The variance of the shock in the HAD process

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by  Cristian F. Coletti

THE VARIANCE OF THE SHOCK IN THE HAD PROCESS
CRISTIAN F. COLETTI, PABLO A. FERRARI AND LEANDRO P. R. PIMENTEL
8
Abstract. WeconsidertheHammersley-Aldous-Diaconis(HAD)processwithsinksand
0
sourcessuchthatthereisamicroscopicshockateverytimet;denoteZ(t)itsposition. We
0
2 showthat the meanandvarianceof Z(t)arelinear functions oft andcompute explicitely
the respective constants in function of the left and right densities. Furthermore, we
n
describe the dependence of Z(t) on the initial configuration in the scale √t and, as a
a
J corollary,prove a central limit theorem.

6
1

]
R
P 1. Introduction
.
h
t Let S and W be one-dimensional Poisson processes and let P be a two dimensional
a
m Poisson process of rate 1. Assume they are homogeneous and mutually independent. The
[ Hammersley-Aldous-Diaconis process [6, 1], or shortly the HAD process, (S,W,P) =
H
(H , s [0,t]) has been constructed by Groeneboom [9] in the square [0,x] [0,t] as a
1 s
∈ ×
v deterministic function of S, W and P as follows. The point configuration H represents
s
6
the position of particles. At time zero the particles start at the points in S, called the
2
5 sources. Then (see Figure 1), at the first s > 0 such that (y,s) is in P for some y [0,x]
∈
2 or s is in W, the closest particle to the right of y jumps to y if (y,s) is in P or to 0 if s is
.
1 in W (points in W are called sinks). If there is no particle to the right of y, then a new
0
particle is added at y at time s. The new configuration does not move until the second s
8
0 such that (y,s) is in P for some y or s is in W, when the second jump occurs, and so on
v: until time t. In other words, we define the process inductively as follows: H0 = S and for
i s > 0,
X
r H if s / W s′, (y,s) P for some y′ [0,x]
a s−
H = ∈ ∪{ ∈ ∈ }
s H R(y,H ) y if (y,s) P (0,s′), s′ W
( s− s−
\{ } ∪{ } ∈ ∪{ ∈ }
n o
where R(y,H) = inf y′ H x : y′ > y for H (0,x). Let N be the positions of
{ ∈ ∪ { } } ⊂
particles at time t, and let E be the set of times a new particle enters the system from the
right through the vertical axis x [0,t).
{ }×
Let λ 0 and ρ 0. Assume that S has intensity λ, W has intensity ρ and P has
≥ ≥
intensity 1. Define the point process S′ by removing the first sink of S and consider the
corresponding HAD process (S′,W,P). For time t 0 the coupled processes (S,W,P)
H ≥ H
and (S′,W,P) will differ at one point denoted Z(t) and called a second class particle
H
(Figure 1).
1
2 CRISTIAN F. COLETTI, PABLO A.FERRARIAND LEANDROP. R. PIMENTEL
(0,t) N (x,t)
W E
(0,0) S (x,0)
Figure 1. The black points represent the Poisson points S (sources), W
(sinks) and P. The polygonal lines represent the trajectory of HAD particles
while the trajectory of the second class particle is represented by the dashed
polygonal line.
Results. Our main results are the computation of the mean and variance of Z(t) and the
dependence of fluctuations on the initial configuration.
Theorem 1. Assume that λρ > 1. Then for all t 0 we have
≥
ρ
E Z(t) = t, (1.1)
λ
(cid:0) (cid:1)
2 ρ 1
Var(Z(t)) = − λ t. (1.2)
(cid:0)λ 1 (cid:1)2
− ρ
(cid:0) (cid:1)
Let
1 1 1 1
R(S) = ρ and R(W) = ρ .
