Table Of ContentSpatial Extent of Branching Brownian Motion
Kabir Ramola,1, Satya N. Majumdar,1, and Gr´egory Schehr1,
∗ † ‡
1Laboratoire de Physique Th´eorique et Mod`eles Statistiques, UMR 8626,
Universit´e Paris-Sud 11 and CNRS, Bˆatiment 100, Orsay F-91405, France
(Dated: March 16, 2015)
We study the one dimensional branching Brownian motion starting at the origin and investigate
5
thecorrelation between the rightmost (X 0) and leftmost (X 0) visited sites up to time
max min
1 ≥ ≤
t. At each time step the existing particles in the system either diffuse (with diffusion constant D),
0
die (with rate a) or split into two particles (with rate b). We focus on the regime b a where
2 ≤
these two extreme values X and X are strongly correlated. We show that at large time t,
max min
r the joint probability distribution function (PDF) of the two extreme points becomes stationary
a
P(X,Y,t ) p(X,Y). Ourexactresultsforp(X,Y)demonstratethatthecorrelationbetween
M →∞ →
X andX isnonzero,eveninthestationarystate. FromthisjointPDF,wecomputeexactlythe
max min
3 stationary PDF p(ζ) of the (dimensionless) span ζ =(Xmax−Xmin)/pD/b, which is the distance
betweentherightmost andleftmost visitedsites. Thisspandistributionischaracterized byalinear
1 behavior p(ζ) 1(1+∆)ζ for small spans, with ∆ = a 1 . In the critical case (∆ = 0)
∼ 2 (cid:0)b − (cid:1)
this distribution has a non-trivial power law tail p(ζ) 8π√3/ζ3 for large spans. On the other
] ∼
h hand, in the subcritical case (∆ > 0), we show that the span distribution decays exponentially
c as p(ζ) (A2/2)ζexp √∆ ζ for large spans, where A is a non-trivial function of ∆ which we
e ∼ (cid:16)− (cid:17)
m compute exactly. We show that these asymptotic behaviors carry the signatures of the correlation
between X and X . Finally we verify our results via direct Monte Carlo simulations.
- max min
t
a PACSnumbers: 05.40.Fb,02.50.Cw,05.40.Jc
st Keywords:
.
t
a
m I. INTRODUCTION x
-
d
Branching Brownian motion (BBM) is a well-known
n
o model that finds applications in several areas of sci-
c ence including physics, mathematics and biology. BBM t
[ arisesnaturallyinthecontextofsystemswherenewpar-
2 ticles are generated at each time step such as models
v of evolution, epidemiology, population growth and nu-
3 clear reactions, and now has a long history [1–16]. In
9 addition, BBM has also been widely used in theoreti-
6
cal physics where it has been studied in the context of
7
reaction-diffusion models, disordered systems amongst
0
others [11, 12]. BBM is also an important model in
. X X
1 probability theory as it combines the long-studied dif- min span max
0
fusive motion with the random branching mechanism of
5
Galton-Watsontrees[17]. Inthispaperweareinterested
1 FIG.1: Arealization ofthedynamicsofbranchingBrownian
in one dimensional BBM. The process begins with a sin-
: motion (left) in the supercritical case (right) in the critical
v gle particle at the position x = 0 at time t = 0. The
case. Thespanof theprocess isdefinedass=X X ,
Xi dynamics proceeds in continuous time, where in a small where X and X are the maximum and mimnaimx−ummdiins-
max min
r timeinterval∆t,eachparticlesplitsintotwoindependent placements of theprocess upto a certain time t respectively.
a particleswithprobabilityb∆t,dieswithwithprobability
a∆t, and with the remaining probability (1 (a+b)∆t)
−
performs a Brownian motion on a line with a diffusion
constantD. Arealizationofthe dynamics ofsucha pro- in general N(t) 0 particles present in the system at a
≥
cess is shown in Fig. 1. particular time t. The parameters b and a in this BBM
modeldefinethreeregimeswithdifferentproperties. The
In a given realization of this BBM process, there are
number of particles N(t) is a random variable whose
statistics depends on a and b. When the rate of birth
is greater than the death rate (a < b), the supercritical
phase, the process is explosive and the average number
Electronicaddress: [email protected]
∗
Electronicaddress: [email protected] of particles in the system grows exponentially with time
†
Electronicaddress: [email protected] N(t) =exp((b a)t). In contrast, when the birth rate
‡ h i −
2
space
is smaller than the death rate (b < a), the subcritical
phase, the process eventually dies and, on an average,
X
there are no particles present in the system as t .
X
At the critical point a = b, the system is charact→eriz∞ed max
by a fluctuating number of particles with N(t) = 1 at
h i span
all times t.
s=X −X
Ifonetakesasnapshotofthesystematagiventimet, max min
thespatialpositionsoftheexistingparticleshappentobe 0 t
time
stronglycorrelated,sincetheparticlesarelinkedbytheir
commongenealogy. One important objectthat has been
X
extensively studied is the order statistics of these parti- min
cles,i.e.,thestatisticsofthepositionx (t)ofsaythek-th −Y
k
rightmostparticle at time t, where the particle positions
on the line are ordered as x (t) >x (t) > .... >x (t)
1 2 N(t)
[4, 6–9, 18, 19]. Another related interesting quantity
is the gap g (t) = x (t) x (t) between the k-th FIG.2: Schematicrepresentationofatrajectory oftheBBM
k k k+1
− confinedinthebox[ Y,X]. NotethatX andX denote
and (k +1)-th particle at time t. Most of these stud- − max min
respectively the maximum and the minimum of the process
ies have thus focused on extreme value questions at
up to time t. The process starts with a single particle at the
a given time t. However, there are other interesting
origin at time t=0 and henceX 0 while X 0.
