Level spacing functions and non linear differential equations

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Submitted on 1 Jan 1993

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equations

G. Mahoux, M. Mehta

To cite this version:

G. Mahoux, M. Mehta. Level spacing functions and non linear differential equations. Journal de

Physique I, EDP Sciences, 1993, 3 (3), pp.697-715. �10.1051/jp1:1993100�. �jpa-00246751�

Classification Physics Abstracts

2.30 2.90 5.90

Level spacing functions and

_non

linear differential equations

G. Mahoux and M-L-

Mehta(*)

S-Ph.T-, C-E-

Saclay,(**)

F-91191 Gif-sur,Yvette Cedex, France

Abs tract The Fredholm determinant of the kernel sin K(z

-v)/K(z -v)

_on the finite interval

(-t, t),

_appears in the theory of random matrices and in _some other problems of mathematical physics. In _a previous article _we studied the functions

S(t), A(t)

and

B(t)

related to this

Fredholm determinant, and derived relations _among and diJferential equations for them. Here

we exploit these relations to deduce the _power series at t ₌ 0, and the asymptotic behaviour _at t ₌ cc, of the various level spacing functions of the random matrix theory.

1 Introduction.

The

theory

of random matrices

provides

_a reasonable

understanding

of the statistical

properties

of energy spectra ^of

heavy

nuclei and of

classically

chaotic systems. Three ensembles of random

matrices, orthogonal, unitary

and

symplectic,

_are

extensively studied; they

_are characterized

by

_a parameter

fl taking

values 1, ² and 4

respectively. Among

various

quantities

of interest,

one considers

Ep(n, t),

the

probability

that _a

randomly

chosen interval of

length 2t,

^measured in units of the local _mean

spacing,

contains

exactly

_n

eigenvalues [I].

The main

object

of this

paper is the

decay

of these

probabilities

when the

length

of the interval _goes to

infinity.

The

Ep(n,tl's

are

essentially

linear combinations of the derivatives of certain Fredholm determinants [2].

Thus,

for the

unitary

ensemble

(fl

₌

2),

_one has

E2(n,t)

₌

( (-))

^~

F(z,t)lz=i, (I.I)

where

~

F(z,t)

=

fl _{(I zli(t)), (1.2)}

I=o

is the Fredholm determinant of the

integral equation lf(x)

=

j~ Il(x,Y)f(Y)

_dY.

(1.3)

(*)

Member of C-N-R-S,, France

(**)

