
In the following, we will denote by R α ( n ), Q α ( n ) and (cid:96) α max ( n ) thequantity defined above for free random walks (1.1) where the superscript α refers todifferent jump distributions (1.2, 1.3, 1.4), where α = d corresponding to discrete jump distribution p d ( η ) , c corresponding to generic continuous jump distribution p c ( η ) , e corresponding to exponential jump distribution p e ( η ) , (1.9) ecord statistics for random walk bridges p d ( η ), p c ( η ) and p e ( η ) are defined respectively in Eq. (1.2), (1.3) and (1.4).Besides we will use the notations R αB ( n ), Q αB ( n ) and (cid:96) α max ,B ( n ) for the number of records,the probability of record breaking (1.7) and the age of the longest lasting record (1.8)for random bridges, where the subscript B refers to bridges.Before turning to our results for such constrained RWs, let us first remind the mainknown results for the record statistics of free RWs, where the walker can end up at anyposition x n after n time steps [5, 6, 7, 8, 9, 10]. We first focus on the case of continuous(symmetric) jump distributions p c ( η ) (1.3). In this case, a remarkable property is thatthe full statistics of records, including the number of records R c ( n ) as well as Q c ( n ) (1.7)and (cid:96) cmax ( n ) (1.8) are completely universal, i.e., independent of the jump distribution p c ( η ) – including L´evy flights – even for finite n [5]. This universal behavior stems fromthe universality of the Sparre Andersen theorem [26]. In particular, for large n , theaverage number of records (cid:104) R c ( n ) (cid:105) behaves for large n as [5] (cid:104) R c ( n ) (cid:105) ∼ A c √ n , A c = 2 √ π , as n → ∞ , (1.10)independently of 0 < µ ≤ p c ( η ) in (1.3). Thisbehavior (1.10) should be compared to the logarithmic growth found for i.i.d. randomvariables [22]. Moreover, it is also possible to compute exactly the full probabilitydistribution P c ( m, n ) = Pr( R c ( n ) = m, n ) of the number of records which, for large n and large m keeping X = m/ √ n fixed, takes the scaling form [5] P c ( m, n ) ∼ √ n ϕ c (cid:18) X = m √ n (cid:19) where ϕ c ( X ) = 1 √ π e − X , X > . (1.11)For free random walks, it is also possible to compute exactly Q c ( n ) in Eq. (1.7) andshow that it converges, for large n , to a non-trivial universal constant given by [5]lim n →∞ Q c ( n ) = Q c ( ∞ ) = (cid:90) ∞ d x
11 + √ π x e x erf( √ x ) = 0 . . . . , (1.12)a constant which also arises in the excursion theory of Brownian motion [27, 28].Interestingly, in the case of free RWs, this probability Q c ( n ) turns out to be relatedto the average value of (cid:104) (cid:96) cmax ( n ) (cid:105) in Eq. (1.8) via the relation [27] (cid:104) (cid:96) cmax ( n + 1) (cid:105) = (cid:104) (cid:96) cmax ( n ) (cid:105) + Q c ( n ) , (1.13)implying in particular thatlim n →∞ (cid:104) (cid:96) cmax ( n ) (cid:105) n = λ cmax = Q c ( ∞ ) = 0 . . . . , (1.14)while the full distribution of (cid:96) cmax ( n ) /n was studied more recently in [29] (see also [30]).For discrete free random walks, with a jump distribution given in Eq. (1.2), it wasshown in [5] that the statistics of records is quantitatively different. In particular, inthis case, the mean number of records (cid:104) R d ( n ) (cid:105) also grows like √ n but with a differentprefactor (cid:104) R d ( n ) (cid:105) ∼ A d √ n , A d = (cid:114) π , as n → ∞ , (1.15) ecord statistics for random walk bridges ↵ = d ↵ = c ↵ = d ↵ = e ↵ = c h R ↵ ( n ) i ⇠ r ⇡ p n ⇠ p ⇡ p n h R ↵ B ( n ) i ⇠ p ⇡ p n ⇠ p ⇡ p n ⇠ A c B ( µ ) p n Q ↵ ( n ) ⇠ Q d ( ) = Q c ( )= 0 . . . . ⇠ Q d ( ) = Q c ( )= 0 . . . . Q ↵ B ( n ) ⇠ Q d B ( ) = Q e B ( )= 0 . . . . ⇠ Q d B ( ) = Q e B ( )= 0 . . . . h ` ↵ max ( n ) i n ⇠ dmax = cmax = Q d , c ( ) ⇠ dmax = cmax = Q d , c ( ) h ` ↵ max ,B ( n ) i n ⇠ dmax ,B = emax ,B = 0 . . . . ⇠ dmax ,B = emax ,B = 0 . . . . Free RW RW Bridge jumpdistribution discrete discretecontinuous continuousexponential ?? Table 1.
Summary of the results for the record statistics of RW bridges, obtainedin the present paper. For comparison, we have also presented, in the left part of thetable, the results for free RW, obtained in Refs. [5]. Note that, for the free RW, theresults for continuous distributions are completely universal, and thus also hold foran exponential distribution, which is not true for the RW bridges. The expression ofthe constants Q d , e B ( ∞ ) and λ d , emax ,B is given in Eqs. (4.9) and (4.30) respectively. Thecomputation of Q c B ( n ) and of (cid:104) (cid:96) cmax ,B ( n ) (cid:105) for a continuous distribution with arbitraryL´evy index µ [see Eq. (1.3)] remains an open question – hence the question marks (?)in the table. which is 1 / √ R d ( n ), P d ( m, n ) = Pr( R d ( n ) = m, n ), takes the following scalingform, for large n and large m keeping X = m/ √ n fixed [5, 8]: P d ( m, n ) ∼ √ n ϕ d (cid:18) X = m √ n (cid:19) , ϕ d ( X ) = (cid:114) π e − X , X > , (1.16)which is simply obtained from its continuous counterpart ϕ c ( X ) in Eq. (1.11) with thesubstitution X → √ X . On the other hand, the results for the probability of recordbreaking Q d ( n ) behaves, for large n , exactly as in the continuous case (1.12), i.e.lim n →∞ Q d ( n ) = Q d ( ∞ ) = Q c ( ∞ ) = (cid:90) ∞ d x
11 + √ π x e x erf( √ x ) = 0 . . . . . (1.17)Moreover, the relation in Eq. (1.13) also holds for the corresponding discrete quantities,from which it follows, using Eq. (1.17), thatlim n →∞ (cid:104) (cid:96) dmax ( n ) (cid:105) n = λ dmax = Q d ( ∞ ) = 0 . . . . . (1.18)For random walk bridges, the situation is quite different. As expected, the statisticsof records for discrete (1.2) and continuous (1.3) jump distributions are still different, as ecord statistics for random walk bridges n , is not universal any more and depends on p c ( η ).One expects however, and we can show it explicitly for the average number of records (cid:104) R c B ( n ) (cid:105) , that, for large n , the various observables characterizing the record statisticsdepend only on the exponent µ (1.3) and not on the further microscopic details of thejump distribution p c ( η ). Another important feature of the record statistics of randomwalk bridges is that it is technically much more involved. Indeed, for free randomwalks, the computations require the full joint distribution of the ages of the records τ , τ , · · · , τ m − , A n but there is no need to keep track of the actual value of the recordat a given time step. The knowledge of the actual value of the record at a given timestep is however required for bridges, where the random walk returns back to the originafter n time steps. This is done here by considering the full joint distribution of the ages τ i ’s and the record increments ρ i ’s (which are the differences between two consecutiverecords), see Fig. 1. The computation of this joint distribution is the main technicalachievement of the present paper [see Eq. (3.21) below]. Finally, yet another noticeabledifference between free RWs and RWs bridges, from the point of view of records, isthat there is no simple relation between Q αB ( n ) and (cid:104) (cid:96) α max ,B ( n ) (cid:105) for bridges, while theyare directly related for free random walks (1.13). Hence, one expects that Q αB ( n ) and (cid:104) (cid:96) α max ,B ( n ) (cid:105) generically lead to two different universal constants, which we can computeexplicitly in the case of the discrete α = d and exponential α = e distributions.Our main results can be summarized as follows (see also Table 1). First, fordiscrete random walk (1.2) we obtain exact results for the full distribution of the records P d B ( m, n ) = Pr( R d B ( n ) = m, n ). In particular, for large n (and n even as a discreterandom walk bridge has necessarily an even number of steps), we show that (cid:104) R d B ( n ) (cid:105) ∼ A d B √ n , A d B = √ π / . (1.19)Hence it also grows like √ n , as for free RW (1.15), but a with a different prefactor: A d B /A d = π/ <
1. On the other hand, for large n and large m , keeping X = m/ √ n fixed, the distribution P d B ( m, n ) takes the scaling form P d B ( m, n ) ∼ √ n ϕ d B (cid:18) X = m √ n (cid:19) , ϕ d B ( X ) = 4 X e − X , X > , (1.20)which is different from its counterpart for free RW (1.16).For continuous jump distribution p c ( η ), we obtain the large n behavior of (cid:104) R c B ( n ) (cid:105) as (cid:104) R c B ( n ) (cid:105) ∼ A c B ( µ ) √ n , as n → ∞ , (1.21)where A c B ( µ ) is given in terms of an integral which depends explicitly on µ [see Eqs. (2.23)and (2.24)]. In the case µ = 2, this amplitude can be evaluated explicitly as A c B ( µ = 2) = √ π , (1.22)which should be compared to the value of the amplitude A c = 2 / √ π , independentlyof µ , for the free RWs (1.10), and hence A c B ( µ = 2) /A c = π/ <
1. By comparing ecord statistics for random walk bridges A d B /A c B ( µ = 2) = 1 / √
2, as for free RWs. Although the analysis of the statisticsof R c B ( n ), beyond the first moment, is quite difficult, it is possible to compute exactlythe full distribution P e B ( m, n ) of the number of records R e B ( n ), for any finite n , for thecase of the exponential distribution p e ( η ) (1.4), which is representative of the case µ = 2[see Eq. (1.3)]. In particular, for large n and large m , keeping X = m/ √ n fixed, thedistribution P e B ( m, n ) takes the scaling form P e B ( m, n ) ∼ √ n ϕ e B (cid:18) X = m √ n (cid:19) , ϕ e B ( X ) = 2 X e − X , X > , (1.23)which is different from its counterpart for free RW (1.11), and also slightly differentfrom the corresponding distribution ϕ d B ( X ) for the discrete bridges (1.20).On the other hand, for the record breaking probability Q αB ( n ), we obtain exactresults for the discrete ( α = d) and exponential ( α = e) jump distributions. Inparticular, for large n , we show that these quantities converge to the same constantwhich can be computed exactly in terms of a single integral given in Eq. (4.9). Thisintegral can be easily evaluated numerically with arbitrary accuracy, yieldinglim n →∞ Q d B ( n ) = lim n →∞ Q e B ( n ) = Q d B ( ∞ ) = Q e B ( ∞ ) = 0 . . . . , (1.24)which is different from the one characterizing free RWs (1.12), Q d B ( ∞ ) > Q d ( ∞ ).Furthermore, for the discrete and exponential distributions, we compute exactly theaverage age of the longest lasting record (cid:104) (cid:96) α max ,B ( n ) (cid:105) and show thatlim n →∞ (cid:104) (cid:96) dmax ,B ( n ) (cid:105) n = lim n →∞ (cid:104) (cid:96) emax ,B ( n ) (cid:105) n = λ dmax ,B = λ emax B = 0 . . . . (1.25)where, again, this constant can be expressed in terms of an integral given in Eq. (4.41).Note in particular that λ α max ,B (cid:54) = Q αB ( ∞ ), while the two corresponding quantities coincidefor free RWs [see Eqs. (1.14), (1.18)]. Although we can not prove it, we expect that ourresults obtained for the exponential case for the large n behavior of the distribution ofthe number of records (1.23), the probability of the record breaking (1.24) and the ageof the longest lasting record (1.25) are valid for any continuous distribution p c ( η ) withfinite σ , i.e., with µ = 2. This conjecture is corroborated by our numerical simulationsof the record statistics of RW bridges with a Gaussian jump distribution.The paper is organized as follows. In section 2, we compute exactly themean number of records for RW bridges, (cid:104) R αB ( n ) (cid:105) , for the discrete and exponentialdistributions (for any value of n in these cases) and for generic continuous distributions(in the large n limit). In section 3 we compute exactly the full distribution of thenumber of records R αB ( n ) for the discrete and exponential distributions for arbitrary n .Section 4 is devoted to the statistics of the ages of the records, where we compute Q αB ( n )and (cid:104) (cid:96) α max ,B ( n ) (cid:105) , again for the discrete and exponential jump distributions. In section 5we compare our exact analytical results to numerical simulations before we conclude insection 6. Some technical details have been left in the appendices A, B and C. ecord statistics for random walk bridges x x B ( i ) timesteps nkx max ,B ( n ) Figure 2.