max min
extreme value observables that concern the history of ≥ ≤
the process over the entire time interval [0,t]. For in-
stance, one can consider the global maximum, X =
max need to know the joint PDF of X and X . In this
max min
max x (τ),x (τ),x (τ)...,x (τ) which rep-
0 τ<t 1 2 3 N(τ) paper we also compute analytically the stationary PDF
resent≤s the{maximum of all the particle p}ositions up to
of the span s in the critical (b = a) and the subcritical
(cid:2) (cid:3)
time t. This has the simple interpretation as the max-
(b < a) cases. Our exact results demonstrate that the
imum displacement of the entire process up to time t
correlation between X and X is also manifest in
max min
(see Fig. 1). This global maximum has appeared in
the stationary span PDF.
a variety of applications including the spread of gene
The rest of the paper is organized as follows. In sec-
populations [4] and the propagationof animal epidemics
tionII,wedefinethe modelpreciselyandsummarizeour
in two dimensions [13]. Similarly the global minimum
main results. In Section III we derive an exact evolution
X = min x (τ),x (τ),x (τ)...,x (τ) is
min 0 τ<t 1 2 3 N(τ) equationfor the joint distribution of X and X . In
another interes≤ting q{uantity that, by symmetry, ha}s the max min
Section IV we derive the stationary joint PDF of X
(cid:2) (cid:3) max
same marginal probability distribution function (PDF)
and X for the critical (b = a) and the subcritical
min
as X .
max (b < a) cases. In Section V we compute the stationary
−
The marginal PDF of Xmax has been studied exten- PDF of the span and extract its asymptotic behaviors
sively for the supercritical [6, 7], critical and the sub- analytically. In Section VI we compare our analytical
critical phases [4, 20]. While the marginal distributions predictions with Monte Carlo simulations. Finally, we
of Xmax, and hence that of Xmin, are well studied, conclude with a discussion in Section VII. Some details
−
much less is known about the correlation between these of computations are relegated to the appendices.
two random variables. In this paper, we study the joint
PDF of X and X . In the supercritical phase, this
max min
jointPDFisalwaystime dependent, andishardtocom- II. THE MODEL AND A SUMMARY OF THE
pute analytically. However, in this case, X and X RESULTS
max min
get separated from each other ballistically in time and
hence become uncorrelated at late times. In contrast, The model and the observables. We consider the
in the critical and subcritical phases (b a), we show BBM on a line starting with a single particle at the ori-
≤
that the joint PDF reaches a limiting stationary form gin at time t = 0. The process evolves via the following
at late times, which we compute analytically. More- continuous time dynamics. In a small time interval ∆t,
over, for b a, our exact results for the stationary joint each existing particle (i) dies with probability a∆t, (ii)
≤
PDF demonstrate that this correlation between Xmax branchesintotwooffspringwithprobabilityb∆tand(iii)
and Xmin remains finite even in the stationary state. diffuses, with diffusion constant D, with the remaining
ThejointPDFofX andX hasthefollowingin- probability 1 (a+b)∆t. A schematic trajectory of the
max min
−
teresting physical application. For instance, in the con- processis showninFig. 2,whereX andX denote
max min
text of epidemic spreads, it is important to characterize respectivelythemaximaldisplacementsoftheprocessup
the spatial extent over which the epidemic has propa- to time t in the positive and the negative direction. It is
gatedup to time t. This is clearly measuredby the span convenienttodefinetheseobservablesintheirdimension-
s=X X of the process upto time t (see Fig. 1) lessformsx =X / D/bandx =X / D/b.
max min max max min min
−
[21–23]. Evidently, to compute the distribution of s, we Sincetheparticlestartsatthe origin,x 0necessar-
max
p ≥ p
3
ily and similarly x 0 necessarily. In the subsequent PDF p(x,y) via the relation
min
≤
discussions we find it convenientto consider the positive
quantitiesx and x asourbasicrandomvariables. ∞ ∞
max − min p(ζ)= dx dyp(x,y)δ(x+y ζ). (7)
−
As mentioned in the introduction, the marginal PDF Z0 Z0
of xmax (and consequently that of xmin), has been ex- We compute p(ζ) exactly for all ∆ 0 and find the
tensively studied for all a and b [4−, 6, 7]. While in the following asymptotic behaviors. ≥
supercritical phase (b > a), this marginal PDF remains Critical case (∆=0): In this case we find
time dependent for all t [6, 7], for b a, it approaches a
≤
stationary form p (x) which is known explicitly. It is ζ
marg
, ζ 0
convenientto expressitin terms of its cumulative distri- 2 →
tbhueticornitRica(xl)ca=seRx∆∞=pm0ar[g4(]x,′)dx′. We set ∆=a/b−1. In p(ζ)∼A , ζ , =8π√3=43.53118... (8)
ζ3 →∞ A
1
R(x)= x 2 . (1)
1+ Subcritical case (∆>0): Here we get
√6
(cid:18) (cid:19)
1
Consequently, p (x) = d (x)/dx has the asymp- (1+∆)ζ , ζ 0
marg
− R 2 →
totic behaviors p(ζ) (9)
2 ∼A2 ζ exp( √∆ζ), ζ .
p (0)= , x 0
marg 3 → 2 − →∞
pmarg(x)∼12 r (2) where A=12∆ 3∆/2+ 1+3∆/2 −2.
, x .
