From the Pearcey to the Airy process

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From the Pearcey to the Airy process

Mark Adler, Mattia Cafasso, Pierre van Moerbeke

To cite this version:

Mark Adler, Mattia Cafasso, Pierre van Moerbeke. From the Pearcey to the Airy process. Elec- tronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2011, 16, pp.1048 - 1064.

�10.1214/EJP.v16-898�. �hal-03031616�

El e c t ro nic

Journ a l of

Pr

ob a b il i t y

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 Uni- versity, 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 (MIS- GAM), a Marie Curie Grant (ENIGMA), Nato, FNRS and Francqui Foundation grants is gratefully acknowledged.

1 Introduction

Putting dynamics into random matrix models leads to finitely many nonintersecting Brownian mo- tions on R for the eigenvalues, as was discovered by Dyson

[

13

]

. Applying scaling limits to the ran- dom 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. Jo- hansson

[

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

p ( t; x, y ) : = 1

p πt e

^−(y−x)

2

t

,

all starting from the origin x = 0 at time t = 0 and such that

ⁿ

2

particles are forced to ±

p_n

2

at t = 1.

For very large n, the average mean density of particles has its support, for t ≤

¹₂

, on one interval centered about x = 0, and for

¹₂

< 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 fluctu- ations behave as the so-called Airy process, which describe the non-intersecting Brownian motions with space stretched by the customary GUE edge rescaling n

^1/6

and time rescaled by the factor n

^1/3

, 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 moving out of the cusp x =

₂₇²

(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 τ

i

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:

Theorem 1.1.

Given finite parameters t

₁

< t

₂

, let both τ

1

, τ

2

→ ∞ , such that τ

2

− τ

1

→ ∞ behaves in the following precise way:

τ

2

− τ

1

2 (t

2

− t

₁

) = ( 3 τ

1

)

^1/3

+ t

₂

− t

₁

( 3 τ

1

)

^1/3

+ 2t

₁

t

₂

3 τ

1

+ O τ

⁻₁^5/3

. (1)

Let also E

₁

, E

₂

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:

P

2

\

i=1

(

P (τ

i

) −

₂₇²

( 3 τ

i

)

^3/2

( 3 τ

i

)

^1/6

∩ − E

_i

= ;

)!

= P

2

\

i=1

A ( t

_i

) ∩ (− E

_i

) = ;

!

1 + O 1 τ

^4/3₁

!

.

(2)

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 multi- time 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 probabilities

¹

are given by the matrix Fredholm determinant for t

₁

< . . . < t

_`

,

P





`

\

i=1

(A (t

i

) ∩ E

_i

= ;)





= det

I − [χ

E_i

K

_t^A

it_j

χ

E_j

]

1≤i,j≤`

(3)

of the matrix kernel in ~ x = (x

1

, . . . , x

_`

) and ~ y = ( y

₁

, . . . , y

_`

) , denoted as follows:

K

^A_t1,...,t`

(~ x , ~ y )

p

d ~ x d ~ y

: =











K

_t^A

1,t₁

( x

₁

, y

₁

)

p

d x

₁

d y

₁

. . . K

_t^A

1,t_`

( x

₁

, y

_`

)

p

d x

₁

d y

_`

.. . .. .

K

_t^A

`,t₁

( x

_`

, y

₁

)

p

d x

_`

d y

₁

. . . K

_t^A

`,t<sub>`</sub>

( x

_`

, y

_`

)

p

d x

_`

d y

_`











,

(4)

where, for arbitrary t

_i

and t

_j

, the extended Airy kernel K

^A_t

i,t_j

(x , y) is given by (see Johansson

[

14, 15

]

)

K

_t^A

i,t_j

( x , y ) : = K ˜

^A_t

i,t_j

(x , y) − 1I (t

i

< t

_j

)p

^A

(t

j

− t

_i

, x , y ) , (5)

1χEis the indicator function for the setE

# of particles

t=1/2 t=1

x t

n/2 n/2

-p

n/2 p

n/2

Figure 1: The Pearcey process.

where

K ˜

_t^A

i,t_j

(x , y) : =

Z _∞

0

e

^{−λ(ti−tj)}

A ( x + λ)A ( y + λ)d λ

= 1

( 2 π i )

² Z

Γ_>

d v

Z

Γ_<

du e

^{−v3/3+y v}

e

^−u3/3+xu

1

(v + t

_j

) − (u + t

_i

)

p

^A

(t, x , y) : = 1 p 4 πt e

^t

3 12−(^x−y)²

4t −₂^t

(

^y+x

) , for t > 0.