A discrete random walk bridge of n = 20 steps. Here the number of recordsis R = x max ,B (20) + 1 = 6.
2. Mean number of records
Let R αB ( n ) be the number of records up to step n for a random walk bridge. We canthen write R αB ( n ) = n (cid:88) k =0 σ α ( k, n ) , (2.1)where σ α ( k, n ) is a binary random variable taking values 0 or 1. The variable σ α ( k, n ) = 1 if a record happens at step k and σ α ( k, n ) = 0 otherwise, for a randomwalker which comes back to the origin after n steps. By convention σ α ( k = 0 , n ) = 1as x B (0) = 0 is a record. On the other hand one has obviously σ α ( k, n ) = 0 for k ≥ n [in particular σ α ( n, n ) = 0 because x B ( n ) = x B (0) = 0 is never a record – as a record isdefined by a strict inequality in Eq. (1.5)]. Hence, the mean number of records up tostep n is given by: (cid:104) R αB ( n ) (cid:105) = n − (cid:88) k =0 (cid:104) σ α ( k, n ) (cid:105) = n − (cid:88) k =0 r α ( k, n ) , (2.2)where r α ( k, n ) = (cid:104) σ α ( k, n ) (cid:105) is the record rate, i.e., the probability that a record happensat step k . Therefore, to compute (cid:104) R αB ( n ) (cid:105) , we will first evaluate r α ( k, n ) and then sumover k (2.2). For n ≥
1, the evaluation of r α ( k, n ) relies on the two following quantities: • The free Green’s function (propagator) G α ( x, x , n ) that denotes the probability(discrete distribution) or probability density (continuous distribution) that arandom walker starting at x arrives at x after n steps. • The constrained Green’s function G α> ( x, x , n ) that denotes the probability (discretedistribution) or probability density (continuous distribution) that a random walkerstarting at x arrives at x after n steps and staying strictly positive in-between.To compute r α ( k, n ), let us suppose that a record happens at step k with a recordvalue x (see Fig. 2). This corresponds to the event that the walker, starting at theorigin at step 0, has reached the level x for the first time at step k and returns backto the origin after n steps – as we are considering random walk bridges. In the time ecord statistics for random walk bridges , k ], the walker propagates from 0 to x , being constrained to stay strictly below x . To compute the corresponding propagator, we take x as the new origin of space andthen reverse both the time and coordinate axis. Hence, we see that on the time interval[0 , k ], the particle propagates with G α> ( x, , k ). On the other hand, between step k andstep n (where the walker ends at the origin) the walker is free and thus propagates with G α (0 , x, n − k ) = G α ( x, , n − k ), as the jump distribution is symmetric. The record rateis then obtained by integrating the probability of this event over x ≥ x ≥ k = 0, is suchthat x = 0). Using the statistical independence of the random walk in the time intervals[0 , k ] and [ k, n ] (being Markovian), one thus has, for n ≥ r α ( k, n ) = 1 G α (0 , , n ) (cid:90) ∞ d x G α> ( x, , k ) G α ( x, , n − k ) , ≤ k ≤ n − , (2.3)where we have divided by G α (0 , , n ) as we are considering random walks that areconditioned to come back to the origin after n time steps (bridges). Note that in thecase of a discrete random walk, i.e., α = d, the integral over x in Eq. (2.3) has to bereplaced by a discrete sum. Let us now analyze this formula (2.3) for different types ofrandom walks. In the case of a discrete random walk, it is convenient to compute (cid:104) R d B ( n ) (cid:105) = (cid:80) nk =0 r d ( k, n ) as (cid:104) R d B ( n ) (cid:105) = (cid:104) R d B ( n ) (cid:105) (0) G d (0 , , n ) (2.4)where, here and in the following, the subscript ‘(0)’ refers to a random walk startingfrom x (0) = 0 and ending at x ( n ) = 0 after n steps and G d (0 , , n ) is the probabilityfor this event. The generating function of the numerator (cid:104) R d B ( n ) (cid:105) (0) is from Eq. (2.3) ∞ (cid:88) n =0 z n (cid:104) R d B ( n ) (cid:105) (0) = ∞ (cid:88) x =0 ˜ G d > ( x, , z ) ˜ G d ( x, , z ) , (2.5)where the propagators can be explicitly computed (see Appendix A):˜ G d ( x, , z ) = ∞ (cid:88) n =0 z n G d ( x, , n ) = 1 √ − z (cid:18) − √ − z z (cid:19) x (2.6)and ˜ G d > ( x, , z ) = ∞ (cid:88) n =0 z n G d > ( x, , n ) = (cid:18) − √ − z z (cid:19) x . (2.7)Hence one obtains by summing over x in Eq. (2.5), using Eqs. (2.6) and (2.7) ∞ (cid:88) n =0 z n (cid:104) R d B ( n ) (cid:105) (0) = 12(1 − z ) + 12 √ − z . (2.8) ecord statistics for random walk bridges (cid:104) R d B ( n ) (cid:105) (0) =
12 + 12 k +1 (cid:0) kk (cid:1) , for n even , n = 2 k , for n odd . (2.9)Using that the denominator in Eq. (2.4) is given by G d (0 , , n ) = 2 − n (cid:0) nn/ (cid:1) if n is evenand 0 otherwise one gets finally (cid:104) R d B ( n ) (cid:105) = (cid:40) n − (cid:0) nn/ (cid:1) − + , n even , , n odd . (2.10)The first terms are (cid:104) R d B (0) (cid:105) = 1, (cid:104) R d B (2) (cid:105) = 3 / (cid:104) R d B (4) (cid:105) = 11 /
6. Finally, for large n itbehaves like (cid:104) R d B ( n ) (cid:105) ∼ √ π / √ n , for n even , , for n odd , (2.11)as announced in Eq. (1.19). In this case, the jumps η i ’s are continuous variables with a distribution given by p e ( η ) = b e −| η | /b . As we will see, the record statistics for the exponential distributionis exactly solvable. Indeed, in this case, both the free and the constrained Green’sfunctions can be computed exactly [31]. The associated generating functions (GFs)˜ G e ( x, , z ) (free) and ˜ G e > ( x, , z ) (constrained) read [31] (see also Appendix B)˜ G e ( x, , z ) = ∞ (cid:88) n =1 z n G e ( x, , n ) = z b √ − z e − | x | b √ − z (2.12)˜ G e > ( x, , z ) = ∞ (cid:88) n =0 z n G e > ( x, , n ) = δ ( x ) + 1 − √ − zb e − | x | b √ − z . (2.13)Note that the definition of the GF of the free propagator ˜ G e ( x, , z ) in Eq. (2.12) doesnot contain the n = 0 term – as it will not enter into the calculations – while this term n = 0 has to be retained for the constrained propagator and produces a delta function.The average number of records is given by (cid:104) R e B ( n ) (cid:105) = (cid:80) n − k =0 r e ( k, n ) where therecords rates r e ( k, n ) are given by Eq. (2.3). To proceed, we write as above in Eq. (2.4): (cid:104) R e B ( n ) (cid:105) = (cid:104) R e B ( n ) (cid:105) (0) G e (0 , , n ) , n ≥ , (2.14)where the numerator (cid:104) R e B ( n ) (cid:105) (0) is given by (cid:104) R e B ( n ) (cid:105) (0) = n − (cid:88) k =0 (cid:90) ∞ d x G e > ( x, , k ) G e ( x, , n − k ) , n ≥ . (2.15)Hence its GF is given by ∞ (cid:88) n =1 z n (cid:104) R e B ( n ) (cid:105) (0) = (cid:90) ∞ d x ˜ G e > ( x, , z ) ˜ G e ( x, , z ) = 14 b (cid:18) z − z + z √ − z (cid:19) , (2.16) ecord statistics for random walk bridges G e ( x, , z ) and˜ G e > ( x, , z ) given in Eqs. (2.12) and (2.13). From Eq. (2.16) we easily extract (cid:104) R e B ( n ) (cid:105) (0) for all n . Finally, from Eq. (2.14) one obtains the mean number of records as (cid:104) R e B (0) (cid:105) = 1 , (cid:104) R e B ( n ) (cid:105) = 12 + 2 n − (cid:0) n − n − (cid:1) for n ≥ , (2.17)where we have used G e (0 , , n ) = (cid:0) n − n − (cid:1) / ( b n − ), for n ≥
1. Interestingly, by comparingEqs. (2.10) and (2.17) one sees that (cid:104) R e B ( n ) (cid:105) = (cid:104) R e B (2 n − (cid:105) . For large n , one findsstraightforwardly from Eq. (2.17) (cid:104) R e B ( n ) (cid:105) ∼ √ π √ n , (2.18)which is √ Now we consider the more general situation where the jumps are continuous randomvariables distributed according to p c ( η ) as given in Eq. (1.3). Although an exactcalculation of (cid:104) R c B ( n ) (cid:105) for any finite n seems quite difficult in this case, one can performa large n asymptotic analysis as follows.We recall that the average number of records is given by (cid:104) R c B ( n ) (cid:105) = (cid:80) nk =0 r c ( k, n ).One can show that this sum over k is dominated by the values of k ∼ O ( n ) which arethus large, when n (cid:29)
1. Hence, to evaluate the record rates r c ( k, n ) given in Eq. (2.3)for large k one can replace the propagators G c ( x, , n − k ) and G c > ( x, , k ) by their scalingforms valid for k, n (cid:29)
1, with k/n fixed, and x (cid:29)
1, with x/n /µ fixed. One has indeed G c ( x, , n − k ) ∼ a ( n − k ) /µ R (cid:18) xa ( n − k ) /µ (cid:19) , (2.19) G c > ( x, , k ) ∼ a √ πk / /µ R + (cid:16) xa k /µ (cid:17) , (2.20)where the scaling functions are normalized, i.e., (cid:82) ∞−∞ d x R ( x ) = 1 and (cid:82) ∞ d x R + ( x ) = 1.The scaling function R ( x ) is a L´evy stable distribution: R ( x ) = 12 π (cid:90) ∞−∞ d k e − ikx e −| k | µ , (2.21)and in particular R (0) = Γ(1 + 1 /µ ) /π . On the other hand, there is no explicitexpression for R + ( x ) for generic µ <
2, while for µ = 2 one has R + ( x ) = 2 x e − x .With such a normalization one can check in particular that by integrating G c > ( x, , k ) in Eq. (2.20) over x one recovers the survival probability q ( k ), which isthe probability that the walker, starting from x (0) = 0 stays positive up to step k : (cid:90) ∞ d x G c > ( x, , k ) = q ( k ) ∼ √ π k , as k → ∞ , (2.22)in agreement with the Sparre Andersen theorem [26]. By inserting these scaling forms(2.19, 2.20) into the expression for r c ( k, n ) in (2.3) one finds, that for large k and n ecord statistics for random walk bridges k/n = y , fixed (with 0 ≤ y ≤ r c ( k, n ) = 1 √ n H (cid:18) y = kn (cid:19) ,H ( y ) = √ π Γ(1 + 1 /µ ) 1 √ y (1 − y ) /µ (cid:90) ∞ d x R + ( x ) R (cid:18) x ( y − − /µ (cid:19) . (2.23)Finally, from this scaling form for the record rate (2.23), one obtains finally (cid:104) R c B ( n ) (cid:105) = n (cid:88) k =0 r c ( k, n ) ∼ A c B ( µ ) √ n , A c B ( µ ) = (cid:90) d y H ( y ) . (2.24)Hence (cid:104) R c B ( n ) (cid:105) also grows like √ n for bridges but with an amplitude which depends on µ ,as announced in Eq. (1.21). In particular, one can check that A c B ( µ = 2) = √ π/
2, whichcoincides, as expected, with the result obtained in the exponential case, see Eq. (2.18).
