J
Electr
on
i
o
u
r
nal
o
f
P
c
r
o
ba
bility
Vol. 16 (2011), Paper no. 36, pages 1048–1064.
Journal URL
http://www.math.washington.edu/~ejpecp/
From the Pearcey to the Airy process
M. Adler∗ †
M. Cafasso ‡
P. van Moerbeke §¶
Abstract
Putting dynamics into random matrix models leads to finitely many nonintersecting Brownian
motions on the real line for the eigenvalues, as was discovered by Dyson. Applying scaling
limits to the random matrix models, combined with Dyson’s dynamics, then leads to interesting,
infinite-dimensional diffusions for the eigenvalues. This paper studies the relationship between
two of the models, namely the Airy and Pearcey processes and more precisely shows how to
approximate the multi-time statistics for the Pearcey process by the one of the Airy process with
the help of a PDE governing the gap probabilities for the Pearcey process.
Key words: Airy process, Pearcey process, Dyson’s Brownian motions.
AMS 2010 Subject Classification: Primary 60K35; Secondary: 60B20.
Submitted to EJP on December 19, 2010, final version accepted May 1, 2011.
∗
Department of Mathematics, Brandeis University, Waltham, Mass 02454, USA.adler@brandeis.edu
The support of a National Science Foundation grant # DMS-07- 04271 is gratefully acknowledged
‡
Centre de Recherches Mathématiques, Université de Montréal C. P. 6128, succ. centre ville, Montréal, Québec, Canada
H3C 3J7 and Department of Mathematics and Statistics, Concordia University 1455 de Maisonneuve W., Montréal, Québec,
Canada H3G 1M8.cafasso@crm.umontreal.ca
§
Département de Mathématiques, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium and Brandeis University, Waltham, Mass 02454, USA. vanmoerbeke@math.ucl.ac.be
¶
The support of a National Science Foundation grant # DMS-07-04271, a European Science Foundation grant (MISGAM), a Marie Curie Grant (ENIGMA), Nato, FNRS and Francqui Foundation grants is gratefully acknowledged.
†
1048
1
Introduction
Putting dynamics into random matrix models leads to finitely many nonintersecting Brownian motions on R for the eigenvalues, as was discovered by Dyson [13]. Applying scaling limits to the random matrix models, combined with Dyson’s dynamics, then leads to interesting, infinite-dimensional
diffusions for the eigenvalues. This paper studies the relationship between two of the models,
namely the Airy and Pearcey processes and more precisely shows how to approximate the Pearcey
process by the Airy process with the help of a PDE [2] governing the gap probabilities for the Pearcey
process. The Airy process was introduced by Prähofer-Spohn [18] and further developed by K. Johansson [14, 15]. A simple non-linear 3rd order PDE for the transition probabilities for this process
was found in [3]; see also [19, 20]. The Pearcey process was introduced in [21, 17] in the context of
non-intersecting Brownian motions and plane partitions, also based on prior work on matrix models
with external source [16, 10, 22, 23, 11, 12, 5, 7, 8, 9].
Consider n nonintersecting Brownian particles on the real line R,
−∞ < x 1 (t) < ... < x n (t) < ∞
with (local) Brownian transition probability given by
( y−x)2
1
p(t; x, y) := p e− t ,
πt
all starting from the origin x = 0 at time t = 0 and such that
n
2
particles are forced to ±
For very large n, the average mean density of particles has its support, for t ≤
1
2
1
,
2
pn
2
at t = 1.
on one interval
centered about x = 0, and for < t ≤ 1 on two intervals symmetrically located about x = 0. The
end points of the interval(s) of support describe a heart-shaped region in (x, t)-space, with a cusp
at (x 0 , t 0 ) = (0, 1/2). The Pearcey process P (τ) is defined (see Figure 1) as the motion of these
nonintersecting Brownian motions for large n, about (x 0 , t 0 ) = (0, 1/2) (i.e., near the cusp), with
space microscopically rescaled by a factor of n−1/4 and time rescaled by a factor n−1/2 , in tune with
the Brownian motion rescaling. A partial differential equation for the Pearcey process was found
by Adler-van Moerbeke [4] and in [2] a much better version was obtained, namely a simple third
order non-linear PDE for the transition probabilities; it was obtained by a scaling limit on a PDE
for non-intersecting Brownian motions with target points. This PDE is related to the Boussinesq
equation and its hierarchy; this is part of a general result on integrable kernels, as explained in the
paper [1].
Near the boundary of the heart-shaped region of Figure 1, but away from the cusp, the local fluctuations behave as the so-called Airy process, which describe the non-intersecting Brownian motions
with space stretched by the customary GUE edge rescaling n1/6 and time rescaled by the factor n1/3 ,
again in tune with the Brownian motion space-time rescaling.
This paper shows how the Pearcey process statistics tends to the Airy process statistics when one is
2
moving out of the cusp x = 27
(3(t − t 0 ))3/2 very near the boundary, that is at a distance of (3τ)1/6
for τ very large, with τ being the Pearcey time. To be precise, in the two-time case, the times τi
must be sufficiently near -in a very precise way- for the limit to hold. The main result of the paper
can be summarized as follows:
1049
Theorem 1.1. Given finite parameters t 1 < t 2 , let both τ1 , τ2 → ∞, such that τ2 − τ1 → ∞ behaves
in the following precise way:
τ2 − τ1
2(t 2 − t 1 )
= (3τ1 )1/3 +
t2 − t1
(3τ1 )1/3
2t 1 t 2
+
3τ1
−5/3 + O τ1
.
(1)
Let also E1 , E2 be two arbitrary finite intervals. The parameters t 1 and t 2 provide the Airy times in the
following approximation of the Airy process by the Pearcey process:
(
)!
2
2
\
P (τi ) − 27
(3τi )3/2
P
∩ −Ei = ;
1/6
(3τ
)
i
i=1
(2)
!
!
2
\
1 .
=P
A (t i ) ∩ (−Ei ) = ;
1 + O 4/3
τ1
i=1
The same estimate holds as well for the one-time case. A similar (but different) result was then
obtained in [6] for the one–time case in the situation when one is moving out of the cusp following
both the two branches of the cusp simultaneously. It is also worth recalling that, in this paper as
well as in [6], the statistics of the processes are studied on intervals rather than on general Borel
subsets.
The proof of this Theorem proceeds in two steps. At first, Proposition 2.1, stated later, shows that,
using the scaling above, the Pearcey kernel tends to the Airy kernel. In a second step, we show,
using the PDE for the Pearcey process [2] and Proposition 2.1, the result of Theorem 1.1.
It is a fact that both the Airy and Pearcey processes are determinantal processes, for which the multitime gap probabilities is given by the matrix Fredholm determinant of the matrix kernel, which will
be described here.
The Airy process is a determinantal process, for which the multi-time gap probabilities1 are given
by the matrix Fredholm determinant for t 1 < . . . < t ` ,


