Roughening and inclination of competition interfaces

0.14 MB·English

by  Pablo A. Ferrari

Roughening and inclination of competition interfaces
Pablo A. Ferrari
Universidade de S˜ao Paulo
6 James B. Martin
0 University of Oxford
0
2 Leandro P. R. Pimentel
n E´cole Polytechinique F´ed´erale de Lausanne
a
(Dated: February 1, 2008)
J

0 Westudy thecompetition interface between twoclusters growing overa random vacantsector of
1 the plane in a simple set-up which allows us to perform formal computations and obtain analitical
solutions. We demonstrate that a phase transition occurs for the asymptotic inclination of this

] interface when the final macroscopic shape goes from curved to non curved. In the first case it
R
is random while in the second one it is deterministic. We also show that the flat case (stationary
P growth) is a critical point for the fluctuations: for curved and flat final profiles the fluctuations
. are in the KPZ scale (2/3); for non curve final profile the fluctuations are in the same scale of the
h
fluctuationsof theinitial conditions, which in ourmodel are Gaussian (1/2).
t
a
m PACSnumbers: 64.60.Ak64.60.Ht
[

2 Introduction The behaviorofthe interfaceof agrow- γ−1(0) = (0,1) as follows. Starting from (0,1), walk
v ing materialhas beeninvestigatedusing the Edenmodel one unit up with probability λ and one unit left with
8 [4], ballistic deposition and other random systems. Typ- probability 1−λ, repeatedly, to obtain γ1 =(γ (j)) .
0 0 j<0
9
ically,the growingregionconvergestoanasymptoticde- Then, starting from (1,0) walk down with probability ρ
1
terministic shape and its fluctuations depend on the ge- and right with probability 1−ρ to get γ2 = (γ (j)) .
2 0 0 j>0
ometry of the initial condition [11, 13]. A less well stud- γ1 has asymptotic orientation (λ − 1,λ) while γ2 has
1 0 0
4 ied phenomenon is the competing growth of two mate- asymptotic orientation (1 − ρ,−ρ). Let C0 be the
0 rials. The interface between two growing clusters (com- sector with boundary γ , containing the first quadrant;
0
/ petition interface) presents a random direction on the its asymptotic angle θ = θ ∈ [90o,270o) is the
h λ,ρ
same scale as the deterministic shape [3, 12, 17]. In this angle between (λ − 1,λ) and (1 − ρ,−ρ). Notice that
t
a letter we describe quite explicitly this phenomenon in a θ ∈[90o,180o) if and only if ρ<λ.
m
simple model. On gronds of universality, this will pro-
The path γ is the growth interface at time 0. The dy-
: vide a guide to understand the interplay between the 0
v namics are then defined as follows. For each z ∈C and
asymptotics of the competition interface and the final 0
Xi macroscopic shape in models with different growth and each t ≥ 0, we have a label σt(z) ∈ {0,1,2}. The la-
bel is 0 if z is unoccupied at time t, and is 1 or 2 if z
r competition mechanisms.
a belongs to cluster 1 or 2 respectively. Once occupied, a
Wedeterminetheinclinationofthecompetitioninterface site remains occupied and keeps the same value forever.
foragrowthmodelcalled“lastpassagepercolation”ina Initially, set σ (z) = 1 for all z ∈ γ1, σ (z) = 2 for all
0 0 0
random sectorof the plane of angle θ. The growthinter- z ∈ γ2 and σ (z) = 0 for all z ∈ C \γ . Independently
0 0 0 0
faces are mapped into particle configurations of the to- each vacant site z ∈ C \γ becomes occupied with rate
0 0
tally asymmetric simple exclusion process in one dimen- 1 provided z − (1,0) and z − (0,1) are occupied. Let
sion(TASEP)[16]. UnderEulerspace-timerescaling,the G(z) be the time at which site z becomes occupied. At
particle density of the TASEP convergesto a solution of this time σ (z) assumes the value σ (z¯) where z¯ is the
t t
the Burgers equation. This equation has travelling wave argumentthatmaximizesG(z−(1,0))andG(z−(0,1)).
solutions (shocks) corresponding to the case θ > 180o, Thuswhenasitebecomesoccupieditjoinstheclusterof
andrarefactionfrontscorrespondingto θ <180o. Aper- whichever of its two neighbours (below and to the left)
turbation at one site of the initial particle configuration became occupied more recently. The label of the site
(called a second class particle) follows a characteristic of (1,1) may be left ambiguous, but we stipulate that site
the equation or the path of a shock. To establish our (1,2) always joins cluster 1, and site (2,1) always joins
results, we map the competition interface linearly onto cluster 2.
the path of the second class particle.