The contour u ∈ Γ

_<

consists of two rays emanating from the origin with angles θ

1

and θ

₁⁰

with the positive real axis, and the contour v ∈ Γ

_>

also consists of two rays with angles θ

2

and θ

₂⁰

with the negative real axis, as indicated in Figure 2. As is well known, one may choose θ

1

, θ

2

, θ

₁⁰

, θ

₂⁰

= π/ 3.

In particular, for t

_i

= t

_j

, this is the customary Airy kernel K

^A_t

i,t_i

:= K

^A

(x , y ) :=

Z _∞

0

dλA( x + λ)A( y + λ) = A(x )A

⁰

( y ) − A( y )A

⁰

( x )

x − y . (6)

The

Pearcey process

is determinantal as well, for which the multi-time gap probability is given by the Fredholm determinant for τ

1

< . . . < τ

_`

,

P





`

\

i=1

(P (τ

i

) ∩ E

_i

= ;)





= det

1I − [χ

E_i

K

_τ^P

iτj

χ

E_j

]

1≤i,j≤`

(7)

for the

Pearcey matrix kernel

in ~ξ = (ξ

1

, . . . , ξ

_`

) and ~η = (η

1

, . . . , η

_`

) , denoted as follows:

K

^P_τ1,...,τ`

(~ξ , ~η)

Æ

d ~ξ d ~η

: =











K

_τ^P

1,τ1

(ξ

1

, η

1

)

p

dξ

1

d η

1

. . . K

_τ^P

1,τ`

(ξ

1

, η

_`

)

p

d ξ

1

dη

_`

.. . .. .

K

_τ^P

`,τ1

(ξ

_`

, η

1

)

p

d ξ

_`

dη

1

. . . K

_τ^P

`,τ`

(ξ

_`

, η

_`

)

p

dξ

_`

dη

_`











,

(8)

where for arbitrary τ

i

and τ

j

, (see Tracy-Widom

[

21

]

) K

_τ^P

i,τj

(ξ , η) = K ˜

_τ^P

i,τj

(ξ , η) − 1I (τ

i

< τ

j

) p

^P

(τ

j

− τ

i

; ξ , η) , (9) with

K ˜

_τ^P

i,τj

(ξ, η) := − 1

4π

² Z

X

d U

Z

Y

d V V − U

e

^−V

4

4+^{τj V}₂²−Vη

e

^−U

4

4+^{τi U}₂²−Uξ

p

^P

(τ ; ξ , η) : = 1

p 2πτ e

^−(ξ−η)

2

2τ

, for τ > 0,

where X and Y are the following contours: X = X

₁

∪ X

₂

consists of four rays emanating from the origin with angles σ

1

, σ

⁰₁

with the positive real axis and σ

2

, σ

⁰₂

with the negative real axis, as given in Figure 2. The contour Y consists of two rays emanating from the origin with angles τ , τ

⁰

with the negative real axis; it is customary to pick σ

1

= σ

2

= σ

₁⁰

= σ

⁰₂

= π/ 4 and τ = τ

⁰

= π/ 2.

p I

R

O

@

@

@

@

@

E E E E E E E EE

U ∈ X

₂

U ∈ X

₂

σ

⁰₂

σ

2

τ

⁰

τ

U ∈ X

₁

U ∈ X

₁

V ∈ Y V ∈ Y

σ

⁰₁

σ

1

@

@

@

@

@

π/8< σ1,σ2,σ⁰₁,σ⁰₂<3π/8 3π/8< τ,τ⁰<5π/8

Pearcey contour

p M

N A A A A A A

v ∈ Γ

_>

v ∈ Γ

_>

θ

₂⁰

θ

2

u ∈ Γ

_<

u ∈ Γ

_<

θ

₁⁰

θ

1

A A

A A

A A

π/6< θ1,θ2,θ₁⁰,θ₂⁰< π/2

Airy contour

Figure 2: Integration paths for the kernels.