`
\
(3)
P  (A (t i ) ∩ Ei = ;) = det I − [χ Ei K tAt χ E j ]
i j
i=1
1≤i, j≤`
of the matrix kernel in ~x = (x 1 , . . . , x ` ) and ~y = ( y1 , . . . , y` ), denoted as follows:
p
KA
x , ~y ) d ~x d ~y
t ,...,t (~
`
1



:= 

p
K tA,t (x 1 , y1 ) d x 1 d y1 . . .
1 1
..
.
p
K tA,t (x ` , y1 ) d x ` d y1 . . .
`
1
p
K tA,t (x 1 , y` ) d x 1 d y`
1 `
..
.
p
K tA,t (x ` , y` ) d x ` d y`
`

(4)


,

`
where, for arbitrary t i and t j , the extended Airy kernel K tA,t (x, y) is given by (see Johansson [14,
i j
15])
K tA,t (x, y) := K̃ tA,t (x, y) − 1I(t i < t j )pA (t j − t i , x, y),
(5)
i
1
j
i
j
χ E is the indicator function for the set E
1050
n/2
# of particles
n/2
t
-
p
p
n/2
n/2
t =1
t = 1/2
x
Figure 1: The Pearcey process.
where
K̃ tA,t (x,
i j
Z
∞
e−λ(t i −t j ) A(x + λ)A( y + λ)dλ
y) :=
0
=
2
(2πi)
1
A
Z
Z
1
p (t, x, y) := p
e
4πt
dv
Γ>
du
Γ<
( x− y )2
t3
− 4t − 2t
12
e−v
3
/3+ y v
3
e−u /3+xu
1
(v + t j ) − (u + t i )
( y+x ) , for t > 0.
The contour u ∈ Γ< consists of two rays emanating from the origin with angles θ1 and θ10 with the
positive real axis, and the contour v ∈ Γ> also consists of two rays with angles θ2 and θ20 with the
negative real axis, as indicated in Figure 2. As is well known, one may choose θ1 , θ2 , θ10 , θ20 = π/3.
In particular, for t i = t j , this is the customary Airy kernel
K tA,t
i i
A
:= K (x, y) :=
Z
∞
dλA(x + λ)A( y + λ) =
A(x)A0 ( y) − A( y)A0 (x)
x−y
0
.
(6)
The Pearcey process is determinantal as well, for which the multi-time gap probability is given by
the Fredholm determinant for τ1 < . . . < τ` ,


`
\
P  (P (τi ) ∩ Ei = ;) = det 1I − [χ Ei KτP τ χ E j ]
(7)
i
i=1
1051
j
1≤i, j≤`
~ = (ξ1 , . . . , ξ` ) and η
for the Pearcey matrix kernel in ξ
~ = (η1 , . . . , η` ), denoted as follows:
Æ
~ η
~ dη
(
ξ,
~
)
dξ
~
KP
τ ,...,τ
`
1



:= 

p
KτP ,τ (ξ1 , η1 ) dξ1 dη1 . . .
1 1
..
.
p
KτP ,τ (ξ` , η1 ) dξ` dη1 . . .
`
p
KτP ,τ (ξ1 , η` ) dξ1 dη`
1 `
..
.
p
KτP ,τ (ξ` , η` ) dξ` dη`
`
1

(8)


,

`
where for arbitrary τi and τ j , (see Tracy-Widom [21])
KτP ,τ (ξ, η) = K̃τP ,τ (ξ, η) − 1I(τi < τ j )pP (τ j − τi ; ξ, η),
i
j
i
j
with
K̃τP ,τ (ξ, η) := −
i
j
1
4π2
Z
Z
dU
X
Y
dV
e−
τjV2
V4
+ 2 −V η
4
V −U
e−
τ U2
U4
+ i2 −Uξ
4
(9)
(ξ−η)2
1
pP (τ; ξ, η) := p
e− 2τ , for τ > 0,
2πτ
where X and Y are the following contours: X = X 1 ∪ X 2 consists of four rays emanating from the
origin with angles σ1 , σ10 with the positive real axis and σ2 , σ20 with the negative real axis, as given
in Figure 2. The contour Y consists of two rays emanating from the origin with angles τ, τ0 with the
negative real axis; it is customary to pick σ1 = σ2 = σ10 = σ20 = π/4 and τ = τ0 = π/2.
V ∈YE
E
E
E
U ∈ X 2 @I
@
σ2 @
E
OE
U ∈ X1
E
σ1
@τ E
@Ep
τ0 @
@ σ0
σ20
@ 1
@
U ∈ X2 R U ∈ X1
@
V ∈ Y v ∈ Γ> A
AM
A
θ A
u ∈ Γ<
θ
1
A Ap
A
θ20 A θ10
A
A
AN u ∈ Γ<
v ∈ Γ> A
2
π/8 < σ1 , σ2 , σ10 , σ20 < 3π/8
π/6 < θ1 , θ2 , θ10 , θ20 < π/2
3π/8 < τ, τ0 < 5π/8
Pearcey contour
Airy contour
Figure 2: Integration paths for the kernels.
1052
In [4, 2], it was shown that, given intervals
(i)
(i)
Ei = (ξ1 , ξ2 ),
the log of the probability