Laboratoire de la Direction des Sciences de la Matibre du Commissariat h

l'Energie

Atomique

with the kernel K

~~~~~~~

~~i~~ ~~

~~'~~

For the

orthogonal

_or

symplectic

ensembles, it is _more convenient [2] ^to work with the

following

linear combinations of El

(n, t)

~+~~'~~ ~~)~~~i)

+

El (2n

1,

t),

>

~

~~'~~

E-

(n, t)

=

Ei(2n, t)

+

Ei(2n

+ 1,

t),

_n 2 0,

(1.6)

so that

E4(n,t)

₌

iE+(n, 2t)

+

E-(n, 2t)1, (1.7)

E+(n,t)

₌

( (-))~ F+(z,t)lz=i, (1.8)

where

F+(z,t)

_are the Fredholm determinants of the

integral equation (1.3)

^with the kernels

K+:

1<+(z,

_{Y) =}

iK(x,

_Y) +

K(-z,

_Y)i

(1.9)

The

integral equation (1.3)

has been known for _a

long

time [3], ^{its solutions} are called

spheroidal functions; they

_are either _{even or}

odd;

the _even

(odd)

solutions

correspond

to the kernel

K+

(I<-).

Suppressing

the

dependence

_{on z,} let _us set

A(t)

₌

( _{(log F+(t)}

+

log

F-

(t)), (1.10)

B(t)

₌ ^~

(log F+(t) log F-(t)) (I-11)

In _a

previous

_paper [4] we studied these functions and derived the

following

relations

(A(t)

= 2

B~(t), (1.12)

((tA)

= z

(S(t)(2, (1.13)

((tB)

_{= z} lie

(S(t)~)

,

(l.14) 2~(tB)

= z Im

(S(t)~)

,

(l.15)

where the

complex

valued function

S(t)

₌

S(z,t)

is

completely

^determined

by

the non-linear differential

equation

~~

~ ~~

2)t~~

^{~~~ ~~~~} _' ^~~'~~~

(S*

^{is the}

complex conjugate

of

S), along

with the initial condition

S(0)

₌ 1.

We also

proved that,

_{as a} consequence of the above

equations, A(t)

and

B(t) satisfy

the _non linear second order diRerential

equations

(t ~j

+

2~)

₊

^16x~t~ (~) _{8~ (t~}

+

A)

⁼

^{0, (1.17)}

~ ~ ~

(t

⁺

^2@

_t ⁺

^{x~tB) 16B~} 1(t

⁺

B)

⁺

^4x~t~B2

^{= 0.}

^(1.18)

~

t

~

Eq.(1.17)

_was first derived

by limbo, Miwa,

Mori and Sato [5].

Note

that,

_{as a} consequence of

equations (1.12)

and

(1.13)

A =

z(S(~ 2tB~. (1.19)

Then,

_a careful examination of

equations (1.12)-(1.15)

and

(1.19)

shows that _{any one} of the three functions

S,

A and B,

completely

determines the two others.

Moreover,

each of these

functions

satisfy

_{a non} linear diRerential

equation, equations (1.16)-(1.18) respectively.

Note also that if S is _a solution of

equation (1.16),

then

iS,

-S and -iS _are others. This four-fold

degeneracy

is of _course removed

by

the initial condition at t ₌ 0.

Changing

S into iS

changes

the

sign

of B, and

interchanges F+

and F-.

Now,

in order to get ^{rid of} as many

clumsy

factors _{x as}

possible

in

coming formulae,

_we decide from _{now on} to shift from the variables _t and _z to the _new variables

r = xi and

(

=

~~

(l.20)

x

We also define _new functions

A(r)

₌

?A(t)

_and

_B(r)

=

?B(t), _(1.21)

and _{we rename}

F+((, r)

and

S((, r)

the functions

F+(z,t)

and

S(z,t),

which _we also write

F+(r)

and

S(r)

for short.

With these

definitions, equations (1.10)-(1.19)

become

A(r)

=

-) (1°g J~+(r) +1°g J~-(r)) (1.1°')

~(T)

=

) _{(1°g J~+(r)}

1°g ^F-

(r))

,

(I.ll')

)A(r)

=

B~(r), (1.12')

~A(r)

=

( (S(r)(~ r~~(r), (1.19')

) _(rB(r))

=

(

Re

(S(r)~)

,

(l.14')

~

2rB(r)

₌

(

Im

(S(r)~

,

(l.15')

(r ^~~

^{+ 2}

_~~

⁺

_4rB) ⁴⁸² 1(r ~~

⁺

)

+

4r~B2

= 0.

(1.18')

T T

~

T

~

Power series of

F((, r)

and

F+((, r)

_{near r =} 0 _are known [2].

They

_are of the form

~~~'T)

_# 1~

f

~~T)~

~

~~~

nj ajn

(-4r~)~

,

~

(l.22)

with the coefficients ajn

given

_as finite _sums

(see appendix B).

The

asymptotic

behaviours of

F(2/x, r)

and

F+(2/x, r) (I.e.

of

Ep(0, t), fl

₌ 1,2 and

4)

for

r » I _are also known [6]:

1°g?

^~

r)

=

log E~(0 t)

₌

log

J~+

~

r)

⁺

^logy _(~ r)

,

~i

~~~

'~ '

" K'

~°~~~ ~~'~)

^{~ ~°~}

_(~

^~

()

~ _~+ ~

~(~l)"Cn (r

+

()

^~

,

(l.24)

n=3

1)

~~ 24 ~

4

~°~~

~ ~'

(l.25)

Here, C is

3/2

times the derivative of the Riemann _zeta function evaluated at -I; ^its _approx- imate value is C ₌ -0.248131716.

