On the Gap and Time Interval between the First Two Maxima of Long Continuous Time Random Walks

On the Gap and Time Interval between the First Two Maxima of Long Continuous Time Random Walks

Philippe Mounaix philippe.mounaix@cpht.polytechnique.fr Centre de Physique Théorique, UMR 7644 du CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France.    Grégory Schehr gregory.schehr@lptms.u-psud.fr Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, Bât 100, 91405 Orsay Cedex, France.    Satya N. Majumdar majumdar@lptms.u-psud.fr Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, Bât 100, 91405 Orsay Cedex, France.
September 3, 2019
Abstract

We consider a one-dimensional continuous time random walk (CTRW) on a fixed time interval where at each time step the walker waits a random time , before performing a jump drawn from a symmetric continuous probability distribution function (PDF) , of Lévy index . Our study includes the case where the waiting time PDF has a power law tail, , with , such that the average time between two consecutive jumps is infinite. The random motion is sub-diffusive if (and super-diffusive if ). We investigate the joint PDF of the gap between the first two highest positions of the CTRW and the time separating these two maxima. We show that this PDF reaches a stationary limiting joint distribution in the limit of long CTRW, . Our exact analytical results show a very rich behavior of this joint PDF in the plane, which we study in great detail. Our main results are verified by numerical simulations. This work provides a non trivial extension to CTRWs of the recent study in the discrete time setting by Majumdar et al. (J. Stat. Mech. P09013, 2014).

pacs:
05.40.Fb, 02.50.Cw

I Introduction and summary of main results

Recently, there has been a resurgence of interest in extreme value questions for random walks and its variants, particularly in the physics literature AS2005 ; DB09 ; DB11 ; MMS2013 ; MMS2014 ; SLD2010 . This is motivated to a large extent by the observation that extreme value statistics (EVS) plays a crucial role in the physics of complex and disordered systems BM97 ; DM01 . Although the EVS of independent and identically distributed (i.i.d.) random variables is very well understood, much less is known for strongly correlated variables, which is the case frequently encountered in physics. From that point of view, random walks offer a very useful framework where the effects of correlations on EVS can be characterized analytically.

The statistics of the maximal displacement of a random walk (RW) after a given number of steps is now quite well understood, thanks for instance to the Pollaczeck-Spitzer formula Poll ; Spitz , from which precise asymptotic estimates in the large limit can be obtained AS2005 . The full order statistics, i.e. the statistics of the maximum for any integer , was investigated in SM2012 in the case of ordinary RWs with a narrow jump distribution (characterized by a Lévy exponent ). Of particular interest is the gap between two consecutive maxima, which makes it possible to characterize the “crowding” of the walker positions near their maximum. Along this line we recently obtained a complete description of the joint distribution of the gap and time interval between the first two maxima of RWs not only for narrow jump distributions (), but also for Lévy flights characterized by a Lévy exponent  MMS2013 ; MMS2014 . Since the characteristic displacement of the walker after time steps grows like , the results of MMS2013 ; MMS2014 are of interest for diffusive () and super-diffusive () processes only, leaving out the important case of sub-diffusive processes. Extending the analysis of MMS2013 ; MMS2014 to sub-diffusive RWs has been one of the motivations for this work.

A common model to describe sub-diffusive processes is the so called continuous time random walk (CTRW). In this model, the walker performs a usual random walk where two consecutive jumps are separated by a certain trapping time . The trapping times are i.i.d. random variables drawn from a common distribution which has a power law tail with . This model, initially suggested by Scher and Montroll in the setting of non-Gaussian transport in disordered electronic systems SM1975 , has been widely used to describe anomalous sub-diffusive dynamics in various complex systems phenomenologically BG90 ; MK2000 . Indeed, for the average trapping time between two successive jumps is infinite and, consequently, CTRWs are characterized by an anomalous growth of displacement which is indeed sub-diffusive if (and super-diffusive for ). The case of a finite average trapping time between two successive jumps corresponds to standard, non sub-diffusive, RWs. The fluctuations of the maximal displacement of CTRWs for and arbitrary was studied in Ref. SLD2010 using real space renormalization techniques (see also CTB2010 for a Feynman-Kac approach). A precise estimate of the expected value of the maximum of the CTRW was obtained in Ref. FM2012 by combining the results of Ref. AS2005 with the so called ’subordination property’. However, nothing is known about the order statistics, beyond the first maximum, for such CTRWs. In this paper, we provide a complete analytic description of the distribution of the gap and the time interval between the first two maxima of such CTRWs for any values of the parameters and , as well as for a finite average trapping time between two successive jumps.

We consider a CTRW starting at the origin, , and evolving according to

(1)

where denotes the walker position between the -th and the -th jumps. The jumps are i.i.d. random variables distributed following a symmetric, bounded and piecewise continuous distribution the Fourier transform of which, , has the small behavior

(2)

where is the Lévy index and is the characteristic length scale of the jumps. For , the variance of the jump distribution is well defined and . On the other hand, for , does not possess a well defined second moment because of its heavy tails, (), and the RW (1) is a Lévy flight of index . The time intervals between the -th and the -th jumps () are i.i.d. random variables, independent of , with PDF the Laplace transform of which, , has the small behavior

(3)

where and is the characteristic time scale of the jumps. For , the mean time between two successive jumps, , exists and . For , the mean time between two successive jumps does not exist. As a function of time, the walker position is thus given by

(4)

with and the total number of jumps in the walk. The solution to (4) is readily found to be given by with given by (1).