λ − λ ρ − λ
(cid:16) (cid:17) (cid:16) (cid:17)
In the next theorem we show that in the scale √t the position of Z(t) is the same as the
following random variable, depending only on S and W:
1 1
−1
N(t) := λ 3 ρ t [0,tR(S)] S + [0,tR(W)] W ,
− ρ − λ − ∩ ∩
h (cid:16) (cid:17) (cid:16) (cid:17)i
(cid:0) (cid:1) (cid:12) (cid:12) (cid:12) (cid:12)
where A denotes the number of points in (cid:12)the (finite) set(cid:12)A.(cid:12) (cid:12)
| |
Theorem 2. Assume that λρ > 1. Then
E Z(t) N(t) 2
−
lim = 0.
t→∞ h(cid:0) t (cid:1) i
THE VARIANCE OF THE SHOCK 3
and
Z(t) E Z(t)
lim − = (0,1) in distribution,
t→∞ N
Var Z(cid:0)(t) (cid:1)
where (0,1) is a standardqnormal random variable.
(cid:0) (cid:1)
N
Brief historical overview. The HAD process has a genealogical relation with the famous
Ulam’s problem, which studies the limit behavior of the longest increasing subsequence L
n
in a random permutation of the numbers 1,...,n. At the beginning of the seventies Ham-
mersley [6] proposed to solve this problem as follows. Assume (x ,t ),...,(x ,t ) are n
1 1 n n
independent random points distributed uniformly in the rectangle [0,x] [0,t]. These
×
points specify a uniform random permutation π by setting the point with the i’th smallest
t-coordinate has the π(i) smallest x-coordinate. The length of the longest increasing sub-
sequence of π equals the maximal number M(x,t) of points on an up-right path from (0,0)
to (x,t) (last-passage percolation). The variable M(x,t) has the so called super-additivity
property [7], which was the key to show that √nL c. Hammersley conjectured that
n
→
c = 2 by presenting a non-rigorous but quite reasonable argument based on hydrodynam-
ical local equilibrium for a related discrete-time particle system. Later Logan and Sheep
[10] and Vershik and Kerov [11] proved c = 2 with combinatorial methods.
In the middle of the nineties, Aldous and Diaconis [1] constructed a continuous time
version of the Hammersley process and use Rost [12] coupling ideas, developed in the
exclusion process context, to show that c = 2. This is the reason we call this process
Hammersley-Aldous-Diaconis process. Groeneboom [9] introduced the construction in the
quadrant we use here. In this setup M(x,t) equals the number of particles in [0,x] t
×{ }
plus the number of particles in 0 [0,t] (we are allowing an up-right path to pick
{ } ×
Poissonian points in the horizontal or vertical lines). Thus, the trajectory of the n’th
particle corresponds to the boundary of the region M(x,t) n 1 (Figure 1).
{ ≤ − }
The macroscopic hydrodynamical behavior of the HAD process is given by the Burgers
equation ∂ u+∂ g(u) = 0, with g(u) = 1 [13]. When the initial conditions are u(r,0) = λ
t x u
for r > 0 and u(r,0) = 1 when r 0, the solution has a drastic changing when the product
ρ ≤
λρ goes from values smaller than 1 to values greater than 1. For λρ = 1, the system is
stationary and the solutions are constants in time. For λρ < 1 it develops a rarefaction
front while for λρ > 1 it develops a (macroscopic) shock. Since our results concern the
shock regime, from now on we assume λρ > 1.
The microscopic structure of a shock is described by the so called second-class particle.
Our main result is the explicit expression (1.2) for the variance of the second class particle
at any given time t. The method, inspired in work by one of the authors and Fontes [4, 5]
for the exclusion process, relates the mean and the variance of the flux of particles through
the originwith the mean andthe variance ofthe position of a single second class particle by
a linear equation. In this way, if we calculate one, we get the other. However those works
show that for the exclusion process the variance of a second class particle is asymptotically
linear in time; the constant can also be computed. In the HAD process context, we show
4 CRISTIAN F. COLETTI, PABLO A.FERRARIAND LEANDROP. R. PIMENTEL
how to refine this method to get an exact formula for the variance of a second class particle
at any given time t.