3. Full distribution of the number of records
In the previous section, we computed the average number of records (cid:104) R αB ( n ) (cid:105) . Here weshow that, in some cases, namely for discrete ( α = d) and exponential ( α = e) jumpdistributions, the full distribution of R αB ( n ) can be computed exactly, for any finite n . In the case of discrete random walks, the distribution of R d B ( n ) can be computed byusing the close relation between R d B ( n ) and the maximum of the random walk up tostep n , denoted by x max ,B ( n ) = max ≤ m ≤ n x B ( m ) [8] (see also Fig. 2): R d B ( n ) = x max ,B ( n ) + 1 . (3.1)To derive this relation (3.1), let us consider the time evolution of the two processes R d B ( n ) and x max ,B ( n ). At the next time step n + 1, if a new site on the positive axisis visited for the first time, the process x max ,B ( n ) increases by 1, otherwise its valueremains the unchanged. On the other hand, when this event happens, then the recordnumber R d B ( n ) is also increased by 1, and otherwise it remains unchanged. Therefore wesee that the two processes are locked with each other at all steps: for any realization ofthe random walk, we thus have x max ,B ( n + 1) − x max ,B ( n ) = R d B ( n + 1) − R d B ( n ). Giventhat, initially one has x max ,B (0) = x B (0) = 0 and R d B (0) = 1 (since, by convention, thefirst position is a record) one immediately obtains the relation in Eq. (3.1). Hence, thisrelation allows us to compute the PDF of R d B ( n ) as P d B ( m, n ) = Pr( R d B ( n ) = m ) = Pr( x max ,B ( n ) = m −
1) (3.2)= P d B ( m, n ) (0) G d (0 , , n ) , P d B ( m, n ) (0) = n (cid:88) k =0 G d > ( m − , , k ) G d ≥ ( m − , , n − k ) (3.3)where G d ≥ ( x, x , k ) is the probability that the random walker, starting at x , arrives at x after k steps, while staying non-negative (i.e., it may touch 0 but not −
1) in-between. ecord statistics for random walk bridges P d B ( m, n ) (0) in Eq. (3.3) then reads ∞ (cid:88) n =0 P d B ( m, n ) (0) z n = ˜ G d > ( m − , , z ) ˜ G d ≥ ( m − , , z ) , (3.4)where ˜ G d > ( m − , , z ) is given in Eq. (2.7) and ˜ G d ≥ ( x, , z ) is given by (see Appendix A)˜ G d ≥ ( x, , z ) = ∞ (cid:88) n =0 G d ≥ ( x, , n ) z n = 2 z (cid:18) − √ − z z (cid:19) x +1 . (3.5)Therefore Eq. (3.4) reads explicitly ∞ (cid:88) n =0 P d B ( m, n ) (0) z n = 2 z (cid:18) − √ − z z (cid:19) m − . (3.6)Here, to perform this computation, we show an alternative method which will alsobe useful to compute the statistics of the ages. To do so, we first introduce the ages ofthe records τ i ’s and write the joint probability distribution of the τ i ’s (see Fig. 1) andthe number of records R d B ( n ), P d ( (cid:96) , · · · , (cid:96) m − , a, m, n ) = Pr( τ = (cid:96) , . . . , τ m − = (cid:96) m − , A n = a, R d B ( n ) = m, n )= P d ( (cid:126)(cid:96), m, n ) (0) G d (0 , , n ) , (3.7)where the numerator P d ( (cid:126)(cid:96), m, n ) (0) is given by P d ( (cid:126)(cid:96), m, n ) (0) = f d ( (cid:96) ) · · · f d ( (cid:96) m − ) G d ≥ ( m − , , a ) δ (cid:32) m − (cid:88) i =1 (cid:96) i + a, n (cid:33) , (3.8)where f d ( (cid:96) ) is the first passage probability that the discrete RW, starting from x , arrivesat x + 1 for the first time at step (cid:96) , and δ ( a, b ) denotes a delta Kronecker function.Since the RW is invariant under translation, this probability is independent of x andfor a discrete RW, its generating function is given by˜ f d ( z ) = ∞ (cid:88) (cid:96) =1 f d ( (cid:96) ) z (cid:96) = 1 − √ − z z , (3.9)from which we deduce that f d ( (cid:96) ) = , (cid:96) even , ( − ( (cid:96) − / √ π − (cid:96)/ / (cid:96)/ , (cid:96) odd . (3.10)This joint distribution (3.7) contains all the information about the observables thatwe wish to compute here. In particular, the probability distribution of the record number R d B ( n ) is obtained by integrating it over the (cid:96) i ’s: P d B ( m, n ) = ∞ (cid:88) (cid:96) =1 ∞ (cid:88) (cid:96) =1 . . . ∞ (cid:88) (cid:96) m − =1 ∞ (cid:88) a =1 P d ( (cid:96) , (cid:96) , · · · , (cid:96) m − , a, m, n ) = P d B ( m, n ) (0) G d (0 , , n ) , (3.11) ecord statistics for random walk bridges ∞ (cid:88) n =0 z n P d B ( m, n ) (0) = [ ˜ f d ( z )] m − ˜ G d ≥ ( m − , , z )= 2 z (cid:18) − √ − z z (cid:19) m − , (3.12)where we have used Eqs. (3.5) and (3.9). Thus we see that this result (3.12) coincideswith the one obtained previously by a quite different method (3.6). From this expression(3.12) we can compute P d B ( m, n ) (0) (using Cauchy’s formula) and finally P d B ( m, n ) as (for n even) P d B ( m, n ) = 2 n +1 (cid:0) nn/ (cid:1) ( − n/ m ( n/ m )! m − (cid:88) j =0 ( − j (cid:18) m − j (cid:19) Γ( j/ j/ − n/ − m ) , (3.13)for 1 ≤ m ≤ n/ P d B ( m, n ) = 0 for m > n/ n/ (cid:104) R d B ( n ) (cid:105) can beobtained, in principle, from the full distribution in Eq. (3.13) it is more convenientto compute them from the GF in (3.12). In particular, one can easily check that onerecovers the result for the average number of records (cid:104) R d B ( n ) (cid:105) as obtained previously(2.4), (2.8). Besides one can compute higher moments of R d B ( n ). For instance, thesecond moment is given by (cid:104) (cid:2) R d B ( n ) (cid:3) (cid:105) = n + 12 + √ π n/ n/ / , n odd , , n even . (3.14)For large n , it behaves as (cid:104) (cid:2) R d B ( n ) (cid:3) (cid:105) ∼ n , n odd , , n even . (3.15)Finally, from Eq. (3.6), one can also extract the expression of P d B ( m, n ) in the large n limit, which can be obtained by studying the limit z → z = e − s and investigate the limit s → P d B ( m, n ) = 0 if n isodd, the left hand side of Eq. (3.6) can be written, in the limit s → ∞ (cid:88) n =0 P d B ( m, n ) (0) e − s n = ∞ (cid:88) k =0 P d B (2 k, n ) (0) e − sk ∼ (cid:90) ∞ d y e − y s P d B ( m, y ) (0) . (3.16)On the other hand, the right hand side of Eq. (3.6) assumes a simpler form in the limit z = e − s → (cid:90) ∞ d y e − y s P d B ( m, y ) (0) = 2 e − m √ s . (3.17) ecord statistics for random walk bridges P d B ( m, y ) (0) can be straightforwardly obtained by Laplace inversion as P d B ( m, n ) (0) ∼ (cid:114) π mn / e − m /n , for n even . (3.18)Finally, using that G d (0 , , n ) ∼ (cid:112) /π n − / , one finds that the distribution P d B ( m, n )takes the scaling form (valid for n even): P d B ( m, n ) ∼ √ n ϕ d B (cid:18) X = m √ n (cid:19) , ϕ d B ( X ) = 4 X e − X , (3.19)as announced in the introduction in Eq. (1.20). Note that this distribution ϕ d B ( X )coincides, up to a scale factor, with the probability distribution function of the maximumof a Brownian bridge on the unit time interval, as expected from the relation stated inEq. (3.1). In particular, at variance with the result for the free random walk (1.16),this PDF is not a half-Gaussian distribution. In addition, it is easy to check that, fromthe moment of this distribution ϕ d B ( X ), one recovers the asymptotic result for (cid:104) R d B ( n ) (cid:105) and (cid:104) (cid:2) R d B ( n ) (cid:3) (cid:105) obtained respectively in Eqs. (2.11) and (3.15). For the exponential jump distribution p e ( η ) (1.4), the starting point of our analysis isthe equivalent of the joint distribution given, for discrete random walks, in Eq. (3.7).However, because p e ( η ) is a continuous distribution, this computation is more delicatethan in the discrete case. Indeed, as we are considering random walk bridges, the weightof the last part of the paths, where the walker comes back to origin, i.e., the last segmentof duration A n = a (see Fig. 1), involves the propagator G e ≥ ( Y, , a ) = G e > ( Y, , a )[see Eq. (3.3)] where Y = x max ,B ( n ) is the actual value of the last record, whichcoincides with the maximum. For discrete random walk the number of records R d B ( n )and x max ,B ( n ) are directly related through x max ,B ( n ) = R d B ( n ) − ρ i ’s and the joint distribution of the τ i ’s, ρ i ’s, R e B ( n ) and x max ,B ( n ) (see Fig. 1):Pr[ { τ i = (cid:96) i , ρ i ∈ [ r i , r i + d r i ] } ≤ i ≤ m − , A n = a, R e B ( n ) = m, x max ,B ( n ) ∈ [ Y, Y + d Y ] , n ]= P e ( { (cid:96) i , r i } ≤ i ≤ m − , a, m, Y )d r · · · d r m − d Y , (3.20)where the joint PDF P e ( { (cid:96) i , r i } ≤ i ≤ m − , a, m, Y ) is given by P e ( { (cid:96) i , r i } ≤ i ≤ m − , a, m, Y ) = 1 G e (0 , , n ) m − (cid:89) i =1 (cid:90) ∞ d y i G e > ( y i , , (cid:96) i − p e ( y i + r i ) × G e > ( Y, , a ) δ (cid:32) m − (cid:88) i =1 r i − Y (cid:33) δ (cid:32) m − (cid:88) i =1 (cid:96) i + a, n (cid:33) , (3.21)where we have used that, for the exponential jump distribution which is continuous, G e ≥ ( x, , n ) = G e > ( x, , n ) for x >