In the subcritical casxe3∆>→0 [4∞]: xSmiginn.atInutreersestoinfgthlyhp,eonceorcraenlpashtioownthbaettwtiheeesne axsmymaxptaontidc
behaviors of p(ζ) for the critical (8) and the subcritical
cases (9) carry the signatures of the correlation between
3∆ √∆ 3∆
(x)= csch2 x+sinh−1 . (3) xmax andxmin (seealsoFigs. 8and9below). Inorderto
R 2 2 r 2 ! demonstrate this, we compute the asymptotic behaviors
of p(ζ) in the hypothetical case where one assumes that
This result can further be simplified to give x and x are completely uncorrelated. Given that
max min
x and x have the same PDF p (x) [obtained
α 1 max − min marg
(x)= − (4) from Eq. (1) for ∆ = 0 and from Eq. (4) for ∆ > 0],
R [ 1+αcosh(√∆x)+ (α2 1)sinh(√∆x)] the span PDF p (ζ), assuming that x and x
− − uncorr max min
are uncorrelated can be obtained by inserting p(x,y) =
where α=1+3∆. p
p (x,y) into Eq. (7) and is given by
uncorr
Consequently, p (x) = d (x)/dx has the asymp-
marg − R ζ
totic behaviors
p (ζ)= p (x)p (ζ x)dx. (10)
uncorr marg marg
−
Z0
2
p (0)= +∆, x 0
marg For small ζ, it behaves as
3 →
p (x) r (5)
marg ∼6∆23e−2sinh−1√32∆ exp( √∆x), x . puncorr(ζ)∼p2marg(0)ζ ,whenζ →0. (11)
− →∞
Substituting pmarg(0) = 2/3+∆ from Eq. (5) in Eq.
By symmetry, x has the same marginal PDF (11) gives
− min p
p (x)for∆ 0. Whilep (x)isthuswellknown,in
marg marg
≥ 2
this paper we compute the joint stationary PDF p(x,y)
p (ζ) +∆ ζ ,whenζ 0. (12)
uncorr
of x=x and y = x for ∆ 0. One of our main ∼ 3 →
max − min ≥ (cid:18) (cid:19)
results is to highlight the nonzero correlation between
ComparingthisresultwiththeexactoneinEqs. (8)and
x and x even in the stationary state. Indeed we
max min
− (9), we see that, while both of them grow linearly for
show that
small ζ, the slopes are different, reflecting the fact that
p(x,y)=puncorr(x,y)=pmarg(x)pmarg(y). (6) xmax and xmin are actually correlated.
6 To investigate the large ζ behavior of p (ζ) in Eq.
uncorr
The stationary PDF of the dimensionless span ζ = (10), we need to treat separately the critical (∆ = 0)
s/ D/b=x x can be computed from the joint and the subcritical (∆>0) cases – see Eqs. (2) and (5).
max min
−
p
4
In the critical case (∆=0), substituting the asymptotic A. Backward Fokker-Planck equation for Q(X,Y,t)
behaviorp (x) 12/x3 fromEq. (2)inEq. (10), one
marg
∼
gets for large ζ
We derive a backward Fokker-Planck (BFP) equa-
tion for Q(X,Y,t), following similar steps as in
24
p (ζ) , for∆=0. (13) Refs. [18, 19]. We investigate how Q(X,Y,t) evolves
uncorr ∼ ζ3
into Q(X,Y,t+∆t). The goal is to derive a differential
equationfortheevolutionofQ(X,Y,t). Forthispurpose
While this uncorrelatedassumptioncorrectlyreproduces
the ζ 3 decay (8), the prefactor 24 is different from the wesplitthetimeinterval[0,t+∆t]intotwosubintervals:
−
[0,∆t] and [∆t,t+∆t]. We then take into account all
exact value = 8π√3 = 43.53118... in Eq. (8), again
A possible stochastic events that take place in the first
reflectingthenonzerocorrelationbetweenx andx .
max min
subinterval [0,∆t]. In [0,∆t], the particle at x=0 can
Onthe otherhand, for∆>0,oneobtains fromEqs. (4)
and (10):
A)splitintotwoparticleswithprobabilityb∆t,result-
∆ ing in two BBM processes that are both confined within
puncorr(ζ)∼ 4A2ζexp −√∆ ζ ,for∆>0. (14) [ Y,X] up to time t+∆t with probability Q2(X,Y,t).
−
(cid:16) (cid:17) The contribution from this term to Q(X,Y,t + ∆t) is
Here also, the assumption of vanishing correlation cor- then b∆t Q2(X,Y,t).
rectly reproduces the ζ-dependence ζexp √∆ ζ
∝ −
B) die with a probability a∆t, leading to no particles
of the right tail (9) but the amplitude is inco(cid:16)rrect by(cid:17)a
at subsequent times. This event automatically ensures
factor ∆/2, reflecting once again the presence of finite
with probability 1 that the process remains confined
correlations between x and x .
max min
within[ Y,X]upto t+∆t. Hence itcontributes aterm
−
a∆t 1 to Q(X,Y,t+∆t).
×
C) diffuse with probability 1 (b + a)∆t, moving
−
a distance ∆x = η(0)∆t in the first time step. This
shifts the process by a distance ∆x at the first time
III. JOINT DISTRIBUTION OF THE
step. The probability that the resulting process is
MAXIMUM AND MINIMUM
confined within [ Y,X] up to time t+∆t is then given
−
by Q(X ∆x,Y +∆x,t) . By the subscript η(0)
η(0)
h − i
We are interested in the spatial extent of the BBM we denote an averaging over all possible values of the
process up to time t. The process begins with a single diffusive jump at the first time step. Hence this term
particle at x = 0 at time t = 0. We recall that the span contributes(1 (b+a)∆t) Q(X ∆x,Y +∆x,t)
η(0)
− ×h − i
of the process up to t, characterizing the spatial extent, to the final probability Q(X,Y,t+∆t).