Theprocess(G1,G2),whereGk isthesetofsitesz ∈C
t t t 0
The growth model The random sector is such that σ (z) = k, describes the competing spatial
t
parametrized by the asymptotic slope of its sides. growth model. The growth interface at time t is the
Let λ ∈ (0,1] and ρ ∈ [0,1) and define a random path polygonal path γ composed of sites z ∈ C such that
t 0
γ0 = (γ0(j))j∈Z ⊆ Z2 with γ1(0)= (1,0), γ0(0) = (1,1), G(z) ≤ t and G(z +(1,1)) > t. The competition inter-
1
Cluster 1 ϕ If η0 is distributed according to the Bernoulli product
γ0 γt measure with density λ for j ≤ 0 and ρ for j > 0, then
theasymptoticbehaviorofX(t)showsaphasetransition
in the line λ=ρ: with probability one,
J(t)
Cluster 2
X(t) 1−ρ−λ if λ≤ρ
lim = (3)
t→∞ t (cid:26)U if λ>ρ
I(t)
where U is a random variable uniformly distributed in
[1−2λ,1−2ρ] ([5, 15, 19] for the deterministic case and
[7, 8, 9, 10] for the random case).
The limits (3) are based on the following hydrodynamic
FIG. 1: Growth and competition interfaces.
limits. Ifη isdistributedwiththeproductmeasurewith
0
densitiesλandρasbefore,thenthemacroscopicdensity
evolution is governedby the Burgers equation:
face ϕ = (ϕn)N is defined by ϕ0 = (1,1) and, for n ≥ 0,
ϕ = ϕ + (1,0) if ϕ + (1,1) ∈ G1 and ϕ =
n+1 n n ∞ n+1
ϕ +(0,1) if ϕ +(1,1)∈G2 . Note that ϕ chooses lo- limǫ f(xǫ)η (x)= f(r)u(r,t)dr (4)
n n ∞ t/ǫ
callytheshortersteptogouporright,sothatϕn+1isthe ǫ→0 xX∈Z ZR
argumentthatminimizes{G(ϕ +(1,0)),G(ϕ +(0,1))}.
n n
This competition interface represents the boundary be- withprobabilityoneforallf :R→Rwithcompactsup-
tween those sites which join cluster 1 and those joining port,whereu(r,t)isthesolutionoftheBurgersequation
cluster 2 (see Figure 1). The process ψ(t) = (I(t),J(t))
∂u(r,t) ∂
defined by ψ(t)=ϕn for t∈[G(ϕn),G(ϕn+1)) gives the + (u(r,t)(1−u(r,t)))=0, r ∈R, t≥0
position of the last intersecting point between the com- ∂t ∂r
petition interface ϕ and the growth interface γ .