Q = Q(τ1 , . . . , τ` ; E1 , . . . , E` ) := log P 
`
\

{P (τi ) ∩ Ei = ;}
i=1
satisfies a non-linear PDE, which we describe here. Given the times and the intervals above, one
defines the following operators:
∂τ :=
X ∂
i
"τ :=
∂ τi
X
∂E :=
,
X
∂
k
∂ ξk
i
τi
i
∂
∂ τi
" E :=
,
∂E :=
,
(i)
∂E
i
XX
i
X
∂
(i)
ξk
k
(i)
∂ ξk
i
.
(10)
Then Q satisfies the Pearcey partial differential equation in its arguments and the boundary of the
intervals2 :
X
¦
©
1
2∂τ3Q + (2"τ + " E − 2)∂ E2Q − (
τi ∂E )∂τ ∂ E Q + ∂τ ∂ E Q, ∂ E2Q
= 0.
i
∂E
4
i
2
(11)
From the Pearcey to the Airy kernel
Define the rational functions
Φ(x, t; u) := −
h(x, t; u) :=
1
4(3u)3
ux
4
−
t
(3u)2
+
x − t2
3u
+
4
3
tx +
u
6
t2 x
(12)
2
(x + 6t ),
and the diagonal matrix
‚
S :=
eΦ(x,t 1 ;z
4
)−h(x,t 1 ;z 4 )
0
0
eΦ(x,t 2 ;z
4
)−h(x,t 2 ;z 4 )
Œ
.
We now state:
Proposition 2.1. Given a parameter z → 0, define two times τ1 and τ2 blowing up like z −6 and
depending on two parameters t 1 and t 2 ,
τi =
2
1
3z 6
(1 + 6t i z 4 ) + O(z 10 ).
Here and below, given two arbitrary functions f , g and a vector field X , we denote { f , g}X := gX ( f ) − f X (g)
1053
(13)
Given large ξi and η j , define new space variables x i and y j , using τi above,
ξi =
ηj =
2
27
2
27
(3τi )3/2 − (3τi )1/6 x i
(14)
3/2
(3τ j )
1/6
− (3τ j )
yj.
With this space-time rescaling, the following asymptotic expansion holds for the Pearcey kernel in powers
of z 4 , with polynomial coefficients in t 1 + t 2 ,
Æ
Æ
€
Š
−1
4
A
~ η
~ η
~xd
~ y + O(z 8 ),
S KP
(
ξ,
~
)
d
ξd
~
S
=
1
+
(t
+
t
)O
(z
)
K
(~
x
,
~
y
)
d
(15)
1
2
1
τ ,τ
t ,t
1
2
1
2
where O1 refers to a differential operator in ∂ /∂ x, ∂ /∂ y, with polynomial coefficients in x, y, t 1 ± t 2
and (t 1 − t 2 )−1 and varying with the four matrix entries of the matrix kernel.
On general grounds, due to the Fredholm determinant formula, this would give Theorem 1.1, but
−4/3
−2/3
with a much poorer estimate in (2), namely instead of O(τ1 ), the estimate would be O(τ1 ).
The existence of the PDE for the Fredholm determinant of the Pearcey process enables us to boot−2/3
−4/3
strap the O(τ1 ) estimate to O(τ1 ), without tears!
Also note that conjugating a kernel does not change its Fredholm determinant. Before proving
Proposition 2.1, we shall need the following identities:
Lemma 2.2. Introducing the polynomial
Ψ(x, s; ω) :=
3
ω4 + ω2 s2 − 4s(xω − ω3 ),
4
2
1
(16)
the Airy kernel (6) and the (double) integral part of the extended Airy kernel (5), at t 1 = s, t 2 = −s,
satisfy the following differential equations,
Ψ(x, s, −∂ x ) − Ψ( y, s, ∂ y ) + 4s K A (x, y)
(17)
1
= (x − y)(x + y + 6s2 )K A (x, y),
4
and
3 ∂
Ψ(x, s, −∂ x ) − Ψ( y, −s, ∂ y ) − s (∂ x − ∂ y ) K̃ A (x, y)
s,−s
2 ∂s
1
= (x − y)(x + y + 6s2 )K̃ A (x, y).
s,−s
4
(18)
Proof. The operator on the left hand side of (17) reads
Ψ((x, s, −∂ x ) − Ψ(( y, s, ∂ y ) + 4s
3
1
= 4s(1 + y∂ y − ∂ y3 + x∂ x − ∂ x3 ) + s2 (∂ x2 − ∂ y2 ) + (∂ x4 − ∂ y4 ).
2
4
(19)
Using the first representation (6) of the Airy kernel, one checks using the differential equation for
the Airy kernel, A00 (x) = xA(x) and thus A000 (x) = xA0 (x) + A(x) and A(iv) (x) = 2A0 (x) + x 2 A(x),
1054
and using differentiation by parts to establish the last equality,
( y∂ y − ∂ y3 ) + (x∂ x − ∂ x3 ) K A (x, y)
Z∞ d
= −
dz z
+ 2 A(x + z)A( y + z) = −K A (x, y).
dz
0
In order to take care of the other pieces in (19), one uses the second representation (6) of the kernel
K A (x, y), yielding
∂ x2 − ∂ y2 K A (x, y) = (x − y)K A (x, y)
∂ x4 − ∂ y4 K A (x, y) = (x 2 − y 2 )K A (x, y).
This establishes the first identity (17) of Lemma 2.2. The operator on the left hand side of the
second identity (18) reads:
3 ∂
Ψ(x, s, −∂ x ) − Ψ( y, −s, ∂ y ) − s (∂ x − ∂ y )
2 ∂s
1 4
3
3 ∂
= (∂ x −∂ y4 ) + s2 (∂ x2 −∂ y2 ) + 4s(x∂ x − ∂ x3 − y∂ y + ∂ y3 ) − s (∂ x −∂ y ),
4
2
2 ∂s
of which we will evaluate all the different terms acting on the integral. Notice at first that, since the
expression under the differentiation vanishes at 0 and ∞, one has
Z∞
Š
∂ € −2sz A
0=
ze
K (x + z, y + z)
dz
∂z
Z0∞
Z∞
dz e−2sz K A (x + z, y + z) −
=
zdz e−2sz A(x + z)A( y + z)
0
0
Z
∞
− 2s
zdz e−2sz K A (x + z, y + z),
0
Again, using the second representation (6) of the kernel K A (x, y), one checks
Z∞
A
∂ x − ∂ y K̃ (x, y) = −(x − y)
dz e−2sz K A (x + z, y + z)
s,−s
∂ x2
2
− ∂ y K̃
A
s,−s
0
(x, y) = (x − y)K̃
A
s,−s
(x, y),
and also
s
∂
∂s
∂ x −∂ y K̃
A
s,−s
(x, y) = 2s(x − y)
Z
∞
zdz e−2sz K A (x + z, y + z)
0
and, using the differential equation x∂ x − ∂ x3 A(x) = −A(x) and (20),
−2s x∂ x − ∂ x3 − y∂ y − ∂ y3 K̃ A (x, y)
s,−s
Z∞
zdz e−2sz K A (x + z, y + z)
= −2s(x − y)
0
1055
(20)
Using A(i v) (x) = 2A0 (x) + x 2 A(x) and the Darboux-Christoffel representation (6) of the Airy kernel,
one checks
∂ x4 − ∂ y4 K̃ A (x, y)
s,−s
Z∞
e−2sz (A(i v) (x + z)A( y + z) − A(x + z)A(iv) ( y + z))dz
=
0
= −2(x − y)
Z
+2(x − y)
∞
dz e−2sz K A (x + z, y + z) + (x 2 − y 2 )K̃ A (x, y)
Z0 ∞
s,−s
zdz e−2sz A(x + z)A( y + z).
0
Adding all these different pieces and using the expression for
Z∞
zdz e−2sz K A (x + z, y + z),
−2s(x − y)
0
given by (20), leads to the statement of Lemma 2.2.
For the sake of notational convenience in the proof below, set