isdefinedass=X X ,whereX andX are
max min max min
−
respectively the maximum and minimum displacements Adding the contributions from these three terms A),
of the process up to time t (see Fig. 1). B) and C), we have
We start by defining the joint cumulative probability Q(X,Y,t+∆t) = b∆t Q2(X,Y,t)+a∆t
+(1 (b+a)∆t) Q(X ∆x,Y +∆x,t) ,(17)
Q(X,Y,t) Pr X <X, X > Y; up to time t . η(0)
max min − h − i
≡ { − }
where η(t) is a Gaussian white noise process with the
Thishasthesimpleinterpretationastheprobabilitythat
properties
theprocessisconfinedwithinthebox[ Y,X]up totime
−
t (see Fig. 2). The marginal cumulative distribution of
the maximum can be obtained by taking the Y hη(t)i=0, hη(t)η(t′)i=2Dδ(t−t′). (18)
→ ∞
limit of Q(X,Y,t). Similarly the marginal cumulative
Taylor expanding Eq. (17), using the properties of the
distribution of the minimum is obtained by taking the
noiseinEq. (18), andtakingthe limit∆t 0,wearrive
X limit. The joint PDF P(X,Y,t) of X = X →
→ ∞ max at the exact BFP evolution equation
and X =Y is then given by
min
−
2
∂ ∂ ∂
∂ ∂ Q(X,Y,t)=D Q(X,Y,t)+a
P(X,Y,t)= Q(X,Y,t) (15) ∂t ∂X − ∂Y
∂X ∂Y (cid:18) (cid:19)
(b+a) Q(X,Y,t)+b Q2(X,Y,t). (19)
where, by definition, X 0 and Y 0. The PDF of the −
≥ ≥
span s=X +Y is then given by
Since attime t=0,boththe maximumandminimumof
the process is at x=0, the initial condition is
∞ ∞
P(s,t)= dXdYδ(X +Y s)P(X,Y,t). (16)
Z0 Z0 − Q(X,Y,0)=Θ(X)Θ(Y), (20)
5
where Θ is the Heaviside step function defined as Similarly we have R(X,Y,t) R(x,y,τ). The distribu-
→
tions of the scaled variables are related to the unscaled
1 for x>0, distributions as
Θ(x)= (21)
(0 for x<0.
1 s
P(ζ,τ)= P ζ = ,τ =bt ,
At any time t > 0, the maximum Xmax 0 and the D/b D/b !
≥
minimum X 0,leadingtotheboundaryconditions
− min ≥ p 1 pX Y
P(x,y,τ)= P x= ,y = ,τ =bt .
0 for X <0, D/b D/b D/b !
Q(X,Y,t)= (22)
(0 for Y <0. p p (29)
It is actually convenient to work with In terms of these scaled variables Eq. (24) takes the
simpler form
R(X,Y,t)=1 Q(X,Y,t), (23)
− 2
∂R(x,y,τ) ∂ ∂
= R(x,y,τ)
which denotes the complementary probability that the ∂τ ∂x − ∂y
maximumorminimumuptotimetisnotwithin[ Y,X]. (cid:18) (cid:19)
− ∆R(x,y,τ) R2(x,y,τ). (30)
Inserting Eq. (23) into Eq. (19) we have − −
Eq. (30) is a non-linear equation whose explicit solution
2
∂R(X,Y,t) ∂ ∂
=D R(X,Y,t) at finite time t is hard to obtain analytically. In the
∂t ∂X − ∂Y supercritical case b > a, we expect this solution to be
(cid:18) (cid:19)
+(b a) R(X,Y,t) b R2(X,Y,t), (24) time dependent at all times t. However, for b a, we
− − ≤
showbelowthatasτ ,Eq. (30)admitsastationary
→∞
with the initial conditions solution for R(x,y,τ) that can be computed explicitly.
Using this solution, and Eqs. (15), (23) and (27), the
R(X,Y,0)=Θ( X)Θ( Y), (25) joint PDF of x and y can then be expressed as
− −
and the boundary conditions ∂ ∂
P(x,y,τ)= R(x,y,τ). (31)
−∂x∂y
1 for X <0,
R(X,Y,t)= (26)
Finally, this joint PDF can be used to evaluate the PDF
(1 for Y <0.
of the dimensionless span of this process defined in Eq.
(16), which is then given by
B. Dimensionless Variables ∞ ∞
P(ζ,τ)= dxdyδ(x+y ζ)P(x,y,τ). (32)
−
Z0 Z0
It is natural to consider the evolution equations in
Atlargetimes,this PDFconvergestothe stationarydis-
terms of dimensionless variables as follows
tributionp(ζ)=P(ζ,τ ),whichweanalyseindetail
→∞
X in section V.
x= ,
D/b
Y
y = p ,
D/b IV. STATIONARY JOINT DISTRIBUTION OF
x AND x FOR b a
τ =bt, max min ≤
p
a
∆= 1. (27)
We now focus on the case b a where the joint distri-
b −
≤
bution R(x,y,τ) in Eq. (30) is expected to approach a
Similarly, we can define the dimensionless span of the stationary limit as τ :
process as →∞
(x,y)=R(x,y,τ ). (33)
s X X R →∞
max min
ζ =x+y = = − . (28)
D/b D/b Setting the left hand side (lhs) of Eq. (30) to 0 in the
stationary limit gives
Our goal in this paperpis to derivepthe stationary joint
PDFofx andy andalsothe stationaryPDFofζ. In or- ∂ ∂ 2
(x,y)=∆ (x,y)+ 2(x,y), (34)
dertoavoidaproliferationofsymbols,wekeepthe same ∂x − ∂y R R R
(cid:18) (cid:19)
notation for the PDF’s of the unscaled and scaled vari-
ables,withP(X,Y,t) P(x,y,τ)andP(s,t) P(ζ,τ). for x 0 ,y 0. Next, it is convenient to make a
→ → ≥ ≥
6
1
ζ = 2
y ζ
0.8 ζ = 3
v 0.6 ζ = 5
(ζ,v)
R
0.4
ζ = 10
0.2
x
0
-10 -5 0 5 10
v
FIG. 3: The change of variables x,y ζ,v .