t with initial condition u(r,0) = λ for r ≤ 0 and ρ for
In [8] we prove that with probability one, r > 0. If λ = ρ the solution is constant, if λ < ρ it is a
shock:
ϕ
lim n =eiα (1)
n→∞|ϕn| u(r,t)= λ if r ≤(1−λ−ρ)t (5)
(cid:26)ρ if r >(1−λ−ρ)t
where α∈[0,90o] is given by
and it is a rarefaction front if λ>ρ:
λρ if ρ≥λ
(1−λ)(1−ρ)
tanα =  (2) λ if r ≤(1−2λ)t
 (cid:0)11+−UU(cid:1)2 if ρ<λ u(r,t)= ρ21 − 2(λr−ρ) iiff r(1>−(21λ−)t2<ρ)rt ≤(1−2ρ)t
andU isa randomvariableuniformly distributedin[1− (6)

2λ,1−2ρ]. ([14, 16] for initial product measures and [18, 19] for ini-
tial measures satisfying (4) with t=0; also [1, 2])
Simple exclusion and second-class particles The
totally asymmetric simple exclusion process (η , t ≥ 0) The characteristics v(a,t), corresponding to the Burg-
t
is a Markov process in the state space {0,1}Z whose el- ers equation and emanating from a, are the solutions of
ements are particle configurations. η (j) = 1 indicates a dv/dt = 1−2u(v,t) with v(0) = a. The solutions are
t
particle at site j at time t, otherwise η (j) = 0 (a hole constant along the characteristics. When two character-
t
is at site j at time t). With rate 1, if there is a particle istics carryinga different solution meet, they give rise to
at site j, it attempts to jump to site j+1; if there is a ashock. Thereisonlyonecharacteristicemanatingfrom
holeatj+1thejumpoccurs,otherwisenothinghappens. locations where the initial data is locally constant and
Thebasiccouplingbetweentwoexclusionprocesseswith there are infinitely many characteristics when there is a
initial configurations η and η′ is the joint realization decreasing discontinuity. In particular, if the initial con-
0 0
(η ,η′) obtainedby using the same potential jump times dition is u(r,0) = λ for r < 0 and u(r,0) = ρ for r ≥ 0,
t t
ateachsiteforthetwodifferentinitialconditions. Letη then the characteristics v (t) emanating from the point
0 r
andη′ beconfigurationsofparticlesdifferingonlyatsite v (0) = r are given by v (t) = r +(1 −2λ)t if r < 0
0 r r
X(0)=0. Withthebasiccoupling,theconfigurationsat and v (t) = r+(1−2ρ)t if r > 0. For r = 0 there are
r
time t differ only at a single site X(t), the position of a two cases. When λ ≤ ρ, the characteristics emanating
so-calledsecond-class particle. Suchaparticlejumpsone frompositive sites areslowerthanthose emanating from
step to its right to an empty site with rate 1, and jumps negative sites. They collide, giving rise to a shock (5)
backwards one step with rate 1 when a (first class) par- traveling at speed 1−λ−ρ. When λ > ρ there are in-
ticle jumps over it. finitely many characteristics emanating from the origin:
2
foreachs∈[1−2λ,1−2ρ]thelinev (t)=stisacharac- exactly the argument that minimizes {G(2,1),G(1,2)};
0
teristic emanating from 0. The limits (3) show that the thus,afterthe firstjumpofthe *pair,itslabelsaregiven
second-classparticlefollowsthecharacteristicwhenthere by ϕ . By recurrence, ϕ gives exactly the labels of the
1 n
isonlyone(thatis,whenλ=ρ),thatitfollowstheshock *pairafteritsnthjump. Thereforethelabelsofthe*par-
whentheinitialconditionhasanincreasingdiscontinuity ticle and *hole are J(t) and I(t), respectively. In addi-
and that it chooses uniformly one of the characteristics tion, J(t)−1 is exactly the number of jumps that the
emanating from a decreasing discontinuity. *pair has made backwards up to time t, and I(t)−1 is
the numberofits jumps forwards. Thisshowsthatifthe
Growth and simple exclusion Rost [16] relates the
simple exclusion process to the growth model as follows.
Consider initial configurations η for the exclusion pro-
cess in which η0(0) = 0 and η0(01) = 1. Elsewhere let bjuemfopre P2 H−1 H0 (H1 P1)∗ H2 P0 H3
η be distributed according to Bernoulli product mea-
0
sure with density λ for j < 0 and ρ for j > 1. De- after P2 H−1 H0 H1 (H2 P1)∗ P0 H3
jump
fine the initial growth interface γ by γ (0) = (1,1)
0 0
and γ (j) − γ (j − 1) = (1 − η (j),−η (j)); then γ
0 0 0 0 0
has the same distribution as before. Label the parti-
FIG. 2: Pair representation of second class particle.