p
p
P
P
Æ
 K11 (ξ1 , η1 ) dξ1 dη1 K12 (ξ1 , η2 ) dξ1 dη2
~ ~) d ξ
~ dη
KP
~=
τ1 ,τ2 (ξ, η

p
p
P
P
K21
(ξ2 , η1 ) dξ2 dη1 K22
(ξ2 , η2 ) dξ2 dη2




(21)
P
A
and similarly for the Airy kernel KA
t 1 t 2 ; the K̃i j and K̃i j refer, as before, to the (double) integral
part.
e A (x, y) and K
e A (x, y),
Proof of Proposition 2.1: Notice that acting with ∂ x and ∂ y on the kernels K
11
12
22
21
amounts to multiplication of the integrand of the kernels with −u and v respectively; i.e., u ↔ −∂ x
and v ↔ ∂ y .
Given the (small) parameter z ∈ R, consider the following (i, j)-dependent change of integration
variables (U, V ) 7→ (u, v) in the four Pearcey kernels KiPj in (21), namely
U=
1
3z 3
(1 + 3uz 4 )(1 + 3t i z 4 ),
V=
1
3z 3
(1 + 3vz 4 )(1 + 3t j z 4 ),
(22)
together with the changes of variables (ξ, η, τi , τ j ) 7→ (x, y, t i , t j ), in accordance with (13) and
(14),
1
1
τi = 6 (1 + 6t i z 4 ) + O(z 10 ),
τ j = 6 (1 + 6t j z 4 ) + O(z 10 )
3z
3z
(23)
2
2
(3τi )3/2 − (3τi )1/6 x,
η=
(3τ j )3/2 − (3τ j )1/6 y.
ξ=
27
27
P
P
P
To be precise, for Kkk
, one sets in the transformations above i = j = k, for K12
and K21
, one sets
i = 1, j = 2 and i = 2, j = 1 respectively. Then, remembering the expressions Φ and Ψ, defined in
1056
(12) and (16), the following estimate holds for small z:
4
e−Φ( y,t j ;z ) e−
e−Φ(x,t i ;z
4)
τjV2
V4
+ 2 −V η
4
4
e
− U4 +
τi U 2
−Uξ
2
=
=
e−z
4
e−z
4 Ψ(x,t
e
e
Ψ( y,t j ;v)
i ;u)
v3
e− 3 + y v
3
e
− u3 +xu
4
−z Ψ( y,t j ;∂ y )
−z 4 Ψ(x,t
i ;−∂ x )
(1+ t i O(z 8 )+ t j O(z 8 ))
(24)
3
e
− v3 + y v
e
− u3 +xu
3
(1+ t i O(z 8 )+ t j O(z 8 ));
replacing in the latter expression u and v by differentiations ∂ x and ∂ y has the advantage that
upon doubly integrating in u and v, the fraction of exponentials in front can be taken out of the
integration. Moreover, setting t 1 = t + s and t 2 = t − s, one also checks3 :
e−Φ( y,t j ;z
4
)
p
P
p
(τ
−
τ
;
ξ,
η)
dξdη
j
i
4
e−Φ(x,t i ;z )
‚
Œ
4
p
z
t
= pA (−2s, x, y) d x d y 1 +
(x − y)(x + y + 6s2 ) + r(x, y, s) + O(z 8 ) .
4
s
(25)
The multiplication by the quotient of the exponentials on the left hand side of the expressions above
will amount to a conjugation of the kernel, which will not change the Fredholm determinant. Setting
t 1 = t + s and t 2 = t − s, with t = 0, one finds for the non-exponential part in the kernel
dUdV
dξdη
V −U
p
d x d y dud v(1 + 27 (t i + t j )z 4 )
=
+ O(z 8 )
v + t j − u − t i + 3z 4 (v t j − ut i )
¨
 p
Š
i = 1, j = 1
d x d y dud v €
4
8