2. Preliminaries, coupling and Burke’s theorem
Notation. For a generic point process M we denote
x
f(y,M)M(dy) := f(x,M) (2.3)
Z0 y∈M∩[0,x]
X
and recall the Slivnyak-Mecke Theorem: if M is a Poisson process with intensity λ, then
x x
E f(y,M)M(dy) = Ef(y,M)λ(y)dy (2.4)
(cid:16)Z0 (cid:17) Z0
The stationary regime and the Burke’s theorem. Let γ > 0 and S and W be
χ χ
Poisson processes with intensity γ and γ−1, respectively. Then χ = (S ,W ,P) is a
χ χ
H
stationaryHADprocess withparameterγ. Cator andGroeneboom[2]proved thefollowing
Burke’s theorem: N and E are independent Poisson processes with the same density as
χ χ
S and W , respectively.
χ χ
Coupling two stationary process. Let λρ > 1. Let S , W , I and J be mutually
η σ
independent one-dimensional Poisson point processes with intensities 1, 1, λ 1 and ρ 1
ρ λ −ρ −λ
respectively. Set
S = S +I and W = W +J . (2.5)
σ η η σ
Thus, S and W are two (independent) one-dimensional Poisson processes with intensity
σ η
λ and ρ respectively. Let P be the rate-1 two-dimensional Poisson process mentioned in
the Introduction.
WerunthreeHAD processes simultaneously withdifferent boundaryconditions butwith
the same P. This is a realization (φ,σ,η) of the so called basic coupling defined by
φ = (S ,W ,P), σ = (S ,W ,P) and η = (S ,W ,P).
σ η σ σ η η
H H H
For χ φ,σ,η we denote by χ(x,t) the number of χ-particles (particles that count for
∈ { }
the process χ) in the interval [0,x] at time t, with the convention that particles escaping
through a sink during the time interval [0,t] are located at 0. The flux of the discrepancies
between σ and η through the space-time line (0,0)-(x,t) is the process defined by
ξ(x,t) = σ(x,t) η(x,t). (2.6)
−
There are two kinds of discrepancies: i) those starting at a point y at time 0 in I (in
the x-axis); ii) those starting at a added sink in a time s in J (in the t-axis). The position
at time t of the second class particle starting at time zero at site y is denoted Z (t), or
(y,0)
shortly Z (t), and the position at time t of the second class particle starting at site 0 at
y
time s is denoted Z (t), or shortly Z (t).
(0,s) s
THE VARIANCE OF THE SHOCK 5
The flux of second class particles defined in (2.6) is also given by
ξ(x,t) = # y I : Z (t) x # s J : Z (t) > x .
(y,0) (0,s)
∈ ≤ − ∈
:= ξ (x,t) ξ (x,t) (2.7)
+(cid:8) − (cid:9) (cid:8) (cid:9)
−
3. The mean and the variance of the flux of second class particles
In this section we show the following.
Proposition 1. For all x,t 0
≥
1 1
E ξ(x,t) = λ x ρ t. (3.8)
− ρ − − λ
(cid:16) (cid:17) (cid:16) (cid:17)
(cid:0) (cid:1)
1 1
Var ξ(x,t) = λ x+ ρ t. (3.9)
− ρ − λ
(cid:16) (cid:17) (cid:16) (cid:17)
(cid:0) (cid:1)
Denote by M¯ := M , the total number of points of a generic finite point process M and
| |
χ = χ(x,t). With this notation,
χ = N¯ +W¯ = S¯ +E¯ , (3.10)
χ χ χ χ
for χ σ,η and
∈ { }
ξ = σ η. (3.11)
−
Proof of Proposition 1. By definition (2.6) and Burke’s theorem,
t x
E ξ(x,t) = E ξ = E σ E η = E N¯ +W¯ (N¯ +W¯ ) = λx+ +ρt .
σ σ η η
− − λ − ρ
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
This shows (3.8).