0. In particular, the joint distribution of the τ i ’s, ecord statistics for random walk bridges A n and R e B ( n ), i.e., the equivalent of Eq. (3.7) for the discrete case, is obtained byintegrating the formula in (3.20) over r i ’s and Y : P e ( (cid:96) , (cid:96) , · · · , (cid:96) m − , a, m, n ) = Pr( τ = (cid:96) , . . . , τ m − = (cid:96) m − , A n = a, R e B ( n ) = m, n )= P e ( (cid:126)(cid:96), m, n ) (0) G e (0 , , n ) , (3.22)where P e ( (cid:126)(cid:96), m, n ) (0) = m − (cid:89) i =1 (cid:90) ∞ d r i T ( (cid:96) i , r i ) × (cid:90) ∞ d Y G e > ( Y, , a ) δ (cid:32) m − (cid:88) i =1 r i − Y (cid:33) δ (cid:32) m − (cid:88) i =1 (cid:96) i + a, n (cid:33) , (3.23)where T ( (cid:96), r ) = (cid:90) ∞ d y G e > ( y, , (cid:96) − p e ( y + r ) . (3.24)Note that this formula (3.23) together with (3.24) is actually valid for any continuousjump distribution p c ( η ) (1.3)– where the superscript ‘e’ is replaced by ‘c’. However,its analysis is in general very hard to do, mainly because the constrained propagator G c > ( x, , n ) does not have any explicit expression, which prevents one to perform theanalysis of this multiple integral. Fortunately, such an explicit expression exists for thecase of an exponential jump distribution p e ( η ) = 1 / (2 b )e −| η | /b , which we now focus on.In this case, the generating function of the constrained propagator G e > ( x, , n ) isgiven by Eq. (2.13), from which one gets that the GF of the building block T ( (cid:96), r ) inEq. (3.24) is given by [using Eq. (2.13)] ∞ (cid:88) (cid:96) =1 T ( (cid:96), r ) z (cid:96) = z (cid:90) ∞ d y ˜ G e > ( y, , z ) p e ( y + r ) = 1 b (1 − √ − z )e − r/b . (3.25)From Eq. (3.25) one obtains T ( (cid:96), r ) = 1 b f e ( (cid:96) ) e − r/b , ∞ (cid:88) (cid:96) =1 f e ( (cid:96) ) z (cid:96) = 1 − √ − z , (3.26)which yields the expression of the coefficients f e ( (cid:96) ) as f e ( (cid:96) ) = ( − (cid:96) +1 √ π / − (cid:96) )Γ( (cid:96) + 1) ∼ √ π(cid:96) / , as (cid:96) → ∞ . (3.27)By comparing this expression for f e ( (cid:96) ) and the one for f d ( (cid:96) ) in Eq. (3.27) we easily seethat f e ( (cid:96) ) = f d (2 (cid:96) − T ( (cid:96), r ) (3.26, 3.27) inEq. (3.23), the joint probability distribution P e ( (cid:126)(cid:96), m, n ) (0) can be written P e ( (cid:126)(cid:96), m, n ) (0) = m − (cid:89) i =1 f e ( (cid:96) i ) (cid:90) ∞ d Y G e > ( Y, , a ) e − Y/b × m − (cid:89) i =1 (cid:90) ∞ dr i b δ (cid:32) m − (cid:88) i =1 r i − Y (cid:33) δ (cid:32) m − (cid:88) i =1 (cid:96) i + a, n (cid:33) . (3.28) ecord statistics for random walk bridges m − (cid:89) i =1 (cid:90) ∞ d r i δ (cid:32) m − (cid:88) i =1 r i − Y (cid:33) = Y m − ( m − , (3.29)which can be easily shown by taking the Laplace transform on both sides of (3.29) withrespect to Y , we obtain an expression for the joint probability of the τ i ’s, A n and R e B ( n )as P e ( (cid:126)(cid:96), m, n ) (0) = m − (cid:89) i =1 f e ( (cid:96) i ) q e ( m, a ) δ (cid:32) m − (cid:88) i =1 (cid:96) i + a, n (cid:33) , (3.30) q e ( m, a ) = 1( m − b m − (cid:90) ∞ d Y e − Y/b Y m − G e > ( Y, , a ) , (3.31)which has thus a structure very similar to the one found in the discrete case (3.8), butwith different building blocks. Furthermore, the GF of q e ( m, a ) in Eq. (3.31) can beobtained explicitly as (see Appendix B)˜ q e ( m, z ) = ∞ (cid:88) a =1 q e ( m, a ) z a = 1 b − √ − z (1 + √ − z ) m − = (1 − √ − z ) m b z m − . (3.32)From this joint distribution (3.30), together with Eqs. (3.26) and (3.32), it ispossible to compute the statistics of all the observables which we want to study in thispaper. In particular, one obtains the distribution of the number of records P e B ( m, n ) bysumming it over (cid:96) , (cid:96) , . . . , (cid:96) m − and a as done before in the discrete case (3.11). Thisyields P e B ( m, n ) = P e B ( m, n ) (0) G e (0 , , n ) ∞ (cid:88) n =0 z n P e B ( m, n ) (0) = [ ˜ f e ( z )] m − ˜ q e ( m, z ) = 1 b (1 − √ − z ) m (1 + √ − z ) m − = 1 b (1 − √ − z ) m − z m − , (3.33)from which one gets P e B ( m, n ) = 2 n − (cid:0) n − n − (cid:1) ( − n + m − ( n + m − m − (cid:88) j =0 ( − j (cid:18) m − j (cid:19) Γ( j/ j/ − n − m ) , (3.34)for 1 ≤ m ≤ n , which is independent of the scale parameter b [the factor 1 /b inEq. (3.33) is indeed cancelled by the denominator in the first line of Eq. (3.33), G e (0 , , n ) = (cid:0) n − n − (cid:1) / ( b n − )]. By comparing this result for the exponential distribution(3.34) with the corresponding one for discrete jumps obtained before in Eq. (3.13), weeasily find that P e ( m, n ) = P d ( m, n − n ≥
1. The moments of R e B ( n ) can beobtained, in principle, from the explicit expression of the full distribution in (3.34). Itis however simpler to compute them from the GF in Eq. (3.33). For instance, from thisexpression, one can easily recover the value for (cid:104) R e B ( n ) (cid:105) obtained by the previous method ecord statistics for random walk bridges R e B ( n ). In particular, the second moment is given by ∞ (cid:88) n =0 z n (cid:104) [ R e B ( n )] (cid:105) (0) = z b (1 − z ) / + z b (1 − z ) (3.35)from which one obtains, for n ≥ (cid:104) [ R e B ( n )] (cid:105) = n −
12 + √ π n )Γ( n − / ∼ n , as n → ∞ . (3.36)Finally, from Eq. (3.33), one can obtain the limiting scaling form of the distribution ofthe number of records P e B ( m, n ) for m (cid:29) n (cid:29)
1, keeping m/ √ n fixed, as P e B ( m, n ) ∼ √ n ϕ e B (cid:18) X = m √ n (cid:19) , ϕ e B ( X ) = 2 X e − X , X > , (3.37)where we have used that G e (0 , , n ) ∼ / (2 b √ πn ), for n (cid:29)
1. This formula (3.37) yieldsthe result announced in the introduction in Eq. (1.23). Here also, one can easily checkthat, from the moments of the distribution ϕ e B ( X ) in Eq. (3.37), one recovers the large n behavior of (cid:104) R e B ( n ) (cid:105) and (cid:104) [ R e B ( n )] (cid:105) obtained respectively in Eqs. (2.18) and (3.36).
4. Statistics of the ages of records
In this section, we study in detail two quantities associated to the ages of the records:(i) the probability Q αB ( n ) (1.7) that the age of the last record is the longest one for arandom walk bridge and (ii) the statistics of (cid:96) α max ,B ( n ) (1.8), which is the age of thelongest lasting record. As in the previous section, we focus on the discrete ( α = d) andthe exponential ( α = e) distributions. Q αB ( n )The computation of Q αB ( n ) for a random walk bridge and for arbitrary jump distribution p ( η ) is in general a very hard task. We show here that it can be computed exactly forthe two special cases studied above: the discrete random walk ( α = d) and the case ofexponential jump distribution ( α = e). The probability Q d B ( n ) (1.7) can be simply computedfrom the full joint PDF of the intervals given in Eq. (3.7) by summing it over theappropriate variables. For given values of the number of records m and of A n = a , wehave to sum the joint PDF over the variables (cid:96) i ’s from 1 to a – as A n = a is the longestinterval (see Fig. 1). Finally, we sum over all the possible values of a ≥ m ≥ Q d B ( n ) = ∞ (cid:88) m =1 ∞ (cid:88) a =1 a (cid:88) (cid:96) =1 · · · a (cid:88) (cid:96) m − =1 P d ( (cid:126)(cid:96), m, n ) , (4.1) ecord statistics for random walk bridges P d ( (cid:126)(cid:96), m, n ) is explicitly given in Eq. (3.7) and (3.8). Again, we separate thenumerator and the denominator and write Q d B ( n ) = Q d B ( n ) (0) G d (0 , , n ) ∞ (cid:88) n =0 Q d B ( n ) (0) z n = ∞ (cid:88) a =1 z a a (cid:88) x =0 (cid:2) h d ( z, a ) (cid:3) x G d ≥ ( x, , a ) , (4.2)where we have made the change of variable x = m − h d ( z, a ) = a (cid:88) k =1 f d ( k ) z k . (4.3)From Eq. (4.2) one can obtain, in principle, the value of Q d B ( n ) (0) using Eq. (4.3)together with the explicit expression of G d ≥ ( x, , a ) given in Eq. (A.5) – for instance usingMathematica – though obtaining a closed form expression for Q d B ( n ) (0) for any n seemsquite difficult. It is however possible to extract the large n asymptotic behavior Q d B ( n ) (0) by analyzing Eq. (4.2) in the limit z = e − s when z →
1, which thus corresponds to s →
0. In this limit, the double sum on the right hand side of Eq. (4.2) is dominated bylarge a and large x . In this limit, keeping x/ √ a fixed, G d ≥ ( x, , a ) admits the followingscaling form (for x + a even): G d ≥ ( x, , a ) ∼ (cid:114) πa √ a g d (cid:18) x √ a (cid:19) , g d ( y ) = y e − y , (4.4)while G d ≥ ( x, , a ) = 0 for x + a odd. Similarly, using the expression of f d ( (cid:96) ) given inEq. (3.10), we obtain that in the scaling limit a → ∞ , s → z = e − s → s a fixed, h d ( z, a ) in Eq. (4.3) takes the scaling form h d ( z, a ) ∼ − √ s F ( as ) , F ( y ) = 1 + 12 √ π (cid:90) ∞ y d uu / e − u (4.5)= erf( √ y ) + 1 √ π e − y √ y . From these asymptotic behaviors in Eqs. (4.4) and (4.5) one obtains that the sum over x in Eq. (4.2), which becomes an integral in the limit s → a → ∞ keeping y = as fixed, takes the scaling form (as a function of the scaling variable y ) a (cid:88) x =0 (cid:2) h d ( z, a ) (cid:3) x G d ≥ ( x, , a ) ∼ (cid:114) πa a (cid:88) x =0 [1 − √ sF ( a s )] x g d (cid:18) x √ a (cid:19) (4.6) ∼ (cid:114) πa (cid:90) a d x e − x √ sF ( a s ) g d (cid:18) x √ a (cid:19) ∼ (cid:114) πa G ( y = a s ) . Note that in the discrete sum over x in the first line of Eq. (4.6) only the terms suchthat x + a is even contribute – while the terms such that x + a is odd are just zero. Hence ecord statistics for random walk bridges / x , as indicated in thesecond line of Eq. (4.6). The function G ( y ) in the third line of Eq. (4.6) is given by G ( y ) = 1 − √ πy F ( y ) exp (cid:2) yF ( y ) (cid:3) erfc [ √ yF ( y )] . (4.7)Finally, one obtains the asymptotic behavior of the generating function of Q d B ( n ) (0) inEq. (4.2), as z → ∞ (cid:88) n =0 Q d B ( n ) (0) z n ∼ c d Q √ − z , c d Q = (cid:114) π (cid:90) ∞ d y √ y e − y G ( y ) , (4.8)such that Q d B ( n ) (0) ∼ c d Q / √ π n as n → ∞ , for n even (while Q d B ( n ) (0) = 0 if n is odd).Using that G d (0 , , n ) = 2 − n (cid:0) nn/ (cid:1) ∼ (cid:112) /π n − / , for n (cid:29)