1 ± 4sz + O(z ) for

v−u
i = 2, j = 2


=
¨

p


i = 1, j = 2
 d x d y dud v 1 ± 3z 4 s(u+v) + O(z 8 ) for
v−u∓2s
v−u∓2s
i = 2, j = 1.
p
(26)
Along the same vein as the remark in the beginning of the proof of this Theorem, one notices
that multiplication of the integrand of the kernel K̃ A
(x, y) with the fraction, appearing in the last
12
21
formula of (26), amounts to an appropriate differentiation, to wit:
±
3
3s(u + v)
v − u ∓ 2s
←→
3 ∂
s (∂ y − ∂ x ).
2 ∂s
The precise expression for r(x, y, s) := (x − y)2 − 8s2 (x + y − 2s2 ) will be irrelevant in the sequel.
1057
(27)
So we have, using (24), (25), (26), and the differential equation for K A of Lemma 2.2,
e
e
−Φ( y,t 1 ;z 4 )
2
−Φ(x,t 1
2
p
K 11 (ξ, η) dξdη
p
− K A (x, y) d x d y
P
;z 4 )
22
t=0
p
€
Š
= z 4 ±4s + Ψ((x, ±s, −∂ x ) − Ψ(( y, ±s, ∂ y ) K A (x, y) d x d y + O(z 8 )
p
z4
=
(x − y)(x + y + 6s2 )K A (x, y) d x d y + O(z 8 );
4
(28)
the upper(lower)-indices correspond to the upper(lower)-signs. Using (27), and the differential
equation for K̃ A
of Lemma 2.2,
12
21
e
e
−Φ( y,t 2 ;z 4 )
1
p
p
A
K̃ 12 (ξ, η) dξdη − K̃ 12 (x, y) d x d y P
−Φ(x,t 1 ;z 4 )
2
21
21
t=0
3 ∂
p
A
4
s (∂ y −∂ x ) + Ψ(x, ±s, −∂ x ) − Ψ( y, ∓s, ∂ y ) K̃ 12 (x, y) d x d y =z
2 ∂s
21
t=0
(29)
+ O(z 8 )
p
z4
2
A
=
(x − y)(x + y + 6s )K̃ 12 (x, y) d x d y 4
21
+ O(z 8 ).
t=0
Also, using (25),
e−Φ( y,t 2 ;z
4
)
p
p
A
p (τ2 − τ1 ; ξ, η) dξdη − p (−2s, x, y) d x d y P
e−Φ(x,t 1 ;z )
p
z4
= (x − y)(x + y + 6s2 )pA (−2s, x, y) d x d y + O(z 8 ).
4
4
t=0
(30)
4
Since h( y, t i , z 4 ) − h(x, t j ; z 4 ) = − z4 (x − y)(x + y + 6s2 ) for arbitrary t i, j = t ± s at t = 0, and thus
e−h( y,t i ;z
e
4
)
−h(x,t j ;z 4 )
=1+
z4
4
(x − y)(x + y + 6s2 ) + O(z 8 )
(31)
Then the following approximations follow upon combining the three estimates above (28), (29),
(30) and using (31):
e
e
e
e
−Φ( y,t 1 ;z 4 )+h( y,t 1 ;z 4 )
2
−Φ(x,t 1
2
2
;z 4 )+h(x,t
1
2
;z 4 )
−Φ( y,t 2 ;z 4 )+h( y,t 2 ;z 4 )
1
−Φ(x,t 1
2
1
;z 4 )+h(x,t
e−Φ( y,t 2 ;z
4
)+h( y,t 2 ;z 4 )
e−Φ(x,t 1 ;z
4 )+h(x,t
1 ;z
4)
1
2
p
K 11 (ξ, η) dξdη
p
− K A (x, y) d x d y = O(z 8 )
P
22
p
K̃ 12 (ξ, η) dξdη
t=0
P
;z 4 )
21
t=0
p
pP (τ2 −τ1 ; ξ, η) dξdη
1058
p
− K̃ A
(x,
y)
d x d y = O(z 8 )
12
t=0
21
p
− pA (−2s, x, y) d x d y = O(z 8 ).
The reader is reminded that this estimate sofar is done at t = 0. It then follows for t 6= 0 from the
estimates (22), (23), (24), (25), (26), (27) that the right hand side of (15) is an asymptotic series
in z 4 with polynomial coefficients in t 1 + t 2 = 2t, proving the claim about O1 .
So far an important point was omitted, namely to analyze how the Pearcey contour turns in the
limit into an Airy contour; a detailed description of the possible contours was given in Figure 2.
The Pearcey-rays X 1 , with angles σ1 , σ10 ∈ (π/8, 3π/8) can be deformed into two acceptable rays
θ1 , θ10 ∈ (π/6, π/2) for the Γ< -Airy contour, since the two intervals have a non-empty intersection.
Also, since (3π/8, 5π/8)∩(π/6, π/2) 6= ;, the Pearcey Y -contour can be deformed into an acceptable
v ∈ Γ> -Airy contour. Again, since the admissible interval (π/8, 3π/8) for σ2 and σ20 in the X 2 3
Pearcey contour contains π/6, one ends up in the limit integrating the function eu /3 along a contour
of the form Γ> , which we may choose to have an angle of π/6 − δ with the negative real axis, for
3
small arbitrary δ > 0. In the sector Γ> , with 0 < θ2 = θ20 ≤ π/6 − δ, the function eu /3 decays