To compute the variance combine first (2.5) and Burke’s theorem to get
¯ ¯ ¯ ¯
Cov S ,W = Cov E ,N = 0. (3.12)
σ η σ η
Hence, by (3.10), (cid:0) (cid:1) (cid:0) (cid:1)
¯ ¯ ¯ ¯
Cov σ,η = Cov S +E ,N +W
σ σ η η
¯ ¯ ¯ ¯
= Cov S ,N +Cov E ,W
(cid:0) (cid:1) (cid:0) σ η σ(cid:1) η
¯ ¯ ¯ ¯ ¯ ¯
= Cov S +I,N +Cov E ,W +J
(cid:0) η (cid:1) η (cid:0) σ(cid:1) σ
= Cov S¯ ,N¯ +Cov E¯ ,W¯ . (3.13)
(cid:0) η η (cid:1) σ(cid:0) σ (cid:1)
Analogous (and simpler) reasoning for(cid:0)χ η(cid:1),σ give(cid:0)s (cid:1)
∈ { }
¯ ¯ ¯ ¯
Var χ = Cov E ,W +Cov N ,S . (3.14)
χ χ χ χ
On the other hand, since χ(cid:0) =(cid:1)N¯ +W¯(cid:0) , (cid:1) (cid:0) (cid:1)
χ χ
Var χ = Var N¯ +Var W¯ +2Cov N¯ ,W¯ (3.15)
χ χ χ χ
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
6 CRISTIAN F. COLETTI, PABLO A.FERRARIAND LEANDROP. R. PIMENTEL
¯ ¯ ¯ ¯
Using W = S +E N and E independent of N ,
χ χ χ χ χ χ
−
Cov N¯ ,W¯ = Cov N¯ ,S¯ +E¯ N¯ ) = Cov N¯ ,S¯ ) Var N¯ .
χ χ χ χ χ χ χ χ χ
− −
which implies
(cid:0) (cid:1) (cid:0) (cid:0) (cid:0) (cid:1)
Var(χ) = Var W¯ Var N¯ +2Cov N¯ ,S¯ . (3.16)
χ χ χ χ
−
Identities (3.14) and (3.16) imply
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
¯ ¯ ¯ ¯ ¯ ¯
Cov E ,W Cov N ,S = Var W Var N (3.17)
χ χ χ χ χ χ
− −
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Finally we compute Var(ξ) using (3.13) and (3.14):
Var ξ = Var σ +Var η 2Cov σ,η
−
= Cov E¯ ,W¯ Cov N¯ ,S¯ +Cov N¯ ,S¯ Cov E¯ ,W¯
(cid:0) (cid:1) (cid:0) (cid:1)η η (cid:0) (cid:1) η (cid:0)η (cid:1) σ σ σ σ
− −
= Va(cid:0)r W¯ (cid:1)Var N¯(cid:0) (cid:1)Var W¯ (cid:0) Var(cid:1)N¯ (cid:0) (cid:1)
η η σ σ
− − −
(cid:16) x t (cid:17) (cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
= ρt λx , (3.18)
− ρ − λ −
(cid:0) (cid:1) (cid:0) (cid:3)(cid:1)
where in the third line we used (3.17).
4. The mean and the variance of a second class particle
Lemma 1. For al x,t 0,
≥
1 x
Eξ(x,t) = λ x P Z(t) > z dz . (4.19)
+
− ρ −
n Z0 o
(cid:0) 1(cid:1) t (cid:0) (cid:1)
Eξ(x,t) = ρ t P Z(u) x du (4.20)
−
− λ − ≤
n Z0 o
(cid:0) (cid:1) (cid:0) (cid:1)
Proof of Lemma 1. By definition (2.7),
x
ξ(x,t) = 1 Z (t) x I(dy)
+ y
{ ≤ }
Z0
where Z (t) denotes the position at time t of a second-class particle that starts from the
y
discrepancy at y [0,x]. By translation invariance and (2.4),
∈
1 x
Eξ(x,t) = λ P Z(t) x y dy. (4.21)
+
− ρ ≤ −
Z0
(cid:0) 1(cid:1) x (cid:0) (cid:1)
= λ P Z(t) z dz
− ρ ≤
Z0
(cid:0) (cid:1) (cid:0) (cid:1)
which proves (4.19). Analogously, (4.20) follows from
t
Eξ(x,t) = E P(Z (t) > x)J(ds) . (4.22)
− s
(cid:16)Z0 (cid:17)
(cid:3)
THE VARIANCE OF THE SHOCK 7
The main step to calculate the mean value of Z is the following.