The goal of the present paper is to provide an exhaustive discussion of the different behaviors of the joint PDF of the gap and time interval between the first two maxima of the walk that may arise in the large limit, depending on the large argument behavior of and . Before entering the details of the calculations, it is useful to summarize our main results. We first show that the joint PDF has a well defined limiting PDF as :

(5)

where the Laplace transform of with respect to (w.r.t.) is given in Eqs. (20) and (17). Then, we perform a detailed analysis of in the plane for and in the whole ranges and , and for the three different main classes identified in MMS2014 : (i) slow, (ii) exponentially, and (iii) fast decreasing at large .

One of the most remarkable result of this study may be the existence of a scaling form for the joint PDF in the case of a fast decreasing jump distribution (), such that for [see Eq. (68) below for a more precise definition], with . This class includes the case of a Gaussian jump distribution for which . This is a new result, characteristic of CTRWs, without any counterpart in the discrete time RW setting considered in MMS2014 . More specifically, we show that there is a scaling regime with fixed , in which takes the scaling form

(6)

with the asymptotic behaviors

(7)

where the amplitudes and are given in Eqs. (80) and (81), respectively. Physically, the switch from the first to the second behavior (7) around corresponds to the cross-over from a ‘concentration’ – or ‘one-step’ – regime (for ) where the walker get stuck for a long time at the second maximum and then jumps directly to the first maximum, to a ‘many-steps’ regime (for ) where she/he travels a long walk of total duration (with many steps) between the second and the first maxima. For , there is no scaling form and (6) is replaced with the uniform expression (84).

An other important result is how the scaling form derived in MMS2014 for a discrete time Lévy flight (see Sec. 4.1 and Appendix D in MMS2014 ) is affected when one switches to a CTRW. Lévy flights have type (i) jump distributions since, for , necessarily has an algebraic, slow decreasing, tail. In this case we show that in the scaling regime with fixed , the joint PDF takes the scaling form

(8)

with the asymptotic behaviors

(9)

and

(10)

where in the first line of (9) denotes the integer part of and the last two lines are for only. It can be checked that for , Eqs. (8), (9), and (10) reduce to Eqs. (87), (88), and (90) of MMS2014 in which the number of jumps between the first two maxima, , is merely replaced with , as expected from simple law of large number arguments giving as . Our results provide thus a non trivial extension of the ones in MMS2014 to the case in which the mean time between two successive jumps does not exist and law of large number arguments cannot be used.

Finally, by integration of the joint PDF over , one finds that the marginal distribution displays a power law tail with logarithmic corrections and an exponent depending only on and . Namely, for one finds

(11)

where the amplitudes , and are given in Eq. (91), and for one has

(12)

where the amplitudes , and are given in Eq. (97). Again, it can be checked that Eq. (11) is nothing but Eq. (112) in MMS2014 with , as expected (cf. the end of the preceding paragraph). Note that for and , one has whatever the distributions and possessing a second and a first moment, respectively. The third lines of (11) and (12) reveal an interesting freezing phenomenon, as a function of , of the large behavior of as decreases past the value . It follows in particular from (11) and (12) that the first moment of is never defined. This means that, although the typical size of is , its average diverges with the total duration, , of the random walk. For one can estimate from (11) and (12) that for , while for and for . Table 1 summarizes the functional dependence of the different asymptotic behaviors of at large in the plane.

Table 1: Functional dependence of the different asymptotic behaviors of at large in the plane.

Note that the marginal distribution of the gap between the first two maxima, , obtained by integration of the joint PDF over , does not depend on (hence on ) and is the same as the one already studied in Sec. 3 of Ref. MMS2014 in the discrete time RW setting.

The outline of the paper is as follows. In Section II we give the Laplace transform of with respect to for a free-end random walk and a random bridge in which the walker is conditioned to return to at the end of the walk. Section III deals with the asymptotic behavior of for large at fixed , from which we derive, in Section IV, the asymptotic behavior of for a Lévy flight at large then large . The asymptotic behavior of for large is investigated in Section V for three classes of jump distributions encompassing a wide range of jumps of practical interest. It is shown in Section V.3 that when the jump distribution has a fast decreasing tail, the behavior of when both and are large takes on a scaling form capturing the existence of two complementary regimes, one dominated by walks with only one step between the first two maxima, and the other dominated by walks with many steps between the first two maxima. In Section VI we give the scaling form of the asymptotic behavior of for a Lévy flight at large and . In Section VII we use the results obtained in Sections III and VI to determine the large behavior of and show its freezing when the random walk is a Lévy flight with index . Finally, Section VIII is devoted to the comparison of our analytical results with numerical simulations.