1, one obtains from Eq. (4.1)together with Eq. (4.8)lim n →∞ Q d B ( n ) = Q d B ( ∞ ) = √ c d Q = 2 √ π (cid:90) ∞ d y √ y e − y G ( y ) = 0 . . . . , (4.9)which is different from the corresponding value for free random walks, Q d ( ∞ ) =0 . . . . [see Eq. (1.17)]. We now turn to the computation of Q e B ( n ) inEq. (1.7) in the case of an exponential jump distribution. In this case, our startingpoint is the joint distribution of the ages of the records given in Eq. (3.22). As beforein the case of discrete random walks (4.1), one has Q e B ( n ) = ∞ (cid:88) m =1 ∞ (cid:88) a =1 a (cid:88) (cid:96) =1 · · · a (cid:88) (cid:96) m − =1 P e ( (cid:126)(cid:96), m, n ) , (4.10)from which, using Eqs. (3.22) and (3.30), one obtains [similarly to the discrete case inEq. (4.2)] Q e B ( n ) = Q e B ( n ) (0) G e (0 , , n ) ∞ (cid:88) n =0 Q e B ( n ) (0) z n = ∞ (cid:88) a =1 z a ∞ (cid:88) m =1 [ h e ( z, a )] m − q e ( m, a ) , (4.11)where f e ( (cid:96) ) is given in Eq. (3.27).To obtain the large n behavior of Q e B ( n ) (0) , we need to analyze its generatingfunction in Eq. (4.11) in the limit z = e − s →
1, i.e. s →
0. In this limit, the sum over a is dominated by large values of a ∼ /s . In this limit, h e ( z, a ) takes the scaling form,for s → a s fixed h e ( z, a ) ∼ (cid:0) − √ sF ( a s ) (cid:1) , (4.12)where the scaling function F ( y ) is given in Eq. (4.5). Furthermore, in the limit z = e − s → s → m in (4.11) is dominated by large valuesof m . In the limit, m → ∞ , a → ∞ , keeping m/ √ a fixed, we obtain, from the GF of q e ( m, a ) given in Eq. (3.32), that it takes the form q e ( m, a ) ∼ b √ π ma / e − m / (4 a ) . (4.13) ecord statistics for random walk bridges ∞ (cid:88) n =0 Q e B ( n ) (0) e − s n ∼ b √ π ∞ (cid:88) a =1 e − sa a / ∞ (cid:88) m =1 m (cid:0) − √ sF ( a s ) (cid:1) m − e − m / (4 a ) . (4.14)The sum over m can then be approximated by an integral, in the limit s → ∞ (cid:88) m =1 m (cid:0) − √ sF ( a s ) (cid:1) m − e − m / (4 a ) ∼ ∞ (cid:88) m =1 m e − ( m − √ sF ( as ) e − m / (4 a ) ∼ (cid:90) ∞ d m m e − ( m − √ sF ( as ) − m / (4 a ) = 2 a G ( a s ) , (4.15)where the function G ( y ) is defined in Eq. (4.7). Hence, finally, approximating theremaining sum over a by an integral (in the limit s →
0) and performing the change ofvariable y = s a one obtains ∞ (cid:88) n =0 Q e B ( n ) (0) z n ∼ c e Q √ − z , c e Q = 1 b √ π (cid:90) ∞ d y e − y √ y G ( y ) , (4.16)which implies that, for large n , Q e B ( n ) (0) ∼ c e Q / √ πn . Hence, using that G e (0 , , n ) ∼ / (2 b √ πn ), for large n , one obtains from the first line of Eq. (4.11) thatlim n →∞ Q e B ( n ) = Q e B ( ∞ ) = Q d B ( ∞ ) , (4.17)which is thus the same constant as the one appearing in the discrete case given inEq. (4.9). The average value of (cid:104) (cid:96) dmax ,B ( n ) (cid:105) defined in Eq. (1.8) canbe computed from its cumulative distribution as follows (cid:104) (cid:96) dmax ,B ( n ) (cid:105) = ∞ (cid:88) (cid:96) =0 (1 − F d ( (cid:96), n )) , F d ( (cid:96), n ) = Pr[ (cid:96) dmax ,B ( n ) ≤ (cid:96) ] , (4.18)where F d ( (cid:96), n ) = Pr[ (cid:96) dmax ,B ( n ) ≤ (cid:96) ] is simply obtained by summing up the joint PDF inEq. (3.7) over (cid:96) , · · · , (cid:96) m − and a from 1 to (cid:96) – as all of them have to be smaller than (cid:96) – and finally over the number of records m as follows: F d ( (cid:96), n ) = Pr[ (cid:96) dmax ,B ( n ) ≤ (cid:96) ] = ∞ (cid:88) m =1 (cid:96) (cid:88) a =1 (cid:96) (cid:88) (cid:96) =1 · · · (cid:96) (cid:88) (cid:96) m − =1 P d ( (cid:96) , (cid:96) , · · · , (cid:96) m − , a, m, n ) . (4.19)Therefore, one has (cid:104) (cid:96) dmax ,B ( n ) (cid:105) = (cid:104) (cid:96) dmax ,B ( n ) (cid:105) (0) G d (0 , , n ) , (4.20)where the numerator is given by (cid:104) (cid:96) dmax ,B ( n ) (cid:105) (0) = G d (0 , , n ) − ∞ (cid:88) m =1 (cid:96) (cid:88) a =1 (cid:96) (cid:88) (cid:96) =1 · · · (cid:96) (cid:88) (cid:96) m − =1 P d ( (cid:96) , (cid:96) , · · · , (cid:96) m − , a, m, n ) (0) (4.21) ecord statistics for random walk bridges ∞ (cid:88) n =0 (cid:104) (cid:96) dmax ,B ( n ) (cid:105) (0) z n = ∞ (cid:88) (cid:96) =0 (cid:32) ˜ G d (0 , , z ) − (cid:96) (cid:88) a =1 z a a +1 (cid:88) m =1 (cid:2) h d ( z, (cid:96) ) (cid:3) m − G d ≥ ( m − , , a ) (cid:33) , (4.22)where the function h d ( z, (cid:96) ) is defined in Eq. (4.3). In the limit z = e − s →
1, i.e. s → G d ≥ ( m − , , a ) by its scaling form given in Eq. (4.4) and h d ( z, (cid:96) ) byits asymptotic behavior given in Eq. (4.5), in the scaling regime where a and (cid:96) arelarge, keeping the products s a and s (cid:96) fixed. The sum over m in Eq. (4.22) can then beanalyzed along the same lines as done before in Eq. (4.6) to yield: a +1 (cid:88) m =1 (cid:2) h d ( z, (cid:96) ) (cid:3) m − G d ≥ ( m − , , a ) ∼ (cid:114) πa (cid:2) − √ πsa F ( s (cid:96) ) exp (cid:2) saF ( s(cid:96) ) (cid:3) erfc (cid:2) √ saF ( s(cid:96) ) (cid:3)(cid:3) , (4.23)where the function F ( y ) is defined in Eq. (4.5). Inserting this asymptotic behavior(4.23) in Eq. (4.22), one finds ∞ (cid:88) n =0 (cid:104) (cid:96) dmax ,B ( n ) (cid:105) (0) z n ∼ c dmax (1 − z ) / , as z → , (4.24)where the amplitude c dmax is given by [remembering that only the even values of n contribute to the left hand side Eq. (4.24)] c dmax = √ (cid:90) ∞ d t (cid:20) − I ( t ) (cid:21) , (4.25)where the function I ( t ) is given by I ( t ) = 1 √ π (cid:90) t d y √ y e − y (cid:16) − √ πyF ( t )e y F ( t ) erfc ( √ yF ( t )) (cid:17) (4.26)= F ( t )e − t + t F ( t ) erfc[ √ tF ( t )] − e − t / √ πt − F ( t ) . (4.27)The behavior in Eq. (4.25) implies that (cid:104) (cid:96) dmax ,B ( n ) (cid:105) ∼ (4 c dmax / √ π ) √ n , for large n (even).Using finally that G d (0 , , n ) ∼ (cid:112) / ( π n ) (again, for n even) one finally obtainslim n →∞ (cid:104) (cid:96) dmax ,B ( n ) (cid:105) n = λ dmax ,B = 2 √ c dmax (4.28)= 4 (cid:90) ∞ d t (cid:18) − I ( t ) (cid:19) (4.29)= 0 . . . . , (4.30)which is strictly smaller than the constant Q d B ( ∞ ) given in Eq. (4.9). The average value of (cid:104) (cid:96) emax ,B ( n ) (cid:105) defined inEq. (1.8) can be computed from its cumulative distribution as follows (cid:104) (cid:96) emax ,B ( n ) (cid:105) = ∞ (cid:88) (cid:96) =0 (1 − F e ( (cid:96), n )) , F e ( (cid:96), n ) = Pr[ (cid:96) emax ,B ( n ) ≤ (cid:96) ] , (4.31) ecord statistics for random walk bridges F e ( (cid:96), n ) = Pr[ (cid:96) emax ,B ( n ) ≤ (cid:96) ] is simply obtained by summing up the joint PDF inEq. (3.22) over (cid:96) , · · · , (cid:96) m − and a from 1 to (cid:96) and finally over the number of records m as follows: F e ( (cid:96), n ) = Pr[ (cid:96) emax ,B ( n ) ≤ (cid:96) ] = ∞ (cid:88) m =1 (cid:96) (cid:88) a =1 (cid:96) (cid:88) (cid:96) =1 · · · (cid:96) (cid:88) (cid:96) m − =1 P e ( (cid:96) , (cid:96) , · · · , (cid:96) m − , a, m, n ) . (4.32)Hence (cid:104) (cid:96) emax ,B ( n ) (cid:105) can be written as (cid:104) (cid:96) emax ,B ( n ) (cid:105) = (cid:104) (cid:96) emax ,B ( n ) (cid:105) (0) G e (0 , , n ) , (4.33) (cid:104) (cid:96) emax ,B ( n ) (cid:105) (0) = ∞ (cid:88) (cid:96) =0 G e (0 , , n ) − ∞ (cid:88) m =1 (cid:96) (cid:88) a =1 (cid:96) (cid:88) (cid:96) =1 · · · (cid:96) (cid:88) (cid:96) m − =1 P e ( (cid:96) , · · · , (cid:96) m − , a, m, n ) (0) . Again, to extract the large n behavior of (cid:104) (cid:96) emax ,B ( n ) (cid:105) (0) , it is convenient to analyze thegenerating function of (cid:104) (cid:96) emax ,B ( n ) (cid:105) (0) , which can be easily done by using the previousanalysis. Indeed, one has˜ G e (0 , , e − s ) = ∞ (cid:88) n =1 G e (0 , , n )e − sn ∼ b √ s , as s → . (4.34)On the other hand, from the analysis performed above in Eqs. (4.12, 4.13, 4.15) onehas ∞ (cid:88) n =1 e − sn ∞ (cid:88) m =1 (cid:96) (cid:88) (cid:96) =1 · · · (cid:96) (cid:88) (cid:96) m − =1 P e ( (cid:96) , · · · , (cid:96) m − , a, m, n ) (4.35) ∼ b √ πa (cid:2) − √ πas F ( s (cid:96) ) exp (cid:2) asF ( s(cid:96) ) (cid:3) erfc (cid:2) √ saF ( s(cid:96) ) (cid:3)(cid:3) , (4.36)in the limit s → a → ∞ , (cid:96) → ∞ , keeping s a and s (cid:96) fixed and where the function F ( y ) is defined in Eq. (4.5). Therefore, using Eqs. (4.34) and (4.35), we obtain ∞ (cid:88) n =1 (cid:104) (cid:96) emax ,B ( n ) (cid:105) (0) z n ∼ c emax b (1 − z ) / , as z → , (4.37)where the amplitude c emax is given by c emax = (cid:90) ∞ d t (cid:18) − I ( t ) (cid:19) , (4.38)where the function I ( t ) is given in Eq. (4.26). From Eq. (4.37), one deduces that (cid:104) (cid:96) emax ,B ( n ) (cid:105) (0) ∼ (2 c emax / ( b √ π )) √ n for large n . Using that G e (0 , , n ) ∼ / (2 b √ πn ), for n (cid:29)