{ }→{ }
FIG. 4: (ζ,v) as a function of v [ ζ,+ζ] for different
R ∈ −
change of variables values of ζ. For fixed ζ, (ζ,v) is a smooth non-monotonic
R
function, symmetric around v = 0 in ζ v +ζ, and has
− ≤ ≤
ζ =x+y, a minimum at v=0. The data plotted abovecorresponds to
the case ∆ = 0 and was obtained by numerically evaluating
v =x y, (35)
− Eqs. (51) and (52).
with ζ [0, ) and v [ ζ,ζ] (see Fig. 3). Note that
∈ ∞ ∈ −
the variable ζ represents the dimensionless span of the
process. In terms of these new variables Eq. (34) be-
where the factor 1/2 in (38) comes from the Jacobian of
comes
the transformation x,y ζ,v (35) such that
{ }→{ }
∂ 2
4 (ζ,v) ∆ (ζ,v) 2(ζ,v)=0, (36) ∞ +ζ
∂v R − R −R dζ dvp(ζ,v)=1. (39)
(cid:18) (cid:19)
Z0 Z−ζ
valid in the regime v [ ζ,+ζ] and ζ [0,+ ) (see
∈ − ∈ ∞
Fig. 3). When v +ζ, i.e., y 0, the boundary con- Fortunately, Eq. (36) can be integrated with respect
→ →
dition R(x,y = 0,t) = 1 given in Eq. (26), translates to v upon multiplying by a factor 2∂R(ζ,v), yielding
∂v
into (ζ,ζ) = 1. Similarly, when v ζ, i.e., x 0,
R → − →
the boundaryconditionR(x=0,y,t)=1translatesinto ∂ (ζ,v) 2 ∆ 1
(ζ, ζ)=1. Inaddition,thesolutionmustbesymmet- R = 2(ζ,v)+ 3(ζ,v)+κ(ζ), (40)
R − ∂v 4 R 6R
ricaroundv =0(correspondingto x=y). Since (ζ,v) (cid:18) (cid:19)
R
is a cumulative probability, 0 (ζ,v) 1. Conse-
whereκ(ζ) isayetunknownintegrationconstant. Tofix
≤ R ≤
quently, for a fixed ζ, as v decreases from ζ we expect
κ(ζ), we use the condition in Eq. (37) and arrive at
that (ζ,v) should decrease from its value (ζ,ζ) = 1.
R R
By symmetry, as v increases from ζ, (ζ,v) should 2
∂ (ζ,v) ∆
decrease from its value (ζ, ζ) =−1. RThus we ex- R = 2(ζ,v) 2(ζ,0)
R − ∂v 4 R −R
pect (ζ,v), as a function of v for fixed ζ, is a smooth (cid:18) (cid:19)
R 1(cid:0) (cid:1)
non-monotonic function, symmetric around v = 0 in + 3(ζ,v) 3(ζ,0) . (41)
ζ v +ζ, and with a minimum at v = 0 (see Fig. 6 R −R
− ≤ ≤
4). Assuming analyticity around the minimum at v = 0 (cid:0) (cid:1)
gives the condition Since the solution (ζ,v), is symmetric about the
R
v = 0 line, it is sufficient to solve Eq. (41) for only the
∂ (ζ,v) region v [0,+ζ] (or alternatively for v [ ζ,0]). We
R =0. (37) ∈ ∈ −
∂v v=0 restrict on v ∈ [0,+ζ] where ∂R(ζ,v)/∂v > 0 (see Fig.
(cid:12) 4). Taking the square root of Eq. (41) and integrating,
(cid:12)
Once we find the solution(cid:12) (ζ,v) of Eq. (36), using we obtain for v [0,+ζ]:
Eqs. (31) and (35) the jointRPDF of ζ and v can be ∈
expressed as R(ζ,v) dr
=v.
p(ζ,v)= 1 ∂ ∂ (x,y) 1 ∂2 ∂2 (ζ,v). ZR(ζ,0) ∆4 (r2−R2(ζ,0))+ 16(r3−R3(ζ,0))
−2∂x∂yR ≡ 2 ∂v2 − ∂ζ2 R q (42)
(cid:18) (cid:19)
(38) This equation can be conveniently expressed as
7
for all ζ and v.
10
∗
C CriticalPoint. Thecomputationsbecomeslightlymore
1
explicit exactly at the critical point a = b, i.e., ∆ = 0.
Inthis case,putting γ =(3∆/2)/ (ζ,0)=0 inEq. (47)
R
10-2 gives
(γ,z)
G 1 (ζ,v) v
10-3 γ=0 (ζ,0)G 0,R(ζ,0) = √6 , (48)
γ=1 R (cid:18) R (cid:19)
10-4 γ=10 where, frompEq. (44), (0,z) is given by
γ=10 2 G
γ=10 3 z dx
10-5 (0,z) =
1 10 102 z 103 104 G Z1 √x3−1
2√πΓ 7 2 1 1 7 1
= 6 F , ; ; ,(49)
Γ 2 − √z2 1 6 2 6 z3
3(cid:0) (cid:1) (cid:18) (cid:19)
FIG. 5: The function (γ,z) for different values of γ. For
large z, (γ,z) saturatGes to a γ dependent constant (γ). where 2F1 is the us(cid:0)ua(cid:1)l Gauss hypergeometric function
G C
The case γ =0 corresponds to the function (0,z) analyzed and z 1. This function has the following asymptotic
G ≥
inthecriticalcase. Thelimitingbehaviorsare (0,z) 0as behaviors
z→1 and G(0,z)→C∗ = √3πΓΓ((2163)) ≈2.4286 asGz →∞→. 2 √z 1 for z 1,
√3 − →
(0,z) (50)
G ∼√πΓ(16) 2 for z .