Two _new facts _{came up}

recently. Firstly,

the

asymptotic

behaviours of

E+ (n, t)

and

E2(n, t)

for _a few small values of _{n were} derived

by

_a combination of the

properties

of

Toeplitz

determi- nants and the diRerential

equation (1.17) ii]. Secondly,

_a

thermodynamic

model of continuous Coulomb

charges

_was

exactly

solved in the limit oft » I; and _a combination of it with _equa- tions

(1.12)-(1.16)

_was used to express the Fredholm determinants _{as a}

product

of _a smooth factor and _an

oscillating factor,

the later

depending

on Jacobian

elliptic

functions [8]. Here _we revisit these

questions by exploiting equations (1.10')-(1.19').

The

present

paper is

organised

_as follows:

In section

2,

_we derive the

Taylor expansions

of the Fredholm determinants _{at r}

= 0. In

section

3,

_{we express} the real and

imaginary

parts of

S(r)

_as well _as

Jt(r)

and

B(r)

in terms of the fifth Painlevd transcendents. The function

B(r)

_can also be

expressed

in terms of _a third

Painlevd transcendent. In section 4, _we derive _an

expansion

of various functions in powers of

(( 2/x).

In the

asymptotic region

_r » I, this method is

simpler

and faster than the earlier _ones, except for _one defect: it leaves undetermined the constants c+, which appear in

log F+((, r)

_as

integration

constants. To fix

them,

_one must _use the initial conditions

F+((, 0)

₌ 1, _a difficult

exercise which _we leave out here.

One _more undetermined constant, ⁱⁿ

S((, r),

is calculable _{once one} knows the solution of the standard

problem

^of

connecting

the behaviour at _{r =} 0 of _a Painlevd transcendent _to its

behaviour at _{r =} cc, a

problem

which _can be solved

by using

the method of "isomonodromic deformations" devised

by

^the

japanese

school of M. Sato.

Unfortunately,

it has not yet ^been solved for all solutions of the fifth Painlevd

equation,

and

particularly

not for those of interest here. We

hope

to come back _to this

question

in _{a near} future. Here _we deduce the value of the above mentioned undetermined _constant from other _sources.

In

appendix A,

_we find that the dominant term of

B(r)

for _r » I is _a Jacobian

elliptic

sine function.

However,

since the initial conditions _{at r}

= 0 _are not taken into account, two constants remain undetermined. In

appendix

B, we list _a few of the

expansion

coefficients ajn of

equation (1.22).

In

appendix C,

_we

give asymptotic expansions

of _a few coefficient functions

appearing

in section 4. In

appendix D,

_we tabulate _a few of the _cn

appearing

in

equation (1.24),

and derive

numerically

for these coefficients _a

large

_n

expansion

which suggests ^that

the functions

F+(2 lx, r)

_are Borel summable at

infinity.

2. Power series of

F+((,r)

for _r «1.

In this brief section _{we use}

equations (1.10')-(1.16')

to derive the _power series

expansions

around _{r =} 0 of

S((, r), B((, r), Jt((, r)

and

F+((, r).

We obtain

successively

~~<,

~~ = i +

<r

+

<2 II ^r~

+

<~ It)

_~~

+

~~ '~~

^~

~~

^{~~ ~}

~~~~

~ i

(r ^r~

⁺

)(r~

⁺

B((, T)

_"

(

+

~~~ ~~ ~

~~ _~

~~ ~~

~~

(2.2)

+

(~

~(~

⁺

(~)

~~ ~

Jt((, r)

~

=

( j)~ _~~~~ ^{~~~ ~)~}

^{~~ ~}

^{~~ ~~~}

^~~

(2.3)

J~~

(<, r)

₌ (r +

(r~

~

(~~~

~

~25 ~~~

~

)~

~~~

~~~~~~

_~~

~

.~~

~

(( r)

=

(r~

₊

((T~ ^~05~~~

^~

_255~~~~

^~

(2.6)

~~ ⁱ

r

+

~(~r~ ~(~~~

⁺

^~05~~~~

^~

The

integration

constants in

log F+

_are determined from the initial conditions

F+((, 0)

₌ ^1.

The last three

equations

_are in agreement with the known

general expressions

recalled in

appendix

B.