Proposition 2. For all x,t 0
≥
x ρ t
P Z(t) > z dz = P Z(u) x du. (4.23)
λ ≤
Z0 Z0
(cid:0) (cid:1) (cid:0) (cid:1)
Proof of Proposition 2. Combining (4.19), (4.20) with (3.8),
1 1
λ x ρ t = E ξ(x,t) = E ξ(x,t) E ξ(x,t)
+ −
− ρ − − λ −
(cid:16) (cid:17) (cid:16) (cid:17)
1 x (cid:0) (cid:1) (cid:0) 1(cid:1) (cid:0) t (cid:1)
= λ x P Z(t) > z dz ρ t P Z(u) x du ;
− ρ − − − λ − ≤
n Z0 o n Z0 o
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
that is,
1 x 1 t
λ P Z(t) > z dz = ρ P Z(u) x du (4.24)
− ρ − λ ≤
Z0 Z0
which is equival(cid:0)ent to ((cid:1)4.23) f(cid:0)or λρ > 1(cid:1). (cid:3) (cid:0) (cid:1) (cid:0) (cid:1)
Proof of (1.1). It follows directly from Proposition 2:
x ρ t ρ
E Z(t) = lim P Z(t) > z dz = lim P Z(u) x du = t.
x→∞ x→∞ λ ≤ λ
Z0 Z0
(cid:3) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
A similar idea works to compute Var Z(t) :
Proposition 3. For all x,t 0 (cid:0) (cid:1)
≥
1 t
ρ t+ P Z(u) x du (4.25)
− λ ≤
(cid:0) (cid:1)(cid:16) Z0 (cid:0)1 x (cid:1) (cid:17) x 2
= λ 2 2 zP Z(t) > z dz P Z(t) > z dz + Var(ξ(x,t) )
−
− ρ −
(cid:16) Z0 (cid:16)Z0 (cid:17) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Proof of (1.2). Taking x in both members of (4.25),
→ ∞
1 1
2
ρ 2t = λ Var(Z(t)),
− λ − ρ
(cid:0) (cid:1) (cid:0) (cid:1)
fromwhere(1.2)follows. (Noticethat,forfixedt 0,ξ (x,t)decreases to0whenx .)
−
(cid:3) ≥ → ∞
Proof of Proposition 3. Noticing that if y¯ y and Z (t) x then Z (t) x,
y y¯
≤ ≤ ≤
x y
ξ(x,t)2 = ξ(x,t) +2 1 Z (t) x 1 y = y¯ I(dy¯)I(dy) (4.26)
+ + { y ≤ } { 6 }
Z0 Z0
8 CRISTIAN F. COLETTI, PABLO A.FERRARIAND LEANDROP. R. PIMENTEL
as we do for the square of a sum of functions with values in 0,1 . The expected value of
{ }
the double integral in the above equation is given by
x y
2E 1 Z (t) x 1 y = y¯ I(dy¯)I(dy)
y
{ ≤ } { 6 }
(cid:16)Z0 Z0 1 x y (cid:17)
= 2 λ 2 P Z(t) x y dy¯dy
− ρ ≤ −
Z0 Z0
(cid:0) 1(cid:1) x (cid:0) (cid:1)
= 2 λ 2 yP Z(t) x y dy,
− ρ ≤ −
Z0
(cid:0) 1(cid:1) x(cid:0) (cid:1) x
= 2 λ 2 x P Z(t) z dz zP Z(t) z dz
− ρ ≤ − ≤
(cid:16) Z0 Z0 (cid:17)
(cid:0) 1(cid:1) (cid:0) x (cid:1) x(cid:0)2 x (cid:1)
= 2 λ 2 x2 x P Z(t) > z dz + zP Z(t) > z dz (4.27)
− ρ − − 2
(cid:16) Z0 Z0 (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
using translation invariance. Take the expectation of (4.26) using (4.19) and (4.27) to get
1 x
E ξ(x,t)2 = λ x P Z(t) > z dz (4.28)
+ − ρ −
(cid:0) (cid:1) (cid:0) (cid:1)(cid:16) 1 Z0 (cid:0) x (cid:1) (cid:17) x
+ λ 2 x2 2x P Z(t) > z dz +2 zP Z(t) > z dz
− ρ −
(cid:16) Z0 Z0 (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
On the other hand, taking the square of (4.19),
1 x x 2
