2.4 Limit processes
For completeness, we shortly recall the definitions of the limit processes A
1
and A
2
appearing above. The notation Aix below stands for the classical Airy function [1].
Definition 2.3 The Airy
1
process . The Airy
1
process A
1
is the process with m-point joint distribu- tions at u
1
u
2
. . . u
m
given by the Fredholm determinant
P
m
\
k=1
{A
1
u
k
≤ s
k
} = det
1
− χ
s
K
A
1
χ
s L
2
{u
1
,...,u
m
}×
R
, 2.24
where χ
s
u
k
, x =
1
x s
k
and the kernel K
A
1
is given by K
A
1
u
1
, s
1
; u
2
, s
2
= − 1
p 4
πu
2
− u
1
exp
− s
2
− s
1 2
4u
2
− u
1
1
u
2
u
1
+Ais
1
+ s
2
+ u
2
− u
1 2
exp u
2
− u
1
s
1
+ s
2
+ 2
3 u
2
− u
1 3
. 2.25
Definition 2.4 The Airy
2
process . The Airy
2
process A
2
is the process with m-point joint distribu- tions at u
1
u
2
. . . u
m
given by the Fredholm determinant
P
m
\
k=1
{A
2
u
k
≤ s
k
} = det
1
− χ
s
K
A
1
χ
s L
2
{u
1
,...,u
m
}×
R
, 2.26
where χ
s
u
k
, x =
1
x s
k
and the kernel K
A
2
is given by
K
A
2
u
1
, s
1
; u
2
, s
2
=
R
R
+
e
−λu
2
−u
1
Ais
1
+ λAis
2
+ λ, u
2
≥ u
1
, −
R
R
−
e
−λu
2
−u
1
Ais
1
+ λAis
2
+ λ, u
2
u
1
. 2.27
3 Finite time kernel
In this section we first derive an expression for the joint distributions of particle positions in a finite system. They are given by Fredholm determinants of a kernel, which is first stated for general jump
rates and initial positions. After that, we specialize to the cases of uniform jump rates in the case of step and flat initial conditions. Flat initial conditions are obtained via a limit of finite systems.
3.1 General kernel for PushASEP
To state the following result, proven in Section 4, we introduce a space of functions V
n
. Consider the set of numbers
{v
1
, . . . , v
n
} and let {u
1
u
2
. . . u
ν
} be their different values, with α
k
being the multiplicity of u
k
v
k
is the jump rate of particle with label k. Then we define the space V
n
= span{x
l
u
x k
, 1 ≤ k ≤ ν, 0 ≤ l ≤ α
k
− 1}. 3.1
Recall that the evolution of particle indexed by n is independent of the particles with index m n.
1388
Proposition 3.1. Consider a system of particles with indices n = 1, 2, . . . starting from positions y
1
y
2
. . .. Denote by x
n
t the position of particle with index n at time t. Then the joint distribution of particle positions is given by the Fredholm determinant
P
m
\
k=1
{x
n
k
t
k
≥ s
k
} = det
1
− ˜ χ
s
K ˜ χ
s ℓ
2
{n
1
,t
1
,...,n
m
,t
m
}×
Z
3.2 with n
1
, t
1
, . . . , n
m
, t
m
∈ S , and ˜ χ
s
n
k
, t
k
x =
1
x s
k
. The kernel K is given by Kn
1
, t
1
, x
1
; n
2
, t
2
, x
2
= −φ
n
1
,t
1
,n
2
,t
2
x
1
, x
2
+
n
2
X
k=1
Ψ
n
1
,t
1
n
1
−k
x
1
Φ
n
2
,t
2
n
2
−k
x
2
3.3 where
Ψ
n,t n
−l
x = 1
2 πi
I
Γ
dzz
x − y
l
−1
e
at z+btz
1 − v
1
z · · · 1 − v
n
z 1 − v
1
z · · · 1 − v
l
z ,
l = 1, 2, . . . , 3.4
the functions {Φ
n,t n
−k
}
n k=1
are uniquely determined by the orthogonality relations X
x ∈
Z
Ψ
n,t n
−l
xΦ
n,t n
−k
x = δ
k,l
, 1
≤ k, l ≤ n, 3.5
and by the requirement span {Φ
n,t n
−l
x, l = 1, . . . , n} = V
n
. The first term in 3.3 is given by φ
n
1
,t
1
,n
2
,t
2
x, y = 1
2 πi
I
Γ
dz z
y −x+1
e
at
1
−at
2
z
e
bt
1
−bt
2
z
1 − v
n
1
+1
z · · · 1 − v
n
2
z
1
[n
1
,t
1
≺n
2
,t
2
]
. 3.6
The notation Γ stands for any anticlockwise oriented simple loop including only the pole at 0.
3.2 Kernel for step initial condition
Kategori