cles sequentially from right to left and the holes from
left to right, with the convention that the particle at
exclusionandthegrowthprocessarerealizedinthesame
site 1 and the hole at site 0 are both labeled 1. Let
space, X(t) =I(t)−J(t). As a consequence of this and
P (0) and H (0), j ∈ Z be the positions of the parti-
j j (3) we get the following behavior for ψ(t) that implies
cles and holes respectively at time 0. The position at
(1) and (2). Almost surely,
time t of the jth particle P (t) and the ith hole H (t)
j i
are functions of the variables G(z) with z ∈ C0\γ0 (de- ψ(t) ((1−ρ)(1−λ),λρ) if λ≤ρ
lim = (8)
fined earlier for the growthmodel) by the following rule: t→∞ t (cid:26) 1((U +1)2,(U −1)2) if λ>ρ
4
at time G((i,j)), the jth particle and the ith hole in-
terchange positions. Disregarding labels and defining where U is a random variable uniformly distributed in
η (P (t))=1, η (H (t))=0, j ∈Z,theprocessη indeed [1−2λ,1−2ρ].
t j t j t
realizes the exclusion dynamics. At time t the particle
To prove (8) for λ > ρ recall that P (t) is the position
1
configuration η and the growth interface γ still satisfy
t t of the 1st particle at time t. Thus J(t) is the number of
the same relation as η and γ . This connection yields
0 0 particles that at time zero were to the left of P (0) = 1
the following shape theorem for the growth model. Al- 1
and at time t are to the right of X(t). Therefore, J(t) is
most surely,
equalto the number of particles between X(t)and P (t)
1
lim γt ={(r,s)∈R2 : s=h(r)} (7) taot1ti−meρta.nBd,ybtyhe(3l)a,wXo(ftl)a/rtgceonnuvmerbgeersst,oPU1(.t)H/etnccoenvJe(rtg)/est
t→∞ t
converges to the integral of the solution of the Burgers
whereh(r)=h (r)isrelatedtothehydrodynamiclimit equation at time 1 (u(r,1) given by (6)) in the interval
λ,ρ
(5,6) by h′(r)=u(r,1)/ 1−u(r,1) . (U,1−ρ). Taking f(r)=1I[r∈[U,1−ρ]] in (4):
(cid:0) (cid:1)
Second class particles and competition interfaces
J(t) 1 P1(t) 1−ρ 1
Akeytoolinproving(1,2)istheobservation[9]thatthe = η (j)−→ u(r,1)dr= (1−U)2
t
process given by the difference of the coordinates of the t t X ZU 4
j=X(t)
competition interface I(t)−J(t) behaves exactly as the
second class particle initially put at the origin. To see Analogously,sinceI(t)isthenumberofholestotheright
this call the particle at site 1 *particle and the hole at of H (0)=0 at time zero and to the left of X(t) at time
1
site 0 *hole, and call this couple *pair. The dynamics t and H (t)/t converges to −λ almost surely, we obtain
1
of the *pair is the following: it jumps to the right when (8) for λ > ρ. For λ ≤ ρ the same argument works by
the *particle jumps to the right, and it jumps to the left substitutingU aboveby1−λ−ρ,thelimitpositionofthe
when a particle jumps from the left onto the *hole. The secondclassparticleinthis case,andtaking the solution
*pair then behaves as a second class particle. The only u(r,1) given by (5).
difference is that it occupies two sites while the second
Fluctuations For θ > 180o the second class particle
class particle occupies only one. The labels of the *par-
has Gaussian fluctuations produced by the initial pro-
ticle and *hole change with time. At time 0 they both
file [6]. This together with the relation above implies
have label 1 and the labels of the *pair are represented
that under a diffusive scaling (I(t),J(t)) converges to a
by the point ϕ = (1,1), the initial value of the com-
0 bidimensionalGaussiandistributionwithanon-diagonal
petition interface. If, say, G(2,1) < G(1,2), then the
covariance matrix computed explicitly [8].