In

[

4, 2

]

, it was shown that, given intervals

E

_i

= (ξ

⁽ⁱ⁾₁

, ξ

⁽ⁱ⁾₂

), the log of the probability

Q = Q (τ

1

, . . . , τ

_`

; E

₁

, . . . , E

_`

) : = log P





`

\

i=1

{P (τ

i

) ∩ E

_i

= ;}





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

i

, ∂

_Ei

: =

X

k

∂

∂ ξ

⁽ⁱ⁾_k

, ∂

_E

: =

X

i

∂

_Ei

"

_τ

: =

X

i

τ

i

∂

∂ τ

i

, "

E

: =

X

i

X

k

ξ

⁽ⁱ⁾_k

∂

∂ ξ

⁽ⁱ⁾_k

. (10)

Then Q satisfies the Pearcey partial differential equation in its arguments and the boundary of the intervals

²

:

2 ∂

_τ³

Q + 1

4 ( 2 "

_τ

+ "

E

− 2 )∂

_E²

Q − (

X

i

τ

i

∂

_Ei

)∂

_τ

∂

E

Q +

¦

∂

_τ

∂

E

Q, ∂

_E²

Q

©

∂E

= 0. (11)

2 From the Pearcey to the Airy kernel

Define the rational functions

Φ(x , t; u) : = − 1

4 ( 3u)

³

− t

( 3u)

²

+ x − t

²

3u + 4

3 t x + u 6 t

²

x h(x , t; u) : = ux

4 ( x + 6t

²

) ,

(12)

and the diagonal matrix S : =

‚

e

^{Φ(x,t1;z4)−h(x,t1;z4)}

0

0 e

^{Φ(x,t2;z4)−h(x,t2;z4)}

Œ

. We now state:

Proposition 2.1.

Given a parameter z → 0, define two times τ

1

and τ

2

blowing up like z

⁻⁶

and depending on two parameters t

₁

and t

₂

,

τ

i

= 1

3z

⁶

( 1 + 6t

_i

z

⁴

) + O(z

¹⁰

) . (13)

2Here and below, given two arbitrary functionsf,gand a vector fieldX, we denote{f,g}X:=g X(f)−f X(g)

Given large ξ

i

and η

j

, define new space variables x

_i

and y

_j

, using τ

i

above, ξ

i

= 2

27 ( 3 τ

i

)

^3/2

− ( 3 τ

i

)

^1/6

x

_i

η

j

= 2

27 ( 3 τ

j

)

^3/2

− ( 3 τ

j

)

^1/6

y

_j

.

(14)

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

,

S K

^P_τ1,τ2

(~ξ , ~η)

Æ

d ~ξd ~η S

⁻¹

=

€

1 + (t

1

+ t

₂

)O

1

(z

⁴

)

Š

K

^A_t1,t2

(~ x , ~ y )

Æ

~ d x ~ d y + O(z

⁸

) , (15) where O

₁

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 with a much poorer estimate in (2), namely instead of O (τ

^−4/3₁

) , the estimate would be O (τ

^−2/3₁

) . The existence of the PDE for the Fredholm determinant of the Pearcey process enables us to boot- strap the O(τ

⁻₁^2/3

) estimate to O(τ

⁻₁^4/3

) , 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) : = 1

4 ω

⁴

+ 3

2 ω

²

s

²

− 4s ( x ω − ω

³

) , (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 )

= 1

4 ( x − y )( x + y + 6s

²

) K

^A

( x , y ) ,

(17)

and

Ψ(x , s, −∂

x

) − Ψ(y, − s, ∂

y

) − 3 2 s ∂

∂ s (∂

x

− ∂

y

)

K ˜

^A

s,−s

( x , y )

= 1

4 (x − y )(x + y + 6s

²

) K ˜

^A

s,−s

(x , y ) .