1, we obtain finallylim n →∞ (cid:104) (cid:96) emax ,B ( n ) (cid:105) n = λ emax ,B = 4 c emax (4.39)= 4 (cid:90) ∞ d t (cid:18) − I ( t ) (cid:19) (4.40)= 0 . . . . , (4.41)which is exactly the same constant that appears in the discrete case in Eq. (4.30). ecord statistics for random walk bridges (cid:96) α max ,B ( n ). We perform here the analysis on the exponential case, but it can also bedone on the discrete case, leading to the same result in the large n limit. For theexponential case, one obtains the GF of F e ( (cid:96), n ) (0) = Pr[ (cid:96) emax ,B ( n ) ≤ (cid:96) ] (0) from Eqs.(4.33) and (3.22) as ∞ (cid:88) n =0 F e ( (cid:96), n ) (0) z n = (cid:96) (cid:88) a =1 z a ∞ (cid:88) m =1 (cid:104) ˜ h e ( z, (cid:96) ) (cid:105) m − q e ( m, a ) . (4.42)Again, to investigate the large n behavior of F e ( (cid:96), n ) (0) , we need to analyze this GFin the limit z = e − s →
1, i.e., s →
0. From the analysis performed before, using inparticular the asymptotic behaviors in Eqs. (4.12) and (4.13) one obtains ∞ (cid:88) n =0 F e ( (cid:96), n ) (0) e − s n = 1 b √ s I ( s (cid:96) ) , (4.43)where the function I ( t ) is given in Eq. (4.26). From Eq. (4.43), we obtain that, for large n , F e ( (cid:96), n ) is a function of the ratio (cid:96)/n , as expected from our previous computationshowing that (cid:104) (cid:96) emax ,B ( n ) (cid:105) ∼ λ emax ,B n (4.41). This expresses the fact that, for large n ,the random variable R ( n ) = (cid:96) emax ,B ( n ) /n becomes independent of n , R ( n ) → R when n → ∞ , as it is also the case for free RWs [29]. The cumulative distribution of thisrandom variable R , F e R ( r ) is given by F e ( (cid:96), n ) = F e R (cid:18) (cid:96)n = r (cid:19) , F e R ( r ) = 2 √ π (cid:90) Γ d u πi e u I ( u r ) √ u , (4.44)where Γ is a Bromwich contour in the complex plane and r ∈ [0 , R , f e R ( r ) = d F e R ( r ) / d r .In particular, one finds that f e R ( r ) is non-analytic at r = 1 / / ,
1] where it takes a simple form f e R ( r ) = 1 + 14 r / , ≤ r ≤ . (4.45)On the other hand, for small r , it has an essential singularity where it behaves asln f e R ( r ) ∼ ln r r , as r → . (4.46)The study of the corresponding PDF for free RW was recently carried out in Ref. [29](see also Ref. [30]). In this case, the PDF also exhibits a non-analyticity at r = 1 / r →
1, while for a bridge,there is no singularity [see Eq. (4.45)]. On the other hand, the PDF for free RW alsoexhibits an essential singularity when r → but without the logarithmiccorrection, which is thus a special feature of the RW bridge. In the next section, wepresent a numerical evaluation of this PDF f e R ( r ) (see the left panel of Fig. 11).
5. Numerical results
In this section, we present results obtained from numerical simulations of random walkbridges, which we compare to our exact analytical computations. We consider discrete ecord statistics for random walk bridges α = d) as well as continuous ( α = c) jump distributions, for µ = 2, with a specialattention on the exponential jump distribution ( α = e). To simulate discrete RW bridges, we generate simple free random walks of n stepsaccording to Eqs. (1.1) and (1.2) and retain only the walks such that x ( n ) = x (0) = 0.The statistics is then performed over this restricted ensemble of RW bridges. In Fig. 3,we show a plot of the average number of records (cid:104) R d B ( n ) (cid:105) , computed numerically, asa function of √ n , which shows a perfect agreement with our exact results in (2.10).In Fig. 3 we have also plotted the asymptotic estimate of (cid:104) R d B ( n ) (cid:105) beyond the leading O ( √ n ) term. Indeed, from the exact formula in Eq. (2.10), one obtains: (cid:104) R d B ( n ) (cid:105) ∼ √ π / √ n + 12 + o (1) , (5.1)which, as can be seen on Fig. 3, is a very good estimate of the exact result for √ n ≥ simulationexact1/2+( (cid:47) n/2 ) free RW, exact h R d B ( n ) i , h R d ( n ) i p n Figure 3.
Average number of records (cid:104) R d B ( n ) (cid:105) as a function of √ n for discrete RWbridges. The full blue circles are the results of numerical simulations. The green emptycircles correspond to the exact results given in Eq. (2.10), while the full red line is aplot of its asymptotic form given in Eq. (5.1). For comparison, we have also plotted,with empty blue circles the exact results for (cid:104) R d ( n ) (cid:105) for the free RW, as obtained in[5], which displays a faster growth (cid:104) R d ( n ) (cid:105) ∼ (2 / √ π ) √ n for n (cid:29) We have also computed numerically the distribution of R d B ( n ), P d B ( m, n ) =Pr( R d B ( n ) = m ) both exactly, for n = 40 and 80 from the analytical formula in Eq. (3.13) ecord statistics for random walk bridges n = 40. As shown in Fig. 4, the numerical results agree perfectlywith our exact analytical results. Besides, for large n and large m , keeping m/ √ n = X n=40, exactn=80, exactasymptoticn=40, simulation ( m / / p n p n P d B ( m , n ) Figure 4.
Plot of √ n P d B ( m, n ) as a function of ( m − / / √ n for discrete RW –see Eq. (5.2) and the discussion below in the text. The filled circles correspond tothe exact values obtained from Eq. (3.13) for different values of n = 40 (blue) and n = 80 (green). The open circles correspond to the results of numerical simulationsfor n = 40, which coincide exactly with the exact results. Finally, the solid line is theexact scaling function ϕ d B ( X ) given in Eq. (1.19). fixed, one expects the scaling form given in Eq. (1.20). Our simulations for finite valuesof n exhibit small finite size corrections to this limiting scaling form. As suggestedin Fig. 4, these finite n actually correspond to a simple shift of the scaling variable m/ √ n → ( m − / / √ n , i.e., P d B ( m, n ) ∼ √ n ϕ d B (cid:18) m − / √ n (cid:19) , (5.2)where the scaling function ϕ d B ( X ) is given in Eq. (1.20). Although we have notperformed a detailed analytical study of the finite n corrections to the limiting PDFin Eq. (1.20), this form in Eq. (5.2), and in particular this shift of 1 /
2, is consistentwith the asymptotic expansion of the average number (cid:104) R d B ( n ) (cid:105) , beyond the leading O ( √ n ) term, given in Eq. (5.2).Then, we have computed numerically the probability of record breaking Q d B ( n ),both from numerical simulations and from our exact formula in Eq. (4.2). As shown inthe left panel of Fig. 5, the agreement between both estimates is very good. Besides,we also see on that figure that the convergence of Q d B ( n ) to its asymptotic value Q d B ( ∞ ) ecord statistics for random walk bridges / √ n , i.e., Q d B ( n ) − Q d B ( ∞ ) ∝ / √ n . Furthermore, the estimate of Q d B ( ∞ ) obtained in this way, by extrapolation, is in good agreement with our exactresults in Eq. (4.9), see the right panel of Fig. 5. exactsimulationQ ∞ exactsimulationQ ∞ / √ nn Q d B ( n ) Q d B ( n ) Q d B ( ∞ ) Q d B ( ∞ ) Figure 5. Left:
Probability of record breaking Q d B ( n ) for the discrete randomwalk bridge. The blue open circles correspond to our numerical simulations whilethe yellow full circles are the exact values of Q d B ( n ), extracted from the generatingfunction in Eq. (4.2). Right:
Slow convergence of Q d B ( n ) to its asymptotic value Q d B ( ∞ ) = 0 . . . . in Eq. (4.9). The full line is a guide to the eyes and correspondsto Q d B ( ∞ ) + const ./ √ n . Finally, in Fig. 6, we show numerical results for (cid:104) (cid:96) dmax ,B ( n ) (cid:105) /n , obtainedfrom numerical simulations and compare them to the exact results which can bestraightforwardly obtained from the generating function in Eq. (4.22). Here also theagreement between numerics and theory is very good. We also notice that, as in thecase of the probability of record breaking (see Fig. 5), the convergence of (cid:104) (cid:96) dmax ,B ( n ) (cid:105) /n to the asymptotic value λ dmax ,B [see Eq. (1.25)] is actually quite slow. µ = 2 . To generate a RW bridge with continuous jump distributions and µ = 2 (i.e., with afinite variance σ ), we first generate a free random walk of n steps, starting from x (0) = 0and evolving according to Eq. (1.1). We subsequently use the following construction x B ( k ) = x ( k ) − kn x ( n ) (5.3)to generate a RW bridge { x B ( k ) } , for 0 ≤ k ≤ n . In particular, Eq. (5.3) impliesobviously x B (0) = x B ( n ) = 0. This simple construction (5.3) holds, for any finite n , forGaussian jump distributions. For other jump distributions, including in particular theexponential jump distribution ( α = e), this construction is only approximate, becomingexact in the limit of large n , where x B ( k ) / √ n , for k ∼ O ( n ), converges to the Brownianbridge. In Fig. 7, we show a plot of the average number of records (cid:104) R αB ( n ) (cid:105) both ecord statistics for random walk bridges simulationexact0.6380 n h ` d m a x , B ( n ) i / n dmax ,B = 0 . . . . Figure 6.