*particle jumps over the second hole before the second
particlejumpsoverthe*hole(seeFigure2). Inthiscase, Tounderstandthefluctuationsforθ ≤180owerelatethe
the labels of the *pairat time G(2,1) are (2,1), which is modelstoadirectedpolymermodel. Foreachz ∈C \γ
0 0
3
let w =G(z)−max{G(z−(1,0)),G(z−(0,1))}. Then enclosed by two semi-infinite maximizing polymers M1
z
(w , z ∈ C \γ ) is a sequence of i.i.d random variables and M2 starting from γ1 and γ2, respectively, and with
z 0 0 0 0
with an exponential distribution of mean 1. Let Π(z,z′) the same inclination [8]. Therefore χ≤ξ in this case.
be the set of all directed polymers (or up-right paths)
Conclusions The connections studied above between
(z ,...,z ) connecting z to z′, and let G(z′,z) be the
1 n the competition interface, the second class particle and
maximum over all π ∈ Π(z′,z) of t(π), the sum of w
z maximalpolymersfitintotheinterplaybetweenthefluc-
along the polymer π. Each site z has energy −w ,
z tuation statistics and the global geometry of the growth
and the polymer π has energy −t(π). Thus −G(z′,z)
interface developed by Prahofer and Spohn [13]. If the
is the minimal energy, or ground state, between z′ and
final macroscopic profile is curved then the competition
z. There exists a unique polymer M(z′,z) in Π(z′,z)
interface follows a random direction (characteristic) in-
thatattainsthemaximum. Wesaythatthesemi-infinite
tersecting the final surface at a point with non-zero cur-
polymer (zn)N is maximizing if for all n < m we have vature. In this case we have the KPZ scaling and the
(z ,...,z ) = M(z ,z ). Every semi-infinite maxi-
n m n m competition interface gets the transversal fluctuations,
mizing polymer (zn)N has an asymptotic inclination eiα indicating the exponent χ = 2/3. If the macroscopic
[9]. In the competition model, G(z) = G(γ ,z) and for
0 profile is not curved we have two different situations. In
k =1,2,Gk isthesetofsitesz suchthatM(γ ,z)orig-
∞ 0 the flat case (stationary growth) the competition inter-
inates from γk. Denote by ξ the roughening exponent of
0 face also follows the characteristics of the associated hy-
semi-infinite maximizing polymers.
drodynamic PDE and we still have the KPZ scaling. In
For θ = 180o (λ = ρ) the process is stationary and the shock case the competition interface gets the longi-
the connection is explicit. Running the process for- tudinal fluctuations which, in this case, are produced by
ward and backward we extend G(z) to all z ∈ Z2; the Gaussian fluctuations (χ = 1/2) of the initial pro-
G+ = (G(z), z ∈ Z2) and G− = (−G(−z), z ∈ Z2) are file. On microscopic grounds one might suggest different
indentically distributed. We define the forward competi- rules for growth and competition. By universality we
tioninterfacestartingatz,ϕz =(ϕzn)N,bysettingϕz0 =z expect that from the knowledge of the curvature of the
andputtingϕz equaltotheargumentoftheminimum final macroscopic shape one can infer the asymptotics of
n+1
betweenG(ϕz +(1,0))andG(ϕz +(0,1)), andthe back- the competition interface. This fits with the exponents
n n
ward semi-infinite polymer starting at z, Mz = (Mnz)N, founded by Derrida and Dickman [3] in the Eden con-
by setting Mz = z and putting Mz equal to the ar- text since, in this case, the macroscopic profile is curved
0 n+1
gument of the maximum between G(Mz − (1,0)) and for angles θ > 180o (we notice that in their simulations
n
G(Mz −(0,1)). Note that ϕ = ϕ(1,1) and that Mz is they have considered periodic initial conditions and so
n
a semi-infinite maximizing polymer. Together with the the longitudinal fluctuations in the shock direction are
duality relation ϕz(G+)=Mz(G−), this shows that the governedby the exponent 1/3 [13]).
forward competition interface has the same law as the
Acknowledgments We thank R. Dickman for calling
backward semi-infinite maximizing polymer and, in par-
our attention to this problem and a referee for a careful
ticular, they have the same fluctuations, so that χ=ξ.
readingandusefulcommentsaboutapreviousversionof
For θ < 180o (λ > ρ) the competition interface ϕ is this paper.
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