(18)

Proof. The operator on the left hand side of (17) reads Ψ((x , s, −∂

_x

) − Ψ(( y, s, ∂

_y

) + 4s

= 4s ( 1 + y ∂

y

− ∂

_y³

+ x ∂

x

− ∂

_x³

) + 3

2 s

²

(∂

_x²

− ∂

_y²

) + 1

4 (∂

_x⁴

− ∂

_y⁴

) . (19)

Using the first representation (6) of the Airy kernel, one checks using the differential equation for

the Airy kernel, A

⁰⁰

( x ) = xA ( x ) and thus A

⁰⁰⁰

( x ) = xA

⁰

( x ) + A ( x ) and A

^{(i v)}

( x ) = 2A

⁰

( x ) + x

²

A ( x ) ,

and using differentiation by parts to establish the last equality, ( y ∂

y

− ∂

_y³

) + (x∂

x

− ∂

_x³

)

K

^A

( x, y )

= −

Z _∞

0

dz

z d dz + 2

A ( x + z ) A ( y + z ) = − K

^A

( x , y ) .

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:

Ψ( x , s, −∂

x

) − Ψ( y, − s, ∂

y

) − 3 2 s ∂

∂ s (∂

x

− ∂

y

)

= 1

4 (∂

_x⁴

−∂

_y⁴

) + 3

2 s

²

(∂

_x²

−∂

_y²

) + 4s(x ∂

_x

− ∂

_x³

− y ∂

_y

+ ∂

_y³

) − 3 2 s ∂

∂ s (∂

_x

−∂

_y

),

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

0 =

Z _∞

0

dz ∂

∂ z

€

ze

^−2sz

K

^A

(x + z, y + z)

Š

=

Z _∞

0

dz e

^−2sz

K

^A

( x + z, y + z ) −

Z _∞

0

zdz e

^−2sz

A ( x + z ) A ( y + z )

− 2s

Z _∞

0

zdz e

^−2sz

K

^A

(x + z, y + z) ,

(20)

Again, using the second representation (6) of the kernel K

^A

(x , y) , one checks

∂

x

− ∂

y

K ˜

^A

s,−s

( x , y ) = −( x − y )

Z _∞

0

dz e

^−2sz

K

^A

(x + z, y + z)

∂

_x²

− ∂

_y²

K ˜

^A

s,−s

( x , y ) = (x − y ) K ˜

^A

s,−s

( x , y ) , and also

s ∂

∂ s ∂

x

−∂

y

K ˜

^A

s,−s

( x , y ) = 2s( x − y )

Z _∞

0

zdz e

^−2sz

K

^A

(x + z, y + z) and, using the differential equation x ∂

x

− ∂

_x³

A ( x ) = − A ( x ) and (20),

− 2s

x ∂

x

− ∂

_x³

− y ∂

y

− ∂

_y³

K ˜

^A

s,−s

( x , y )

= − 2s(x − y )

Z _∞

0

zdz e

^−2sz

K

^A

( x + z, y + z)

Using A

^{(i v)}

( x ) = 2A

⁰

(x ) + x

²

A ( x ) and the Darboux-Christoffel representation (6) of the Airy kernel, one checks

∂

_x⁴

− ∂

_y⁴

K ˜

^A

s,−s

( x , y )

=

Z _∞

0

e

^−2sz

(A

^{(i v)}

(x + z)A( y + z) − A( x + z)A

^{(i v)}

( y + z))dz

= − 2 ( x − y )

Z _∞

0

dz e

^−2sz

K

^A

( x + z, y + z ) + ( x

²

− y

²

) K ˜

^A

s,−s

( x , y ) + 2 (x − y )

Z _∞

0

zdz e

^−2sz

A(x + z)A( y + z) . Adding all these different pieces and using the expression for

− 2s( x − y )

Z _∞

0

zdz e

^−2sz

K

^A

(x + z, y + z), given by (20), leads to the statement of Lemma 2.2.