Plot of (cid:104) (cid:96) dmax ,B ( n ) (cid:105) /n as a function of n for the discrete random walk bridge.The open circles correspond to the exact analytical value computed from the generatingfunction in Eq. (4.22) while the full circles correspond to the estimates obtained fromdirect numerical simulations. The solid line corresponds to the asymptotic value λ dmax ,B = 0 . . . . , see Eq. (1.25). for an exponential distribution, α = e, and for a Gaussian distribution, both of thembelonging to the class of continuous jump distributions α = c as in Eq. (1.3) with µ = 2.For large n , one expects that, in both cases, (cid:104) R e B ( n ) (cid:105) ∼ (cid:104) R c B ( n ) (cid:105) ∼ A c B ( µ = 2) √ n with A c B ( µ = 2) = √ π/ (cid:104) R e B ( n ) (cid:105) in Eq. (2.17),yielding (cid:104) R e B ( n ) (cid:105) ∼ √ π √ n + 12 + o (1) . (5.4)Note that the first O (1) correction, namely 1 /
2, is the same as in the discrete case (5.1).For smaller values of n , the data for the exponential case, α = e, in Fig. 7 show a slightdiscrepancy between the numerical results and our exact formula for (cid:104) R e B ( n ) (cid:105) given inEq. (2.17). This is due to the fact that our construction of the RW walk bridge (5.3) isin this case only approximate for small values of n . Note that to simulate exact bridgesin this exponential case, one could use the Monte-Carlo method used in Ref. [32], whichhowever requires more numerical efforts and goes beyond the scope of the present work.In both cases, exponential and Gaussian jump distributions, we have computed the ecord statistics for random walk bridges exponential1/2+(n (cid:47) ) /2Gaussianexponential, exactfree RW Figure 7.
Average number of records (cid:104) R αB ( n ) (cid:105) obtained by simulations of a RWwith an exponential jump distribution, α = e, (blue circles) and for a Gaussian jumpdistribution, α = c, (green circles), compared to the exact asymptotic value (fullblack line), given in Eq. (5.4). The red circles correspond to the result for theexponential case in Eq. (2.17). The slight discrepancy between the exact and thenumerical results in the exponential case for small values of n is due to the fact thatour numerical procedure to generate a bridge (5.3) is only approximate for finite n . Forcomparison, we have also plotted (with cyan circles) the exact results for a free RWwith a continuous jump distribution obtained in [5], which displays a faster growth (cid:104) R c ( n ) (cid:105) ∼ / √ π √ n for n (cid:29) distribution of the number of records P e B ( m, n ) and P c B ( m, n ). The results are shownin Fig. 8. In the left panel, we show the results for an exponential jump distribution, α = e, which show a very good agreement between our numerical estimates and the exactformula for P e B ( m, n ) given in Eq. (3.34). These data for finite n show some deviationfrom the asymptotic result in Eq. (1.23) valid for n (cid:29)
1. As suggested in Fig. 4, andsimilarly to the discrete case (5.2), these finite n corrections actually correspond to asimple shift of the scaling variable m/ √ n → ( m − / / √ n , i.e., P e B ( m, n ) ∼ √ n ϕ e B (cid:18) m − / √ n (cid:19) , (5.5)where the scaling function ϕ e B ( X ) is correctly predicted by the expression given inEq. (1.23). Note that this shift of 1 / (cid:104) R e B ( n ) (cid:105) beyond the leading order in Eq. (5.4). In the right panel of Fig. 8, weshow numerical data for P c B ( m, n ) for a Gaussian jump distribution. These data areparticularly instructive, as we do not have any analytical result for this quantity for ecord statistics for random walk bridges asymptoticn=40, exponential stepsn=80, exponential stepsn=40, exact asymptotic (exponential)n=160, Gaussian stepsn=320, Gaussian steps ( m − / / √ n ( m − / / √ n √ n P c B ( m , n ) √ n P e B ( m , n ) Figure 8. Left:
Plot of √ n P e B ( m, n ) as a function of ( m − / / √ n for an exponentialjump distribution. The filled circles correspond to simulations for n = 40 (blue circles)and n = 80 (green circles). The empty circles correspond to the exact analytical resultgiven in Eq. (3.34) for n = 40. The solid line corresponds to the exact scaling function ϕ e B ( X ) given in Eq. (1.23). Right:
Plot of √ n P c B ( m, n ) as a function of ( m − / / √ n for a Gaussian jump distribution for n = 160 and n = 320. The solid line correspondsto the exact scaling function ϕ e B ( X ) given in Eq. (1.23) Gaussian jumps. Our data actually suggest that the distribution of the number ofrecords in this case is also described, for large n , by the same scaling form as for theexponential jump distribution in Eq. (5.5), with the same scaling function ϕ e B ( X ). Thesedata support our conjecture that this result in Eq. (1.23), which we can explicitly showonly for the exponential case, should actually hold for any continuous jump distributionas in Eq. (1.3) with µ = 2.We have also computed numerically the probability of record breaking Q e B ( n ) foran exponential jump distribution, which we have plotted in Fig. 9. Here again, we see avery good agreement between our exact results [extracted from the generating functionin Eq. (4.11)] and the numerical estimates. These data also corroborate the asymptoticbehavior Q e B ( n ) → Q e B ( ∞ ) = 0 . . . . when n → ∞ , where Q e B ( ∞ ) = Q d B ( ∞ ) isgiven in Eq. (4.9) and coincides with the asymptotic value Q d B ( ∞ ) found for the discreteRW. Note that our numerical results indicate that the convergence to the asymptoticvalue is ∝ /n , which is thus faster than for the discrete case, where the convergence is ∝ / √ n (see the right panel of Fig. 5). In Fig. 9, we have also plotted the estimatesof Q c B ( n ) for random walk bridges, with Gaussian jump distribution, obtained fromnumerical simulations, using Eq. (5.3). The data for exponential and Gaussian jumpsare, for n sufficiently large, indistinguishable, which reinforce our claim that all ourasymptotic results for the exponential case should also hold for any continuous jumpdistributions with µ = 2, in the limit n → ∞ .Similarly, in Fig. 10, we show a plot of (cid:104) (cid:96) emax ,B ( n ) (cid:105) /n , for an exponential jumpdistribution and of (cid:104) (cid:96) cmax ,B ( n ) (cid:105) /n , for a Gaussian jump distribution. In the exponentialcase, we have plotted both our exact results, which can be derived from Eq. (4.33), andthe estimates obtained by simulating the random walk bridge, using Eq. (5.3). We seethat the agreement is very good. Interestingly, we also see that the data for exponential ecord statistics for random walk bridges exact, exponential stepssimulation, exponential stepssimulation, Gaussian steps0.654304 n Q e B ( n ) , Q c B ( n ) Q e B ( ) = 0 . . . . Figure 9.
Probability of record breaking for an exponential jump distribution, Q e B ( n ),and for a Gaussian jump distribution, Q c B ( n ). In the exponential case, we showdata corresponding to the exact result (open circle), obtained from the generatingfunction in Eq. (4.11) and data obtained from direct numerical simulations (full circles)using Eq. (5.3). The data for a Gaussian jump distribution correspond to numericalsimulations using Eq. (5.3). The convergence towards the exact asymptotic value Q e B ( ∞ ) = 0 . . . . is fast ( ∝ /n ). and Gaussian jump distributions are almost indistinguishable for n sufficiently large.Finally, we present the numerical evaluation of the distribution f e R ( n ) = d F e R ( r ) / d r of (cid:96) emax ,B ( n ) /n in the large n limit [see Eq. (4.44)]. The numerical estimates of f e R ( r )can be obtained by evaluating F e ( (cid:96), n ) from the generating function that was explicitlycomputed in Eq. (4.42), which we expand close to z = 0. The same procedure was usedby us in Ref. [29] to plot the corresponding PDF for the free RW. The plot of f e R ( r ), forRW bridges with an exponential jump distribution, obtained this way is shown in theleft panel of Fig. 11. The non-analyticity at r = 1 / f e R ( r ) and the exactresult for 1 / ≤ r ≤ V = 1 / R , whose PDF turns out to be simpler tostudy (see Ref. [29] as well as Appendix C).We conclude this section by noticing that a natural way of simulating numericallyrandom walk bridges would be to write an effective local equation of motion for x B ( k )which would take into account the conditioning that the walker has to return back tothe origin after n time steps. For Brownian motion, which is continuous both in space ecord statistics for random walk bridges simulation, exponential stepsexact, exponential stepssimulation, Gaussian steps0.63806 h ` e m a x , B ( n ) i / n , h ` c m a x , B ( n ) i / n n emax ,B = 0 . . . . Figure 10.
Plot of (cid:104) (cid:96) emax ,B ( n ) (cid:105) /n , for an exponential jump distribution, and (cid:104) (cid:96) cmax ,B ( n ) (cid:105) /n , for a Gaussian jump distribution, as a function of n . For the exponentialcase, the exact results are obtained from Eq. (4.33), while in both cases, the numericalestimates are obtained by simulating the random walk bridges from Eq. (5.3), withthe appropriate jump distribution. n=30n=401+r -3/2 /4 -2 +v -1/2 /4 f e R ( r ) f e V ( v ) r v Figure 11. Left:
Probability density of the ratio R ( n ) = (cid:96) max ( n ) /n for an exponentialjump distribution, for n = 30 and n = 40. The points were obtained from the exactexpression for the generating function of the cumulative distribution F e ( (cid:96), n ) (4.42).The red curve is the analytical prediction for the limiting PDF f e R ( r ) (4.45) valid for1 / ≤ r ≤ Right:
Probability density of V ( n ) = n/(cid:96) max ( n ) for an exponentialjump distribution of steps, for n = 30 and n = 40, obtained from the left panel. and time, this can indeed be done by writing an effective Langevin equation for theBrownian bridge [33, 34]. Extending this approach to discrete time random walks –both with discrete and continuous jumps – is a very interesting open problem. ecord statistics for random walk bridges
6. Conclusion
To conclude, we have studied the record statistics of RW bridges, for different typesof symmetric jump distributions. Our results show that the record statistics of suchconstrained random walks, which are constrained to start and end at the origin after n steps, are quantitatively different from their counterpart for free RWs, which canend up anywhere on the real axis. We first showed that the statistics of the numberof records R αB ( n ) is not only different for discrete ( α = d) and continuous ( α = c)distributions but, even for continuous jump distributions, also depends on the detailsof this distribution. We obtained exact results for the average number of records (cid:104) R αB ( n ) (cid:105) ∼ A αB √ n and showed in particular that A c B ≡ A c B ( µ ) depends continuouslyon the L´evy index µ characterizing the RW. This is quite different from the free RWcase where R c ( n ) ∼ A c √ n where A c = 2 / √ π , independently of µ . Furthermore, we havecomputed exactly the full record statistics for two different types of jump distributions:the discrete RW ( α = d) and the exponential distribution ( α = e). We emphasize thatthese calculations are technically much harder than for free RWs as the records statisticsof RW bridges requires to keep track not only of the ages of the records but also of theincrements between successive records [see Fig. 1 and Eq. (3.21)]. For these two jumpdistributions, we have also computed the probability of record breaking Q αB ( n ) and theaverage age of the longest lasting record (cid:104) (cid:96) α max ,B ( n ) (cid:105) and shown that they give rise totwo new non-trivial constants which we have computed explicitly (see Table 1).Although the records statistics for a continuous jump distribution depends, for finite n , on the details of this distribution, one expects that, for large n , it only depends onthe L´evy index µ in Eq. (1.3). Although we can not prove this statement, our numericaldata indicate that this should indeed be the case, at least for µ = 2. This implies thatour asymptotic results (see Table 1) obtained in the exponential case, which we couldsolve exactly, should describe the large n limit of any continuous jump distribution with µ = 2. The generalization of our results to arbitrary value of the L´evy index 0 < µ < Acknowledgments
SNM and GS acknowledge the Indo-French Centre for the Promotion of AdvancedResearch under Project 4604-3.