3 Γ(2) − √z →∞
1(ζ,0)G 3(∆ζ/,02),R((ζζ,,v0)) = √v6, (43) Similarly, putting ∆ =3 0 in Eq. (47) determines (ζ,0)
R (cid:18)R R (cid:19) R
implicitly as the solution of
where thepbivariate function is defined by the integral
G
1 1 ζ
z dx 0, = . (51)
(γ,z)= . (44) (ζ,0)G (ζ,0) √6
G (x3 1)+γ(x2 1) R (cid:18) R (cid:19)
Z1 − − p
Dividing Eq. (48) by (51) gives
p
Theabovefunction (γ,z)canthenbe expressedas(us-
G
ing the identity 3.138 of Ref. [30]): (ζ,v) v 1
0,R = 0, , (52)
G (ζ,0) ζG (ζ,0)
(γ,z)= (cid:18) R (cid:19) (cid:18) R (cid:19)
G
where (0,z) is given explicitly in Eq. (49). The so-
1 F 2 tan−1 z−1 ,1 2 3+γ , lution oGf Eq. (52) thus determines the cumulative joint
(3+2γ)1/4 " s√3+2γ 2r − 3+2γ# distribution (ζ,v) in the critical regime.
R
(45)
where z 1, γ 0 and V. STATIONARY DISTRIBUTION OF THE
≥ ≥ SPAN
φ dθ
F(φ,k)= (46)
Z0 1 k2sin2θ The solution R(ζ,v) obtained from Eq. (43) can next
− beusedtocomputethestationaryjointPDFp(ζ,v)from
is the elliptic integralofthepfirstkind. In Fig. 5, weplot Eq. (38). The stationaryPDF ofthe dimensionless span
the function (γ,z)asafunctionofz fordifferentvalues ζ, denoted by p(ζ), is then obtained, for all ∆ 0, by
G ≥
ofγ. Next, insertingthe boundarycondition (ζ, ζ)= integrating over v as:
R ±
1 in Eq. (43) we have
+ζ
p(ζ)= p(ζ,v)dv . (53)
1 3∆/2 1 ζ
(ζ,0)G (ζ,0), (ζ,0) = √6. (47) Z−ζ
R (cid:18)R R (cid:19) It is then easy to extract p(ζ), for all ∆ 0, numeri-
p ≥
This is an implicit equation for (ζ,0), the solution of cally exactly from Eqs. (43), (47), (38) and (53). As an
R
whichcanthenbeinjectedinEq. (43)tosolvefor (ζ,v) example,weplotthisPDFp(ζ)asafunctionofζ forthe
R
8
1 1
1
10-1 0)10-1
,
10-2 1 ζ 2 10-2 ζ(R1100--32 ∼18(1+∆)ζ2
R(ζ,0)10-3 ζ,(0)1100--21 ∼B − R(ζ,0) 1−1100--54
10-4 1−R1100--43 ∼ 18 ζ2 10-4 10-2 10-1 1 ζ 10 102
10-5 10-5 (cid:0) (cid:1) ∆=0.1
10-2 10-1 1 ζ 10 102 ∆=1
Aexp √∆ζ
10-610-3 10-2 10-1 1 ζ 10 102 103 104 10-6 0 20 ∼ 40 (cid:16)− 260 (cid:17) 80 ∆ 10=010 120 140
ζ
FIG. 6: The function (ζ,0) versus ζ in the critical regime
R FIG. 7: Asymptotic behavior of (ζ,0) in the subcritical
derived using Eq. (51), showing the limiting behaviors R
regime. The plot shows (ζ,0) for different values of ∆
R(ζ,0) → 1 as ζ → 0 and R(ζ,0) → ζB2 as ζ → ∞ (dashed derived using Eq. (47). TRhe dashed lines representing the
(li6n1e)).aIsnsperetd:iPctleodtoinf1Eq. ((6ζ0,)0.)Bsh≈ow3in5g.3t9h0e1liismditeifinngebdehinavEioqr. asymptotic exponential behavior R(ζ,0)∼Aexp(cid:16)−√2∆ζ(cid:17) as
1 (ζ,0) 1 ζ2 a−sRζ 0 (dashed line) as predicted in ζ derived in Eq. (71) are indistinguishable from the
Eq−.R(56). ∼ (cid:0)8(cid:1) → the→ore∞tically obtained curves as they match exactly. Inset:
Plotof1 (ζ,0)showingthelimitingbehavior1 (ζ,0)
1(1+∆−)ζR2 asζ 0(dashedlines)aspredictedi−nREq. (56∼).