3. Fifth and third Painlev4 transcendents.

In this section _we will show that

(I)

The real part ^of

S(r)

is

expressed

in _terms of _a fifth Painlevd transcendent

(P5) (here

^and

in what follows _a

prime

denotes the derivative with respect to

r),

Y" "

()

+ Y'~ + ~~

~~~~

_~YY

+

))

₊

7)

+

b~ ) ^~~~, _(3.I)

witho=-fl=1/32,7"0,b=-2,andnearr=0,

(2) _The part _of

S(r)

_is in ^rms _of

a

P5 _{with the}

same lues

y,(r)

₌ -1 2

((r)~'~ 2(r 4((r)~'~ 6(~r~

₊

O(r~'~). (3.3)

Any

_one ^{of the} above two P5's _can be

simply expressed

in _terms of the other.

(3)

^{The function}

Jt(r)

is

expressed

in terms of _a P5,

Eq.(3.I),

with either _{a =}

1/2, fl

= 7 = 0,

b =

8,

and _{near r =} 0,

Yai = -<r

<~r~ <~

l<I ^r~

⁺

^{°(r~) (3.4)}

orwitha=fl=0,7=-4i,b=8,andnearr=0,

y~~ = 4ir + 4

(it I) ^r~

+

16(r~

₊

O(r~). (3.5)

This latter _{case was} studied in references [5] and

ill].

(4)

^{The function}

B(r) (which

is almost the

imaginary

part ^of

S~ (r))

is

expressed

in terms of _a

P5, equation (3.I),

with _o

= =

0, fl

₌

-1/8,

₇

" 1; ^and near r ₌

0,

Ybi =

(2r)~'~

₊ 4r

<

+

(2r)~'~

4

<~

1°) ^r~

⁺

°(r"~). _(3.6)

It is also

expressed

in terms of _a P3

Y" "

~ ~

+ ~~ ~

~

+ 7Y~ +

~, (3.7)

witha=fl=-7=b=-I,andnearr=0,

y~~ = + +

<2

))

r +

o(r2). (3.8)

To start

with,

consider

equation (1.16').

Let _us prove that its

general

solution is _a homc-

graphic

function of _a P5.

Writing

S ₌

(R

+

it) /vf,

where R and I _are real, and

separating

the real and

imaginary

parts, _we get

R ₌

~'~

~,

I =

f~ _(3.9)

This

pair

of

equations

_expresses R in _terms of I and

I',

and I in _terms of R and R'.

Elimination of I

gives

R"

=

~~~ ^R'~ ~(~

^~ +

~(R~ _{r). (3.10)}

r r r r

Elimination of R

gives

^the _same

equation

for I.

equation (3.10)

has

singular points

at

+@,

and

cc. To

displace

them to

0,

^and _cc we set yr =

(R+@) /(R-@),

and _y;

=

(I+jfl/(I- jfl,

then

yr(r)

and

yi(r) satisfy

the fifth Painlevd

equation (3.I)

with the parameters a =

-fl

=

1/32,

_{7 =} 0 and b

= -2. Near the

origin

_{r =} 0,

S(r)

is

given by equation (2.I),

which

implies

that for _{r <}

I, yr(r)

and

y;(r)

_are

given by equations (3.2)

and

(3.3).

We _now turn to

equation (1.17').

Its solution is also _a

homographic

function of _a P5. To

see

this,

note that it is almost

homogeneous

in Jt. Therefore let [9]

Jt ₌

e~~,

_{u =}

rw', (3.ll)

and for convenience let [10]

~

v = u-1- ^~"

(3.12)

u

Then

A'

=

-uA/r, (rAl'

₌

^rA'+

A

=

-(u I)A, (3.13)

(rA)"

=

rA"

₊

2A'

=

(u'A

₊

(u I)A')

₌

"~A. _(3.14)

r

Disregarding

the _non-zero

factors, equation (I. Ii')

_now reads

(~ i)2

v~ ₊

16r~

_{+ 4r A}

= 0.

(3.15)

u

The derivative of this last

equation gives

vv' ₊ 16r + 2r ~'~

i

^~~~

(- l

+

t

+

ii

A

= 0.

(3.16)

Eliminating

Jt between the last two

equations,

we get

vv'+

16r (v~ ⁺

16r~) (-~

+

~'~ i)

= 0.

(3.17)

'~ '~ r

Using equation (3.12),

_one finds

by elementary algebraic manipulations

that _v factorizes in the left hand side of

equation (3.17).

