Then Gx
N
, . . . , x
1
; t 2.2
=
N
Y
n=1
v
x
n
− y
n
n
e
−atv
n
e
−btv
n
det
F
k,l
x
N +1 −l
− y
N +1 −k
, at, bt
1 ≤k,l≤N
, where
F
k,l
x, a, b = 1
2 πi
I
Γ
dzz
x −1
Q
k −1
i=1
1 − v
N +1 −i
z Q
l −1
j=1
1 − v
N +1 − j
z e
bz
e
a z
, 2.3
where Γ is any anticlockwise oriented simple loop with including only the pole at z = 0.
2.2 Space-like paths
The computation of the joint distribution of particle positions at a given time t can be obtained from Proposition 2.1 by adapting the method used in [4] for the TASEP. However, one of the main
motivation for this work is to enlarge the spectrum of the situations which can be analyzed to what we call space-like paths. In this context, space-like paths are sequences of particle numbers and
times in the ensemble
S = {n
k
, t
k
, k ≥ 1|n
k
, t
k
≺ n
k+1
, t
k+1
}, 2.4
where, by definition, n
i
, t
i
≺ n
j
, t
j
if n
j
≥ n
i
, t
j
≤ t
i
, and the two couples are not identical. 2.5
The two extreme cases are 1 fixed time, t
k
= t for all k, and 2 fixed particle number, n
k
= n for all k. This last situation is known as tagged particle problem. Since the analysis is of the same
degree of difficulty for any space-like path, we will consider the general situation. Consider any smooth function
π, w = πw
1
, in the forward light cone of the origin that satisfies |π
′
| ≤ 1, |w
1
| ≤ πw
1
. 2.6
These are space-like paths in
R
×
R
+
, see Figure 1. The first condition the space-like property is related to the applicability of our result to sequences of particles in
S . The second condition just reflects the choice of having t
≥ 0 and n ≥ 0. Time and particle number are connected with the variables w
1
and w by a rotation of 45 degrees. To avoid unnecessary
p 2’s, we set
¨ w
1
=
t −n
2
w =
t+n 2
« ⇐⇒
¨ t
= w
+ w
1
n = w
− w
1
« 2.7
We want to study the joint distributions of particle positions in the limit of large time, where uni- versal processes arise. Since we consider several times, we can not simply use t as large parameter.
Instead, we consider a large parameter T . Particle numbers and times under investigation will have a leading term proportional to T . In the w
1
, w plane, we consider w
1
around θ T for a fixed θ ,
while w = T πw
1
T . From KPZ we know that correlations are on T
2 3
scale. Therefore, we set the scaling as
w
1
u = θ T − uT
2 3
, w
u = πθ T − π
′
θ uT
2 3
+
1 2
π
′′
θ u
2
T
1 3
. 2.8
1384
n t
w
w
1
π
Figure 1: An example of a space-like path. Its slope is, in absolute value, at most 1.
Notice that w u is equal to T πw
1
uT up to terms that remain bounded, and they become ir- relevant in the large T limit, since the fluctuations grow as T
1 3
. Coming back to the n, t variables, we have
tu =
πθ + θ T − π
′
θ + 1uT
2 3
+
1 2
π
′′
θ u
2
T
1 3
, nu =
πθ − θT + 1 − π
′
θ uT
2 3
+
1 2
π
′′
θ u
2
T
1 3
. 2.9
In particular, setting πθ = 1 − θ we get the fixed time case with t = T , while setting πθ = α + θ
we get the tagged particle situation with particle number n = αT .
2.3 Scaling limits