8 →
criticalcase(∆=0)inFig. 8andforthesubcriticalcase
(∆ = 1) in Fig. 9. From the numerical plots for differ-
in Eq. (43) with both ζ and v small gives
entvalues of ∆, one finds that forsmall ζ, p(ζ) increases
linearly with a slope that depends on ∆. In contrast,for 1
large ζ, p(ζ) has an algebraic tail p(ζ) ζ−3 for ∆ = 0 R(ζ,v)=1− 8(1+∆) ζ2−v2 +O(ζ4,v4). (57)
∝
while it has an exponential tail for ∆ > 0. In the next
(cid:0) (cid:1)
two subsections, we show that these asymptotic behav- Next, using Eq. (38), the joint PDF p(ζ,v) is given by
iors of p(ζ), both for small and large ζ, can actually be
1 ∂2 ∂2
extracted analytically for all ∆ 0.
p(ζ,v) = (ζ,v)
≥ 2 ∂v2 − ∂ζ2 R
(cid:18) (cid:19)
1
= (1+∆)+ (ζ2,v2). (58)
4 O
A. Asymptotic behavior of p(ζ) for ζ 0
Substituting this expression in Eq. (53) gives
→
1
As ζ 0 we have (ζ,0) 1. Therefore we write p(ζ)= (1+∆)ζ+ ζ3 , (59)
→ R → 2 O
(ζ,0)=1 ǫ(ζ), (54) (cid:0) (cid:1)
R − which yields the small ζ behavior announced in Eq. (8),
for ∆ = 0, and in Eq. (9) for ∆ > 0. The asymptotic
where ǫ(ζ) is small. Substituting this in Eq. (47) and
linear growth for small ζ is shown, for the critical case
expanding (γ,z) in Eq. (44) around z = 1, we obtain
G (∆ = 0), in the inset of Fig. 8 and for the subcritical
(using the notation ǫ ǫ(ζ))
≡ case (for ∆ = 0.1,1,10) in the inset of Fig. 9 a.. As
2√ǫ ζ discussed in section II, the amplitude of this linear term
+ (ǫ3/2)= . (55) in (59) carries the signature of the correlation between
3(1+∆) O √6
x and x .
max min
Therefore to lepading order in ζ we have
1
(ζ,0)=1 (1+∆)ζ2+ (ζ4). (56) B. Asymptotic behavior of p(ζ) for ζ
R − 8 O →∞
This limiting behavior for the critical case (∆ = 0) is In this subsection we extract analytically the large ζ
illustrated in the inset of Fig. 6, and for the subcritical tails of p(ζ) both for the critical (∆ = 0) as well as for
case in the inset of Fig. 7. Performingthe same analysis the subcritical case (∆>0).
9
1. Critical point (∆=0) 0.3
1
0.25
We start by analyzing Eq. (51) in the limit ζ .
In(0t,hzi)s floimriltarRge(ζz,.0)Inisserstminagll.thWe eastyhmenptnoetiecdbteohaavn→iaolry∞zoef 0.2 p(ζ)10-2 ∼(cid:0)12(cid:1)ζ ζ 3
G(0,z) in Eq. (50) into Eq (51), we obtain p(ζ) 10-4 ∼A −
G
0.15 10-6
1
(ζ,0)= B + , (60)
R ζ2 O ζ4 0.1 10-8
(cid:18) (cid:19) 10-2 1 ζ 102 104
where
0.05 puncorr(ζ)
√πΓ(1)
=6 2 35.3901, with = 6 . (61)
B C∗ ≈ C∗ 3 Γ(2) 0
3 0 5 10 15 20
ζ
This asymptotic behavior of (ζ,0) is illustrated in
R
Fig. 6. Having thus determined (ζ,0) for large ζ, we
R
nowinvestigate (ζ,v)forlargeζ. Ouraimistoextract FIG.8: TheoreticalstationaryPDFofthedimensionlessspan
R
p(ζ) for large ζ from Eqs. (38) and (53). We note that p(ζ) (solid line) in the critical regime (∆ = 0) calculated by
Eq. (53) involves an integral over v and this integral numerically evaluating Eqs. (51), (52), (38) and (53). The
is dominated by v ζ. Hence we need to investigate (blue)dottedlinecorrespondstoaplotofpuncorr(ζ)evaluated
(ζ,v) in the scalin∼g limit ζ , v but keeping from Eqs. (1) and (10) which is the result for the span dis-
R → ∞ → ∞ tribution obtained undertheassumption that x andx
ζ/v fixed. max min
areuncorrelated. TheInsetdisplaysthesamedatainlog-log
Whenζ , (ζ,0) 0asinEq. (60). Asaresult, scaleshowingthetwolimitingbehaviors(dashedlines),linear
the argume→nt∞1/R(ζ,0) →of (0,1/ (ζ,0)) on the right p(ζ) 1 ζ forsmallspans,andp(ζ) /ζ3 forlargespans
hand side of Eq. R(52) goes tGo . FRrom Eq. (50), we see (8), w∼it(cid:0)h2(cid:1) =8π√3 43.53118.... ∼A
∞ A ≃
that (0,z ) = where is given in Eq. (61).
∗ ∗
G → ∞ C C
Hence, in the scaling limit ζ , v keeping ζ/v
→ ∞ → ∞
fixed, Eq. (52) becomes
gives finally
(ζ,v) v
G 0,R(ζ,0) =C∗ζ . (62) p(ζ)= 1 B C∗dz 6 (z)+6z ′(z)+
(cid:18) R (cid:19) −ζ3 (cid:18)C∗(cid:19)Z0 h F F
z2 2 (z) . (66)
Inverting the above Eq. (62), we get −C∗ F′′
(cid:16) (cid:17) i
(ζ,v) v Hence, we obtain
R = ∗ , (63)
(ζ,0) F C ζ
R (cid:18) (cid:19)
p(ζ) A for large ζ, (67)
where (z) is defined as the inverse function of (0,z). ∼ ζ3
SubstitFuting (ζ,0) /ζ2 fromEq.(60)givesthGefinal
R ∼B with
scaling limit expression of the joint cumulative distribu-
tion
= 6 C∗dz 6 (z)+6z (z)+(z2 2) (z) .