The solution _{v =} 0

gives

_{u =} I

/(I cr),

^with c an

integration

constant.

However,

this cannot

correspond

to the value

ofJt(r)

at r = 0. The other

possibility

~

~'~ _~

~~

~

u

~'

l

~2r ~) ^~~~

^~

^~'

^~~'~~~

or

using equation (3.12)

~ ~

v~

₊

ii~2'

^~~'~~~

Equation (3.12)

expresses v in _terms of _~ and

~',

while

equation (3.19)

_{expresses ~} in _terms of

v and v'.

On

writing

_v in terms of _{~ e yai, one} obtains the Painlevd

equation

of the fifth

kind, equation (3.I),

with the parameters a =

1/2, fl

_{= 7 =} 0 and b

= 8. And _on

writing

_u in terms of _{v, one} obtains

~"

v2

~16r2

^~'~

~~

+

~~~

v2

/)6r2

^~

^~2~

~~~ ^~ ~~~~~ ~~'~~~

Setting

_v

=

4irf

_we get

<" =

fi<'~ [

+

~(<~

i)

4<(<~

i). (3.21)

This last

equation

has

singular points

at I, I and _cc. As

before,

_one sets

f

₌

(ya2+1)/(ya2-1),

then _ya2 satisfies the fifth Painlevd

equation (3.I)

with the parameters _{a =}

fl

₌

0,

_{7 =} -4i and b

= 8. This

equation

has been

investigated

in references [5] and

[11].

Near _r

= 0, Jt is

given by equation (2.3),

which

implies

that _yai and ya2 are

given by equations (3A)

and

(3.5).

Finally,

consider

equation (1.18').

We write it _as

b" + 4b ₌

@,

_b

+ 2rB.

(3.22)

To _see that this is _an

algebraic

transform of _a Painlevd

transcendent,

we follow Bureau

[12],

and consider the

pair

of

equations

~

~b

~~r'

^{~~ ~' ~}

~'

~~~~~

This

pair

is like

equations (3.9).

If _we eliminate _{~b, we}

get equation (3.22)

and if _we eliminate b, we get

~"

~(

^~

~b

2r) ~~ ~b

~2r~

^~

~~~ ^~~~'

^~~'~~~

To put this last

equation

in the canonical form, let

~ =

'~

j~~,

~b =

-j, _(3.25)

so that

~

~"' 2~a

~

~a

~

~"~

~r2j~

~ ~~" ~~'~~~

This is almost the fifth Painlevd

equation.

To put ^{it in standard}

form,

take

r~/2

_as the

new

independent

variable.

Eq.(3.26)

transforms into the fifth Painlevd

equation (3.I)

with

a = b

=

0, fl

₌

-1/8

and ₇

= 1.

As b

=

0,

_{we can} transform it into _a third Painlevd

equation (3.7).

To do

it,

_we follow Gromak

[13],

^{and consider the}

pair

of

equations

~~

q'+q~+1'

^~~

T~2'-~2+1'

^~~'~~~

Eliminating

_{q, we get}

equation (3.26)

for _~a; and

eliminating

_{~a, we get} for

J~ the third Painlevd

equation

in its standard form

(3.7),

with _{a =}

fl

= -7 = b

= -1.

4.

Asymptotic expansions

for

large positive

_r.

Any

_one of the functions

S((, r), Jt((, r)

_or

B((, r)

is

completely

determined

by

the diRerential

equation

it

satisfies,

and initial conditions at _{r =} 0. In

particular,

its

asymptitic

behaviour at r = cc is also

completely

determined. But

deriving

this

asymptotic

behaviour is _a difficult

problem,

^which

^implies connecting

the two

points

_{r =} 0 and _r

= cc. We do not know yet how

to solve it for the _one

parameter

^{classes of solutions defined}

by

the above functions.

So,

what _we do is to

postulate

a

particular

series

expansion

for

S((, r)

around _{r = cc}

(see (4A) below),

^and we find that the

resulting

series for

log F+

coincides with

Dyson's asymptotic

expansion

^of

log F+

at

(

=

2/x. Then,

_we

expand S((, r)

around

(

₌

2/x.