Ii Laplace transform of with respect to for a free-end random walk and a random bridge

Let denote the joint PDF of the gap, , and the time interval, , between the first two maxima of a continuous time random walk after steps. Since we will restrict ourselves to without loss of generality. Let denote the number of jumps between the first to maxima. For one has

(13)

where has the Laplace transform

(14)

Taking the Laplace transform of (13) with respect to and using (14) one gets

(15)

where is the generating function of with respect to . The large limit of (15) can then be readily obtained from the main result of MMS2014 according to which exists and is given by

(16)

with

(17)

where the functions and are the inverse Laplace transforms of and , respectively:

(18)

with

(19)

Injecting this result into the large limit of (15) yields

(20)

Thus, converges to a limiting distribution as whose Laplace transform with respect to , , is given by (20).

It was proved in MMS2014 that the expression (16) of actually holds for both free-end random walks and random bridges in which the walker is conditioned to return to at the end of the walk. Since, for a given , is entirely determined by through (20), it follows immediately that the expression of , hence the one of the limiting joint PDF , is exactly the same for free-end walks and bridges. Therefore, all the results obtained in the following apply to free-end random walks as well as to random bridges without any modification.

Iii Asymptotic behavior of for large at fixed

Sections III to VI are devoted to the asymptotic behavior of the joint PDF when either or (or both) is large. From the results obtained for the large behavior of it will then be possible to derive the large behavior of the marginal distribution , which will be the subject of Sec. VII.

First, we consider the limit at fixed . As we will see, the behavior of in this limit depends on the value of , leading to new, non trivial, asymptotic expressions if .

iii.1

In this case the mean time between two successive jumps, , exists and by a simple law of large number argument, the large behavior of can be expected to be given by , where is large. Let us check this result explicitly by comparing the large behavior of and the large behavior of .

The behavior of at large is determined by the one of at small . Thus, from (20) and (3) with , one has

(21)

where we have made the change of variable . Similarly, for large the behavior of is determined by the one of near . Making the change of variable one gets

(22)

Comparing Eqs. (21) and (22), one obtains the expected expression

(23)

Finally, Eq. (65) of MMS2014 together with Eq. (23) above yield, for all ,

(24)

iii.2

If the mean time between two successive jumps, , does not exist and the large behavior of is no longer related to by the simple expression (23). Following the same line as for Eq. (21) with , one now has

(25)

where we have made the change of variable . It remains to compute the inverse Laplace transform on the right-hand side of (25). One finds three different regimes depending on the value of .

iii.2.1

For the behavior of near is given by, (see Eq. (63) in MMS2014 ),

(26)

with . Injecting (26) into (25) yields

(27)

Performing then the inverse Laplace transform in (27) and using the reflection formula one obtains, for all ,

(28)

Note that the exponent of in (28) depends on and (28) tends to (24) as .

iii.2.2

For the behavior of near reads, (see Eqs. (63) and (C9) in MMS2014 ),

(29)

with

(30)

where is defined by its Laplace transform

(31)

with . Equations (25) and (29) now yield

(32)

which replaces Eq. (28). Performing the inverse Laplace transform in (32) and using the reflection formula one obtains, for all ,

(33)

Note that the exponent of in (33) does not depend on , unlike its counterpart in Eq. (28). It follows in particular that (33) does not tend to (24) as . This exponent ‘freezing’ from to as decreases past the critical value (at fixed ) is due to a switch of leading terms in the asymptotic expansion of near from for , to for . (See Eqs. (63) and (C9) in MMS2014 ). Although a term , as regular, does not contribute to the large behavior of , it gives a term in the asymptotic expansion of near in (25) which is singular if and thus contributes to (and dominates) the large behavior of for .

iii.2.3 and summary

Finally, at the critical value , the equation (C9) in MMS2014 gives

(34)

which, together with Eq. (25), yields

(35)

Using then

and the reflection formula , one obtains

(36)

Defining

(37)

and

(38)

we can gather the equations (24), (28), (33), and (36) in a more concise form

(39)

which summarizes the large behavior of at fixed and . The term in the first line of (39) denotes the integer part of and the last two lines are for only.

Iv Asymptotic behavior of for a Lévy flight at large then large

The random walk is a Lévy flight if , in which case the variance of does not exist and at large . In this section we investigate the asymptotic behavior of for a Lévy flight in the limit then .

In the cases with and with , this asymptotic behavior follows readily from Eqs. (24), (28), (36), and the large behavior of :

(40)

(see Eq. (48) in MMS2014 ). One finds,

(41)

for with and with , and

(42)

for with .

It remains the case with . Injecting the asymptotics

(43)

(see Eq. (41) in MMS2014 ), and

(44)

into the right-hand side of (31), one gets

(45)

the inverse Laplace transform of which gives

(46)

To determine the large behavior of we make the change of variable in Eq. (30), let , and use (46) and the large behavior of :

(47)

(see Eq. (46) in MMS2014 ). One finds

(48)

where the integral over in the second line is equal to . Thus, from (33), (40), and (48) one obtains

(49)

Finally, writing , , and , one has