∗ ′ ∗ ′′
A − C F F −C F
Z0 h (68i)
v It turns out that the integral in Eq. (68) can be per-
(ζ,v)= B ∗ . (64)
R ζ2F C ζ formed explicitly (see Appendix A) and the final answer
(cid:18) (cid:19)
for the amplitude is amazingly simple
Inserting this expression into Eq. (38) we arrive at the
joint PDF =8π√3=43.53118... . (69)
A
6 v v v Thus the leading asymptotic behavior for large ζ is
p(ζ,v)= B ∗ +6 ∗ ′ ∗
−2 ζ4F C ζ C ζ5F C ζ
h + ∗(cid:18)2 v2(cid:19) 1 ′′ (cid:18)∗v (cid:19). (65) p(ζ)∼ 8πζ√33 , (70)
C ζ6 − ζ4 F C ζ
(cid:18) (cid:19) (cid:18) (cid:19)i
as announced in Eq. (8). This large ζ behavior of the
Substituting this expression for p(ζ,v) in Eq. (53) span PDF is shown in Fig. 8 where we plot the numer-
10
ically exact distribution p(ζ) (extracted from Eqs. (51), 0.5
(52), (38) and (53)), along with the asymptotic power
puncorr(ζ)
law tail derived analytically in Eq. (70). At large ζ,
0.4 1
the exact distribution shows a good agreement with the
p(ζ)
asymptotic behaviour. We also display p (ζ), the
uncorr
scpoarrnelPaDteFd,owbhtaicinheddeacsasyusmwinitghtthhaetsxammaxeapnodwxerm,inbuatrewuitnh- p(ζ) 0.3 10-1 ∼12(1+∆)ζ
a different prefactor provided in Eq. (13). The ampli- 0.2
tude A = 8π√3 is thus nontrivial due to the remnant 10-210-2 10-1 ζ 1 10
non-vanishing correlation between x and x in the
max min 0.1
stationary state (see also the discussion at the end of
section II).
0
a. 0 2 4 ζ 6 8 10
2. Subcritical case (∆>0) 1
10-16 A2ζe √∆ζ
In the subcritical regime, the large ζ analysis can be ∼ 2 −
10-10
done along the same lines as before for the critical case 10-18
p(ζ)
but the analysis is a bit more complicated and most of
10-20
theAdseitnaitlhsehcarvieticbaelencarseel,eghaetreedaltsooAwpepfiernsdtixneBed. to com- 10-20 p(ζ) ∼∆4A2ζe−√∆ζ
pute the large ζ behavior of (ζ,0) from Eq. (47) with 40 45 50 55
ζ
R
∆ > 0. This is done in Appendix B. We find (see Eq. 10-30
(B12))
√∆ 10-40
(ζ,0) Aexp ζ , as ζ b. 1 40 80 ζ 120 160
R ∼ − 2 ! →∞
12∆
where A= . (71)
2 FIG.9: TheoreticalstationaryPDFofthedimensionlessspan
3∆/2+ 1+3∆/2 p(ζ) in the subcritical regime. In a. we display p(ζ) (solid
hp p i line), for ∆ = 1, calculated by numerically evaluating Eqs.
In Fig. 7 we show a plot of (ζ,0) obtained by numer- (43), (47), (38) and (53). The (blue) dotted line corresponds
ically evaluating Eq. (47) foRr different values of ∆. For to puncorr(ζ) evaluated from Eqs. (4) and (10) which is the
result for the span distribution obtained under the assump-
large ζ, this shows a good agreement with the asymp-
tionthatx andx areuncorrelated. TheInsetdisplays
totic behavior in Eq. (71). Using this asymptotic tail max min
the same data in log-log scale showing the approach to the
of (ζ,0) from Eq. (71), we then analyse the asymp-
R ζ 0 linear behavior (dashed line) derived in Eq. (59). In
totic behavior of (ζ,v) in Eq. (43), in the scaling limit →
R b. wedisplaythestationaryspanPDFp(ζ)andpuncorr(ζ)for
ζ , v , keeping ζ/v fixed. We find
∆=1inlog-linearscale. Thedashedlineshowingtheζ
→∞ →∞ asymptotic behavior A2ζexp( √∆ ζ) derived in Eq.→(7∞3)
(ζ,v)=H (ζ,v)+ 2(ζ,0)H (ζ,v)+ 3(ζ,0) . ∼ 2 −
R 0 R 1 O R is indistinguishable from the theoretically obtained curve as
(72) they match exactly. The Inset shows the same data high-
(cid:0) (cid:1)
where both H0(ζ,v) and H1(ζ,v) are functions of u = lighting thedifferencebetween theasymptoticdecaysof p(ζ)
ζ v only, whose expressions are provided in Eq. (B16) andp (ζ)(theasymptoticdecayofp (ζ)isprovided
− uncorr uncorr
and Eq. (B17) respectively. From (ζ,v), we can then in Eq. (14)).
R
obtain the joint PDF p(ζ,v) from Eq. (38) and eventu-
ally p(ζ) from Eq. (53). Following this procedure, we
finally obtain the large ζ behavior of p(ζ) for ∆>0 (see
Appendix B for details):
A2
p(ζ)= ζe √∆ ζ 1+ (ζ 1) , (73)
− −
2 O
(cid:0) (cid:1)
with A givenin Eq. (71), as announcedin Eq. (9). Here son,we also show a plotof p (ζ) obtained fromEqs.
uncorr
again, as discussed in section II, the amplitude A2/2 in (4) and (10) which corresponds to the PDF of the span
Eq. (73)bearsthe signaturesofthecorrelationsbetween obtained by assuming that x and x are indepen-
max min
x and x . In Fig. 9 we show a plot of p(ζ), for dent. We also display the agreement of the asymptotic
max min
the subcriticalcase(for∆=1),obtainedbynumerically behaviorsforζ 0andζ derivedinEqs. (59)and
→ →∞
evaluating Eqs. (43), (47), (38) and (53). For compari- (73) with these numerically exact PDFs.