3
exponentially fast. Therefore, the contribution, due to the u-integration of eu /3 × (lower order
terms) over Γ> , having an angle of π/6−δ with the negative real axis, vanishes by applying Cauchy’s
Theorem in that sector. This ends the proof of Proposition 2.1.
3
From the Pearcey to the Airy statistics
This section concerns itself with proving Theorem 1.1.
Proof of Theorem 1.1: For any intervals E1 and E2 , the function
2
\
Q(τ1 , τ2 ; E1 , E2 ) = log P
!
(P (τi ) ∩ Ei = ;)
(32)
i=1
satisfies the Pearcey PDE (11). Reparametrizing, without loss of generality,
τ1 = τ + σ, τ2 = τ − σ, E1 = (ξ + η + µ, ξ + η − µ),
E2 = (ξ − η + ν, ξ − η − ν),
(33)
leads to a manageable PDE for the function
F (τ, σ; ξ, η, µ, ν) := Q(τ1 , τ2 ; E1 , E2 ),
(34)
namely the PDE:
2
∂ 3F
∂ τ3
+
1
4
2(σ
∂
∂σ
−τ
∂
∂ 2F
+ν
−2
∂τ
∂ξ
∂η
∂µ
∂ν
∂ ξ2
¨
«
∂ 3F
∂ 2F ∂ 2F
−σ
+
,
= 0.
∂ τ∂ ξ∂ η
∂ τ∂ ξ ∂ ξ2 ∂
)+ξ
∂
+η
∂
+µ
∂
∂
(35)
ξ
To go from the Pearcey PDE (11) to the PDE (35), one notices that the operators (10), appearing in
the Pearcey PDE (11), have simple expressions in terms of the variables τ, σ, ξ, η, µ, ν,
∂τQ =
∂F
∂τ
,
∂E Q =
∂F
∂ξ
,
"E Q = ξ
1059
∂
∂ξ
+η
∂
∂η
+µ
∂
∂µ
+ν
∂ ∂ν
F,
"τQ := σ
∂
∂σ
+τ
∂ ∂τ
X
F,
τi ∂E Q = τ
i
i
∂
+σ
∂ξ
∂ ∂η
F.
The change of variables, considered in (13) and (14), combined with a linear change of variables,
in parallel with (33),
¨
1
t1 = t + s
4
τi = 6 (1 + 6t i z ) with
t2 = t − s
3z
¨
(36)
2
Ẽ1 = (x + y + u, x + y − u)
3/2
1/6
(3τi ) − (3τi ) Ẽi with
Ei =
Ẽ2 = (x − y + v, x − y − v)
27
yields a z-dependent invertible map,
T : (t, s, x, y, u, v) 7→ (τ, σ, ξ, η, µ, ν),
(37)
and thus the function F in (34) leads to a new z-dependent function G:
F (τ, σ; ξ, η, µ, ν) = F (T (t, s, x, y, u, v)) =: G(t, s, x, y, u, v),
4
of which we compute, in principle, the series in z. From Proposition 2.1,
p it follows that the z -term
in the asymptotic expansion in (15) vanishes when t = 0; so, omitting d x d y, one has
S
for some kernel
KPτ ,τ S−1 = KAt ,t
1
1
2
2
+ z4
K1 +
∞
X
z 4i
Ki ,
with
K1 = t H,
2
H, with the Ki polynomial in t. Then defining
−1
L i := (I − KA
t ,t ) Ki ,
1
(38)
2
one finds:
G(t, s, x, y, u, v)
= F (τ, σ; ξ, η, µ, ν)
2
\
= log P
{P (τi ) ∩ Ei = ;}
(39)
i=1
KPτ ,τ )
= log det(I − KA
t ,t )
= log det(I −
1
2 E1 ×E2
where
− z 4 Tr L1 − z 8 Tr(L2 +
1
L2) − . . .
2 1
=: G0 (s; Ẽ1 , Ẽ2 ) − G1 (t, s; Ẽ1 , Ẽ2 )z 4 − G2 (t, s; Ẽ1 , Ẽ2 )z 8 + O(z 12 ),
1
2 Ẽ1 × Ẽ2
KAt ,t )
G0 (s; Ẽ1 , Ẽ2 ) = log det(I −
G1 (t, s; Ẽ1 , Ẽ2 ) = t Tr((I −
1
K
2 Ẽ1 × Ẽ2
A
−1
t 1 ,t 2 )
H)
,
(40)
Ẽ1 × Ẽ2
.
Note G0 (s; Ẽ1 , Ẽ2 ) is t-independent, since the Airy process is stationary. Then setting
F (τ, σ; ξ, η, µ, ν) = G(T −1 (τ, σ; ξ, η, µ, ν)) = G(t, s, x, y, u, v)
1060
into the PDE (35) yields, using differentiation by parts, a new PDE for G(t, s, x, y, u, v). Upon setting
the expansion (39) of G for small z,
G(t, s, x, y, u, v) = G0 (s; Ẽ1 , Ẽ2 ) − G1 (t, s; Ẽ1 , Ẽ2 )z 4 + O(z 8 ),
into this new PDE leads to
∂ 3 G0
‚
z
2
2s
∂ s∂ x 2
−
∂ 3 G1
Œ
+ O(z 6 ) = 0,
∂ t∂ x 2
(41)
3
with G1 being polynomial in t. From the term ∂∂ τF3 in the PDE (35), it would seem like the leading
term would be of order z −6 ; in fact, by explicit but non-enlightening calculations, there are two
consecutive cancellations, so that the first non-trivial term has order z 2 , thus leading to a simple
PDE connecting G1 with G0 . But since G0 is t-independent, from equation (41) we deduce