For the sake of notational convenience in the proof below, set

K

^P_τ1,τ2

(~ξ , ~η)

Æ

d ~ξ d ~η =









K

₁₁^P

(ξ

1

, η

1

)

p

dξ

1

dη

1

K

₁₂^P

(ξ

1

, η

2

)

p

d ξ

1

dη

2

K

₂₁^P

(ξ

2

, η

1

)

p

d ξ

2

d η

1

K

₂₂^P

(ξ

2

, η

2

)

p

d ξ

2

d η

2









(21)

and similarly for the Airy kernel K

^A_t1t2

^{; the ˜ K}

_{i j}^P

^{and ˜ K}

_{i j}^A

refer, as before, to the (double) integral part.

Proof of Proposition 2.1: Notice that acting with ∂

x

and ∂

y

on the kernels K

e^A

11 22

( x , y ) and K

e^A

12 21

( x , y ) , 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 K

_{i j}^P

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 , τ

i

, τ

j

) 7→ ( x , y, t

_i

, t

_j

) , in accordance with (13) and (14),

τ

i

= 1

3z

⁶

( 1 + 6t

_i

z

⁴

) + O ( z

¹⁰

) , τ

j

= 1

3z

⁶

( 1 + 6t

_j

z

⁴

) + O ( z

¹⁰

) ξ = 2

27 ( 3 τ

i

)

^3/2

− ( 3 τ

i

)

^1/6

x, η = 2

27 ( 3 τ

j

)

^3/2

− ( 3 τ

j

)

^1/6

y.

(23)

To be precise, for K

_kk^P

, one sets in the transformations above i = j = k, for K

₁₂^P

and K

₂₁^P

, one sets

i = 1, j = 2 and i = 2, j = 1 respectively. Then, remembering the expressions Φ and Ψ , defined in

(12) and (16), the following estimate holds for small z:

e

^{−Φ(y,tj;z4)}

e

^{−Φ(x,ti;z4)}

e

^−V

4

4+^{τj V}₂²−Vη

e

^−U

4

4 +^{τi U}₂²−Uξ

= e

^{−z4Ψ(y,tj;v)}

e

^{−z4Ψ(x,ti;u)}

e

^−v

3 3+y v

e

^−u

3

3+xu

( 1 + t

_i

O ( z

⁸

)+ t

_j

O ( z

⁸

))

= e

^{−z4Ψ(y,tj;∂y)}

e

^{−z4Ψ(x,ti;−∂x)}

e

^−v

3 3+y v

e

^−u

3

3+xu

( 1 + t

_i

O(z

⁸

)+ t

_j

O(z

⁸

)) ;

(24)

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 checks

³

:

e

^{−Φ(y,tj;z4)}

e

^{−Φ(x,ti;z4)}

p

^P

(τ

_j

− τ

_i

; ξ, η)

p

dξdη

= p

^A

(− 2s, x , y )

p

d x d y

‚

1 + z

⁴

4

(x − y )(x + y + 6s

²

) + t

s r (x , y, s)

+ O(z

⁸

)

Œ

.

(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

p

dξd η d U d V V − U

=

p

d x d y dud v ( 1 +

⁷₂

( t

_i

+ t

_j

) z

⁴

)

v + t

_j

− u − t

_i

+ 3z

⁴

(v t

_j

− ut

_i

) + O(z

⁸

)

=











p

d x d y dud v v−u

€

1 ± 4sz

⁴

+ O(z

⁸

)

Š

for

¨

i = 1, j = 1 i = 2, j = 2 p

d x d y dud v

v−u∓2s

1 ±

^3z_v⁴₋^s(u+v)_u∓2s

+ O(z

⁸

) for

¨

i = 1, j = 2 i = 2, j = 1.