Appendix A. Useful formulas for discrete random walks
Here we consider a discrete RW, starting from x (0) = 0 and evolving via x ( k ) = x ( k −
1) + η ( k ) , (A.1)where the jump variables η ( k )’s are i.i.d. random variables distributed according to p d ( η ) = δ ( η + 1) + δ ( η − W ( x, n ) the number of lattice RW (A.1) ecord statistics for random walk bridges x (0) = 0 and ending in x after n time steps. One has W ( x, n ) = (cid:40) (cid:0) n n + x (cid:1) , if n + x is even0 if n + x is odd . (A.2)From W ( x, n ), one obtains immediately the discrete propagator G d ( x, x , n ) as G d ( x, x , n ) = W ( x − x , n )2 n n (cid:0) n n + x − x (cid:1) , if n + x − x is even0 if n + x − x is odd . (A.3)The generating function ˜ G d ( x, x , n ) of G d ( x, x , n ) can be computed from this explicitexpression (A.3) and it yields:˜ G d ( x, x , n ) = ∞ (cid:88) n =0 G d ( x, x , n ) z n = ∞ (cid:88) k = (cid:100) x − x (cid:101) (cid:16) z (cid:17) k − ( x − x ) (cid:18) k − ( x − x ) k (cid:19) = 1 √ − z (cid:18) − √ − z z (cid:19) x − x , (A.4)where (cid:100) u (cid:101) is the smallest integer not less than u and where the last equality can beobtained using, for instance, Mathematica.Furthermore, from the expression of G d ( x, x , n ) (A.2), one can also compute theconstrained propagator G d ≥ ( x, , n ) using the method of images. One has indeed: G d ≥ ( x, , n ) = G d ( x, , n ) − G d ( x, − , n ) (A.5)= n (cid:16)(cid:0) n n + x (cid:1) − (cid:0) n n + x +1 (cid:1)(cid:17) , if n + x is even0 , if n + x is odd . (A.6)The generating function ˜ G d ≥ ( x, , z ) of G d ≥ ( x, , n ) can then be obtained from this explicitexpression (A.5) as˜ G d ≥ ( x, , z ) = ∞ (cid:88) n =0 G d ≥ ( x, , n ) z n = ∞ (cid:88) k = (cid:100) x (cid:101) (cid:16) z (cid:17) k − x (cid:18) k − xk (cid:19) − ∞ (cid:88) k = (cid:100) x +22 (cid:101) (cid:16) z (cid:17) k − ( x +2) (cid:18) k − ( x + 2) k (cid:19) = 2 z (cid:18) − √ − z z (cid:19) x +1 . (A.7)Note that this result (A.7) which we obtained here by counting paths can also beobtained using a backward Fokker-Planck equation.Finally, one can also compute the propagator G d > ( x, , n ) associated to a RW whichis constrained to stay strictly above 0. To do so, one can simply analyze the firststep. This first step is necessarily a step +1, such that η (1) = +1, which happenswith probability 1 /
2. After this first step, the remaining propagator is just given by ecord statistics for random walk bridges G d ≥ ( x − , , n − G > ( x, ,
0) = δ x, (A.8) G > ( x, , n ) = 12 G d ≥ ( x − , , n − , n ≥ . (A.9)One thus obtains the generating function ˜ G > ( x, , z ) as˜ G > ( x, , z ) = ∞ (cid:88) n =0 G > ( x, , n ) z n = (cid:18) − √ − z z (cid:19) x . (A.10) Appendix B. Useful formulas for RWs with exponential jump distribution
Here we consider a RW, starting from x (0) = 0 and evolving via x ( k ) = x ( k −
1) + η ( k ) , (B.1)where the jump variables η ( k )’s are i.i.d. random variables distributed according to p e ( η ) = b e −| η | /b . A remarkable feature of the exponential jump distribution is that theconstrained Green’s function G e > ( x, x , n ) can be computed exactly [31].To begin with, it is useful to compute the Fourier transform of the jump distributionˆ p e ( q ) = (cid:90) ∞−∞ d η p e ( η ) e i q η = 11 + ( b q ) , (B.2)from which we easily obtain the free propagator G e ( x, x , n ) = (cid:90) ∞−∞ d q π e − i q ( x − x ) b q ) ] n . (B.3)Finally, the GF of G e ( x, x , n ) is given by˜ G e ( x, x , z ) = ∞ (cid:88) n =1 G e ( x, x , n ) z n = z (cid:90) ∞−∞ d q π e − i q ( x − x ) − z + ( b q ) , (B.4)where the Dirac delta function δ ( x ) comes from the term n = 0, while the second onecomes from the sum from n = 1 to ∞ of the geometric series. Finally, the integral over q in (B.4) can be explicitly evaluated to yield the result given in Eq. (2.12)˜ G e ( x, , z ) = ∞ (cid:88) n =1 z n G e ( x, , z ) = z b √ − z e − | x | b √ − z . (B.5)We now come to the constrained propagator G e > ( x, , n ), which is harder tocompute. The easiest way to compute it is to use the so-called Hopf-Ivanov formula[35] (see also [36] for a detailed derivation of this result) which gives the followingexpression ∞ (cid:88) n =0 z n (cid:90) ∞ d x G e > ( x, , n ) e − λ x = φ ( λ, z ) , (B.6)where φ ( λ, z ) is given by φ ( λ, z ) = exp (cid:18) − λπ (cid:90) ∞ d q ln (1 − z ˆ p e ( q )) λ + q (cid:19) , (B.7) ecord statistics for random walk bridges p e ( q ) is the Fourier transform of the jump distribution given in (B.2). The function φ ( λ, z ) is thus given by φ ( λ, z ) = exp − λπ (cid:90) ∞ d q ln (cid:16) − z +( b q ) b q ) (cid:17) λ + q . (B.8)The integral over q in Eq. (B.8) can be explicitly evaluated, using the formula 4.295-7of Ref. [37] (cid:90) ∞ d x ln ( a + b x ) c + d x = πc d ln (cid:18) a d + b cd (cid:19) , for a, b, c, d > , (B.9)to yield finally the identity ∞ (cid:88) n =0 z n (cid:90) ∞ d x G e > ( x, , n ) e − λ x = 1 + λ b √ − z + λ b = 1 + 1 − √ − z √ − z + λ b . (B.10)It is then easy to inverse the Laplace transform with respect to x to obtain ˜ G e > ( x, , z )given in Eq. (2.13)˜ G e > ( x, , z ) = ∞ (cid:88) n =0 z n G e > ( x, , z ) = δ ( x ) + 1 − √ − zb e − | x | b √ − z . (B.11)Note finally that, by differentiating the relation in Eq. (B.10) ( m −
2) times with respectto λ and setting λ = 1 /b , one obtains straightforwardly the relation given in Eq. (3.32). Appendix C. Analysis of the distribution of (cid:96) emax ,B ( n ) /n In this appendix, we derive the behavior for the PDF f e R ( r ) first for r ∈ [1 / ,
1] andthen for r → Appendix C.1. Density f e R ( r ) in the interval [1 / , (cid:88) n ≥ e − s n F e ( (cid:96), n ) (0) = 1 b √ s I ( s(cid:96) ) , (C.1)valid in the limit s → (cid:96) → ∞ , keeping the product x = s (cid:96) fixed and where thefunction I ( t ) is given in Eq. (4.26). In the limit s →
0, the discrete sum in Eq. (C.1)can be replaced by an integral and F e ( (cid:96), n ) (0) can be written as F e ( (cid:96), n ) (0) = (cid:90) Γ d s π e ns b √ s I ( s(cid:96) ) , (C.2)where Γ is a Bromwich contour. Note that by definition, we have F e ( (cid:96), n ) (0) = Prob( (cid:96) emax ,B ( n ) < (cid:96) ) (0) = Prob (cid:16) n(cid:96) emax ,B > n(cid:96) (cid:17) (0) . (C.3)In the limit of large n this probability converges to the complementary distributionfunction of the random variable V = 1 / R = lim n →∞ n/(cid:96) max ( n ),¯ F e V ( v ) (0) = Prob (cid:16) V > v (cid:17) (0) = √ vb √ n (cid:90) d x π e xv √ x I ( x ) , (C.4) ecord statistics for random walk bridges v = n/(cid:96) and x = s (cid:96) as above. Finally, dividing by G e (0 , , n ) ∼ / (2 b √ πn ),¯ F e V ( v ) = 2 √ πv (cid:90) d x π e xv √ x I ( x ) . (C.5)We want to analyze the behavior of F e R ( r ) for r close to 1, or equivalently the small v behavior of of ¯ F e V ( v ), which is dominated by the large x behavior of the integrand inthe Bromwich integral in Eq. (C.5). Thus we need to expand I ( x ) given by (4.26) inthe small parameter (cid:15) ( x ) = e − x . Formally (cid:90) d x π e xv √ x I ( x ) ≈ (cid:90) d x π e xv √ x (cid:0) I ( x ) + I ( x ) e − x + I ( x ) e − x + . . . (cid:1) , (C.6)where the functions I k ( x ) admits an expansion in powers (both positive and negative)of √ x . Therefore the first term ∝ e − x contributes to the function for v >
1, the secondterm, ∝ e − x , for v >
2, and so on. At first order in (cid:15) ( x ) = e − x we have I ( x ) ≈
12 + 14 (cid:18) (4 x −
1) e − x √ πx − (4 x + 1)erfc √ x (cid:19) + O (e − x ) , (C.7)which, divided by √ x , can be Laplace inverted to give for ¯ F V ( v )¯ F e V ( v ) = 12 + 1 v − √ v ≤ v ≤ , (C.8)and therefore f e V ( v ) = − dd v ¯ F e V ( v ) = 1 v + 14 √ v (1 ≤ v ≤ . (C.9)Finally, using f e R ( r ) = f e V (1 /r ) /r , one obtains f e R ( r ) = 1 + 14 r / (cid:16) ≤ r ≤ (cid:17) , (C.10)as announced in the text in Eq. (4.45). Appendix C.2. Small r behavior of f e R ( r )We now study the small r behavior of f e R ( r ), or equivalently the large v behavior of f e V ( v ) (recalling that V = 1 / R ). One can show that the function I ( x ) / √ x is an entire inthe complex x plane. Hence the integral (C.5) is expected to be dominated by a saddlepoint in x . Let us define φ ( x ) = ln I ( x ) √ x . (C.11)The saddle-point equation is v + φ (cid:48) ( x ) = 0, thus φ (cid:48) ( x ) is large and negative, hence x islarge and negative too. Setting x = − y we obtain the estimate the large negative x behavior of F ( x ) in Eq. (4.5) as F ( x = − y ) ≈ i e y √ πy , (C.12)which yields φ ( x = − y ) ∼ e y πy = e − x πx , (C.13) ecord statistics for random walk bridges x ( F ( x )) in I ( x ). The saddle-point equation now gives v ∼ e − x , (C.14)hence, for v large, using estimates at exponential order, we find f e V ( v ) = − ddv ¯ F e V ( v ) ∼ e − v ln v , (C.15)i.e., a super-exponential decay for this quantity. As a consequence one can also predictthat the essential singularity at the origin of f e R ( r ) = f e V (1 /r ) /r is given by f R ( r ) ∼ e ln r/ (2 r ) , (C.16)as given in the text in Eq. (4.45). References [1] D. Gembris, J. G. Taylor and D. Suter, Sports statistics: Trends and random fluctuations inathletics,
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