∂ ∂2
∂
G
−
2ts
G
(42)
1
0 =0
∂ t ∂ x2
∂s
From (40), one finds 4
G1 − 2ts
∂
∂s
G0 = t Tr((I −
=
X̀
K
−1
A
t 1 ,t 2 )
H)
Ẽ1 × Ẽ2
− 2s
∂
∂s
G0
t i ai (s; x, y, u, v),
1
which substituted back in (42), leads to the PDE’s for the ai , namely
∂ 2
ai (s; x, y, u, v) = 0,
∂x
(43)
implying
ai (s; x, y, u, v) = x bi (s; y, u, v) + ci (s; y, u, v).
(44)
Letting the intervals Ẽ1 and Ẽ2 go to ∞, while keeping their relative position 2 y fixed and widths
−2u and −2v fixed as well, is achieved by letting x → ∞, as follows from (33). Remember
t 1 , t 2 , Ẽ1 , Ẽ2 from (36). But in the limit x → ∞, the expressions G0 , namely
G0 (s; Ẽ1 , Ẽ2 ) = log det(I −
KAt ,t )
1
2 Ẽ1 × Ẽ2
tends to 0 exponentially fast using the exponential decay of the four components of the matrix Airy
kernel (5); note that the term pA (t, x, y) tends to 0 as well, when x and y tend to ∞, due to the
presence of e−t(x+ y)/2 . Next, we sketch the proof that
G1 (t, s; Ẽ1 , Ẽ2 ) = t Tr((I −
KAt ,t )−1H)
1
2
Ẽ1 × Ẽ2
tends to 0 exponentially fast, when x → ∞. Indeed, using the identity (R :=
the resolvent of the Airy kernel A
t ,t )
K
(I −
4
K
The series is finite as
A
−1
t 1 ,t 2 )
K
A
t 1 ,t 2
1
KAt ,t (I − KAt ,t )−1 is
1
2
1
2
2
H = H + KAt ,t (I − KAt ,t )−1H = H + KAt ,t (I + R)H,
1
2
1
2
depends on t 2 − t 1 = −2s, G0 is t–independent and
1061
1
2
H, by Prop. 2.1 is a polynomial in t.
one computes
KAt ,t )−1K1)
−1
= t Tr((I − KA
t ,t ) H)
+ t Tr(KA
= t Tr H
t ,t H̃)
G1 = Tr((I −
1
Ẽ1 × Ẽ2
2
1
Ẽ1 × Ẽ2
2
Ẽ1 × Ẽ2
1
2
Ẽ1 × Ẽ2
where
H
Ẽ1 × Ẽ2
KAt ,t H̃)
Ẽ1 × Ẽ2
Tr
Tr(
1
2
=
=
, with
2
X
Z
KAt ,t (I − KAt ,t )−1
1
2
1
2
K tAt (u, v)H̃ ii (v, u)dud v
i i
Ẽi × Ẽi
ZZ
K tAt (u, v)H̃21 (v, u)dud v
1 2
Ẽ1 × Ẽ2
+
R :=
H ii (u, u)du
i=1 Ẽi
2 ZZ
X
i=1
+
H̃ := (I + R)H,
ZZ
K tAt (u, v)H̃12 (v, u)dud v.
2 1
Ẽ2 × Ẽ1
H
Each of these integrals tend to 0 because each of them contains the Airy kernel and also because
is obtained by acting with differential operators on the Airy kernel, as explained in section 2. This
ends the proof that G1 (t, s; Ẽ1 , Ẽ2 ) → 0, when the Ẽi tend to ∞. This fact together with the form
(44) of the ai imply bi (s; y, u, v) = ci (s; y, u, v) = 0 for i ≥ 1, and thus ai = 0 for i ≥ 1, implying
G1 = 2ts
∂
∂s
G0 .
2
Summarizing, this implies that for Ei = 27
(3τi )3/2 − (3τi )1/6 Ẽi , substituting G1 = 2ts(∂ G0 /∂ s) into
(39),
(
)!
2
2
\
(3τi )3/2 €
P (τi ) − 27
Š
∩ − Ẽi = ;
log P
1/6
(3τ
)
i
i=1
= G0 (s; Ẽ1 , Ẽ2 ) − 2z 4 ts
∂ G0
+ O(z 8 )
∂s
= G0 (s − 2tsz 4 ; Ẽ1 , Ẽ2 ) + O(z 8 )
= log P
!
2
\
¦
©
A (t i (1 − t i z 4 )) ∩ (− Ẽi ) = ;
+ O(z 8 ),
i=1
in view of the change of variables given in (36), one has that s−2tsz 4 = 21 (t 1 (1−t 1 z 4 )−t 2 (1−t 2 z 4 )).
Then, substituting t i = ui (1 + ui z 4 ) into t i (1 − t i z 4 ) and τi = (1 + 6t i z 4 )/(3z 6 ) + O(z 10 ), as in (13),
yields
t i (1 − t i z 4 ) = ui − 2u3i z 8 − u4i z 12 and τi =
1
3z 6
+
2ui
z2
+ 2u2i z 2 + O(z 10 ) for i = 1, 2.
Then eliminating z between the two expressions above for τ1 and τ2 , by first expressing z as a series
in τ1 , yields (1) and (2), with t i replaced by ui . This ends the proof of Theorem 1.1.
1062
References
[1] M. Adler, M. Cafasso, and P. van Moerbeke. Fredholm determinants of general (1, p)–kernels
and reductions of non–linear integrable PDE’s. arXiv:1104.4268, 2011.
[2] Mark Adler, Nicolas Orantin, and Pierre van Moerbeke. Universality of the Pearcey process.