(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 ˜ K

^A

12 21

(x , y ) with the fraction, appearing in the last formula of (26), amounts to an appropriate differentiation, to wit:

± 3s(u + v)

v − u ∓ 2s ←→ 3 2 s ∂

∂ s (∂

y

− ∂

x

) . (27)

3The precise expression forr(x,y,s):= (x−y)²−8s²(x+y−2s²)will be irrelevant in the sequel.

So we have, using (24), (25), (26), and the differential equation for K

^A

of Lemma 2.2, e

^{−Φ(y,t12;z}

4)

e

^{−Φ(x,t12;z}

4)

K

^P₁₁

22

(ξ , η)

p

d ξdη

t=0

− K

^A

(x , y)

p

d x d y

= z

⁴€

± 4s + Ψ((x , ± s, −∂

x

) − Ψ((y, ± s, ∂

y

)

Š

K

^A

(x , y )

p

d x d y + O(z

⁸

)

= z

⁴

4 (x − y )(x + y + 6s

²

)K

^A

(x , y)

p

d x d y + O(z

⁸

);

(28)

the upper(lower)-indices correspond to the upper(lower)-signs. Using (27), and the differential equation for ˜ K

^A

12 21

of Lemma 2.2,

e

^{−Φ(y,t21;z}

4)

e

^{−Φ(x,t12;z}

4)

K ˜

^P

12 21

(ξ , η)

p

d ξdη − K ˜

^A

12 21

(x , y)

p

d x d y

t=0

= z

⁴

3 2 s ∂

∂ s (∂

y

−∂

x

) + Ψ( x , ± s, −∂

x

) − Ψ( y, ∓ s, ∂

y

) K ˜

^A

12 21

( x , y )

p

d x d y

t=0

+ O ( z

⁸

)

= z

⁴

4 (x − y )(x + y + 6s

²

) K ˜

^A₁₂

21

( x , y )

p

d x d y

t=0

+ O(z

⁸

) .

(29)

Also, using (25),

e

^{−Φ(y,t2;z4)}

e

^{−Φ(x,t1;z4)}

p

^P

(τ

2

− τ

1

; ξ , η)

p

d ξ d η − p

^A

(− 2s, x , y )

p

d x d y

t=0

= z

⁴

4 (x − y )(x + y + 6s

²

)p

^A

(− 2s, x , y )

p

d x d y + O(z

⁸

) .

(30)

Since h( y, t

_i

, z

⁴

) − h(x , t

_j

; z

⁴

) = −

^z₄⁴

( x − y)(x + y + 6s

²

) for arbitrary t

_i,j

= t ± s at t = 0, and thus e

^{−h(y,ti;z4)}

e

^{−h(x,tj;z4)}

= 1 + z

⁴

4 ( x − y )( x + y + 6s

²

) + O ( z

⁸

) (31) Then the following approximations follow upon combining the three estimates above (28), (29), (30) and using (31):

e

^{−Φ(y,t12;z}

4)+h(y,t1 2

;z⁴)

e

^{−Φ(x,t12;z}

4)+h(x,t1 2

;z⁴)

K

^P

11 22

(ξ , η)

p

d ξ d η

t=0

− K

^A

( x , y )

p

d x d y = O ( z

⁸

)

e

^{−Φ(y,t21;z}

4)+h(y,t2 1

;z⁴)

e

^{−Φ(x,t12;z}

4)+h(x,t1 2

;z⁴)

K ˜

^P

12 21

(ξ, η)

p

dξdη

t=0

− K ˜

^A

12 21

( x , y )

p

d x d y = O(z

⁸

) e

^{−Φ(y,t2;z4)+h(y,t2;z4)}

e

^{−Φ(x,t1;z4)+h(x,t1;z4)}

p

^P

(τ

2

−τ

1

; ξ , η)

p

d ξdη

_t=0

− p

^A

(− 2s, x , y )

p

d x d y = O(z

⁸

) .