Eξ(x,t) 2 = λ 2 x2 2x P Z(t) > z dz + P Z(t) > z dz (4.29)
+
− ρ −
n Z0 (cid:16)Z0 (cid:17) o
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Substracting (4.29) from (4.28):
1 x
Var(ξ(x,t) ) = λ x P Z(t) > z dz
+
− ρ −
(cid:0) (cid:1)(cid:16) 1Z0 (cid:0) x (cid:1) (cid:17) x 2
+ λ 2 2 zP Z(t) > z dz P Z(t) > z dz
− ρ −
(cid:16) Z0 (cid:16)Z0 (cid:17) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
that is, using (3.9) and Var(ξ(x,t)) = Var(ξ(x,t) )+Var(ξ(x,t) ),
+ −
1 1 x
ρ t = λ P Z(t) > z dz
− λ − − ρ
(cid:0) (cid:1) (cid:0) (cid:1)(cid:16)Z10 (cid:0) x (cid:1) (cid:17) x 2
+ λ 2 2 zP Z(t) > z dz P Z(t) > z dz
− ρ −
(cid:16) Z0 (cid:16)Z0 (cid:17) (cid:17)
+(cid:0)Var(ξ(x(cid:1),t) ) (cid:0) (cid:1) (cid:0) (cid:1)
−
Use (4.23) to get
1 t
ρ t+ P Z(u) x du
− λ ≤
(cid:0) (cid:1)(cid:16) Z10 (cid:0) x (cid:1) (cid:17) x 2
= λ 2 2 zP Z(t) > z dz P Z(t) > z dz + Var(ξ(x,t) )
−
− ρ −
(cid:16) Z0 (cid:16)Z0 (cid:17) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
THE VARIANCE OF THE SHOCK 9
(cid:3)
which shows (4.25).
5. The dependence on the initial condition
Recall that
1 tR(S) tR(W)
−1
N(t) = λ c t S(dz)+ W(ds)
0
− ρ −
h (cid:16)Z0 Z0 (cid:17)i
(cid:0) (cid:1)
where
1 1 1 1 1
c = 3 ρ , R(S) = ρ and R(W) = ρ .
0
− λ λ − λ ρ − λ
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Thus,
ρ
E N(t) = E Z(t) = t and Var N(t) = Var Z(t) = Dt (5.30)
λ
where (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
2 ρ 1
D = − λ .
(cid:0)λ 1 (cid:1)2
− ρ
(by calculating N(t) and using Theorem 1). The next step is to prove:
(cid:0) (cid:1)
Lemma 2.
Cov Z(t),N(t)
lim = D
t→∞ t
(cid:0) (cid:1)
Proof of Lemma 2. The proof follows in the same lines of an analogous result proved by
Ferrari[5]intheTASEPcontext. Tosketchtheidea,denotebyZz|1(t) respectively,Zz|0(t)
the position at time t of a second class particle with respect to an initial configuration con-
(cid:0) (cid:1)
ditioned to have a particle at position z (respectively, conditioned not to have a particle
at position z). Then we can get (by coupling)
tR(S) tR(S)
Cov Z(t), S(dz) = λ E Zz|0(t) Zz|1(t) dz (5.31)
− −
Z0 Z0
(cid:0) (cid:1) (cid:0) (cid:1)
and
tR(W) tR(W)
Cov Z(t), W(ds) = ρ E Zs|0(t) Z′s|1(t) ds. (5.32)
− −
Z0 Z0
(cid:0) (cid:1) (cid:0) (cid:1)
Now, we claim that for all ǫ > 0
1
lim sup E Zz|0(t) Zz|1(t) = 0. (5.33)
t→∞z∈[0,t(R(.)−ǫ)]| − − λ− 1ρ |
(cid:0) (cid:1)
To sketch the idea, call (see Section 3) Zz|1(t) the positio(cid:0)n at t(cid:1)ime t of the second class
n
particles starting at ξ conditioned to have one particle at z. Let Y be the position at time
t
t where the processes with initial configuration being different only at z differ by time t.