Physica D, 239:(924-941), 2010. MR2639611
[3] Mark Adler and Pierre van Moerbeke. PDEs for the joint distributions of the Dyson, Airy and
sine processes. Ann. Probab., 33(4):1326–1361, 2005. MR2150191
[4] Mark Adler and Pierre van Moerbeke. PDEs for the Gaussian ensemble with external source
and the Pearcey distribution. Comm. Pure Appl. Math., 60(9):1261–1292, 2007. MR2337504
[5] Alexander I. Aptekarev, Pavel M. Bleher, and Arno B. J. Kuijlaars. Large n limit of Gaussian random matrices with external source. II. Comm. Math. Phys., 259(2):367–389, 2005.
MR2172687
[6] M. Bertola and M. Cafasso. The Riemann-Hilbert approach to the transition between the gap
probabilities from the Pearcey to the Airy process. International Mathematics Research Notices,
doi: 10.1093/imrn/rnr066, 2011.
[7] P. M. Bleher and A. B. J. Kuijlaars. Random matrices with external source and multiple orthogonal polynomials. Int. Math. Res. Not., (3):109–129, 2004. MR2038771
[8] Pavel Bleher and Arno B. J. Kuijlaars. Large n limit of Gaussian random matrices with external
source. I. Comm. Math. Phys., 252(1-3):43–76, 2004. MR2103904
[9] Pavel M. Bleher and Arno B. J. Kuijlaars. Large n limit of Gaussian random matrices with external source. III. Double scaling limit. Comm. Math. Phys., 270(2):481–517, 2007. MR2276453
[10] Mark J. Bowick and Édouard Brézin. Universal scaling of the tail of the density of eigenvalues
in random matrix models. Phys. Lett. B, 268(1):21–28, 1991. MR1134369
[11] E. Brézin and S. Hikami. Correlations of nearby levels induced by a random potential. Nuclear
Phys. B, 479(3):697–706, 1996. MR1418841
[12] E. Brézin and S. Hikami. Level spacing of random matrices in an external source. Phys. Rev. E
(3), 58(6, part A):7176–7185, 1998. MR1662382
[13] Freeman J. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J.
Mathematical Phys., 3:1191–1198, 1962. MR0148397
[14] Kurt Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math.
Phys., 242(1-2):277–329, 2003. MR2018275
[15] Kurt Johansson. The arctic circle boundary and the Airy process. Ann. Probab., 33(1):1–30,
2005. MR2118857
[16] Gregory Moore. Matrix models of 2D gravity and isomonodromic deformation. In Random
surfaces and quantum gravity (Cargèse, 1990), volume 262 of NATO Adv. Sci. Inst. Ser. B Phys.,
pages 157–190. Plenum, New York, 1991. MR1214388
1063
[17] Andrei Okounkov and Nicolai Reshetikhin. Random skew plane partitions and the Pearcey
process. Comm. Math. Phys., 269(3):571–609, 2007. MR2276355
[18] Michael Prähofer and Herbert Spohn. Scale invariance of the PNG droplet and the Airy process.
J. Statist. Phys., 108(5-6):1071–1106, 2002. Dedicated to David Ruelle and Yasha Sinai on the
occasion of their 65th birthdays. MR1933446
[19] Craig A. Tracy and Harold Widom. A system of differential equations for the Airy process.
Electron. Comm. Probab., 8:93–98 (electronic), 2003. MR1987098
[20] Craig A. Tracy and Harold Widom. Differential equations for Dyson processes. Comm. Math.
Phys., 252(1-3):7–41, 2004. MR2103903
[21] Craig A. Tracy and Harold Widom. The Pearcey process. Comm. Math. Phys., 263(2):381–400,
2006. MR2207649
[22] P. Zinn-Justin. Random Hermitian matrices in an external field. Nuclear Phys. B, 497(3):725–
732, 1997. MR1463645
[23] P. Zinn-Justin. Universality of correlation functions of Hermitian random matrices in an external field. Comm. Math. Phys., 194(3):631–650, 1998. MR1631489
1064