Let τ be the first time the trajectories of Y and Zz|1(t) meet each other. Note that τ
1 t 1
10 CRISTIAN F. COLETTI, PABLO A.FERRARIAND LEANDROP. R. PIMENTEL
is finite with probability one. Before τ ,Zz|1(t) = Zz|0(t). After τ ,Zz|1(t) = Zz|1(t) and
1 1 1
z|0 z|1
Z = Z (t). Since
t 2
1
E Zz|1(t) Zz|1(t) = ,
2 − 1 λ 1
− ρ
(cid:0) (cid:1) (cid:3)
we get (5.33). Thus, Lemma 2 follows by combining (5.31), (5.32) together with (5.33).
(cid:0) (cid:1)
Proof of Theorem 2. By (5.30),
E Z(t) N(t) 2 = 2Dt 2Cov Z(t),N(t) ,
− −
and hence, Theorem 2 hfo(cid:0)llows from L(cid:1)emima 2. (cid:3) (cid:0) (cid:1)
References
[1] Aldous and Diaconis. Hammersley’s interacting particle process and longest increasing subsequences.
Probab. Theory Relat. Fields 103 (1995) 199-213.
[2] E. Cator and P. Groeneboom. Hammersley process with sources and sinks. Ann. Probab. 33 (2005)
879-903.
[3] C.F.ColettiandL.P.R.Pimentel.Onthe collisionbetweentwoPNGdroplets.J.Stat.Physics126
(2007) 1145-1164.
[4] P. A. Ferrari. Shocks in the Burgers equation and the asymmetric simple exclusion. Probab. Theory
Relat. Fields 91 (1992) 81-101.
[5] P. A. Ferrari and L. R. G. Fontes. Current fluctuations for the asymmetric simple exclusion process.
Ann. Probab. 22 (1994) 820-832.
[6] J.M.Hammersley. A few seedlings of research,Proc. Sixth Berkeley Symp. Math. Statist. and Proba-
bility. 1 (1972) 345-394 University of California Press.
[7] J. M. Hammersley. Postulates for subadditive processes. Ann. Probab. 2 (1974) 652-680.
[8] P. Groeneboom. Ulam’s problem and Hammersley’s process. Ann. Probab. 29 (2001) 683-690.
[9] P. Groeneboom. Hydrodynamics methods for analyzing longest increasing subsequences. J. Comut.
Appl. Math. 142 (2002) 83-105.
[10] B. F. Logan and L. A. Shepp. A variational problem for random Young tableaux. Adv. Math. 26
(1977) 206-222.
[11] A. M. Vershik and S. V. Kerov. Asymptotics of the Plancherel measure of the symmetric group and
the limiting form of Young tableaux. Soviet Math. Dokl. 18 (1985) 21-31.
[12] H. Rost. Nonequilibrium behavior of many particle process: Density profile and local equilibrium. Z.
Wahrsch. Verw. Gebiete 58 (1981) 41-53.
[13] T. Sepp¨al¨ainen. A microscopic model for the Burges equation and longest increasing subsequences.
EJP 1 (1996) 1-51.
InstitutodeMatema´tica, UniversidadedeSa˜oPaulo,RuadoMata˜o1010blocoD05311-
970 Sa˜o Paulo SP Brazil,
E-mail address: [email protected], [email protected], [email protected]