More

precisely,

_we will show that, ^if one writes the

Taylor expansions

in

(,

around

(

₌

2/x,

of the functions

S,

Jt and B _as follows

~

C° ~ ~2r ^"

S((> ~)

_"

2/~

(° (( p) ~j _{Sn(T)> (4.1)}

~=o

A(<,

_{T) #} 2

_io ~

^Y

^< )) jj

^{~~ ~}

_{An(T), (4.2)}

B((>

T) _" ²

~

^Y

(( )) jj

En

(T), (4.3)

~o

~~ ~

where _a is _a constant,

then,

for

r

positive

and » I, the coefficient functions

Sn(r), Jtn(r)

and

Bn(r)

have

asymptotic expansions

in powers of r~~ A few terms of these

expansions

for low values of _{n are} listed in

appendix

C.

To derive these

expansions,

_we first look for _a solution of

Eq.(1.16')

of the form

r'~jsjr~i,

co

_(4.4)

j=o

with 7 and

sj's complex

constants. It turns out

that,

_up to the four-fold

degeneracy

mentioned in the

introduction,

the solution is

unique

and

given by

the

right

hand side of

equation (Cl).

From this

particular solution, using successively Eqs.(1.15'), (1.19'), (1.10')

and

(I.ll'),

_we

construct the

corresponding

functions

Jt,

B and

log

F+. The first two of them _are

given by

equations (C6)

and

(Cll).

As for the last _one, it coincides with the

asymptotic expansion

of

log F+(2/x, r)

found

by Dyson

_[6],

equation (1.24). Thus,

the above

particular

solution is

indeed the

asymptotic expansion

of

S(2/x, r).

Let _us make two remarks. When

calculating log F+(I, t)

with

equations (1.10')

and

(I.ll'),

one has to

perform integrations,

and

then,

in order _to determine the

integration

constants, one

has to _use the initial conditions

log F+(1,0)

₌ 0. This determination is _a difficult

problem,

which

we do _not solve here. Its solution has been

given

first

by Dyson, using

the

theory

of inverse

scattering

[6].

As _a second remark, note that the

right

hand side of

equation (Cl),

divided

by (I

+

I),

has real and

imaginary

parts ^which are

respectively

_even and odd in _{r, a} property which is not shared

by S(2/x, r) (see equation (2.I)).

This is _a

sign

that in the

large

_r

asymptotic expansion

of

S(2/x, r), (probably exponentially)

sub-dominant terms _are not _seen.

Next,

let

us

expand S((, r)

_as follows

S((,

_{T) =}

) ^f _{(( ))}

^~

_iRn(T)

+

iIn(r)1, (4.5)

where Rn ^and In are real functions of _r, and let _us define the two component ^column

#n(r)

₌

(j))() (4.6)

Then, equation (I. lo') implies

that in

(r)

satisfies for _n > I the linear diRerential

equation

~~~

=

fi4#n

+

tin, (4.7)

where rid is the 2 _x 2 matrix

fiq~~~ 2RoIj/r ^Rl/T

^l

l

Io IT -2RoIo/r

~

l 13

~

l

~ l

~

=

4r2

64r(

^2r

1(r3 jjr4

,

(4 8)

2r ^~

l/r3

^~ _16r4 ^{~ ~ ~}

4r2 ^~ 64r4 ^~

and where the

inhomogeneous

term ~bn is _a function of

lo, Ii, In-

_I

For _{n =} I, the

equation

is

homogeneous,

~bi " 0. Two

linearly independent particulir

solutions _are

~

(Re ^Si(T)

^~~'~~

#t(T)

= e ^~

Im

si(T)

^'

Im

s~(-r)

(4.1°) ii (r)

₌ ^e~~~

(Re _Si(-r)

where

Si(r)

is

given by equation (C2).

The second _one has _a sub-dominant behaviour at

infinity

_as

compared

with the first _one. So _we write

ji(r)

₌ 2 _a

jt(r), (4.ii)

where

a is _a constant

which,

_{once more,} has to be determined

by using

^the initial condition _at

the

origin, Si (0)

₌

(see equation (2.I)). Although

the

problem

is

linear,

it is

complicated by

the fact that the

equation

involves the solution So of _{a non} linear

(Painlevd) equation.

For _n

=

2,

one has

2RoRiIi

+

IOR(

(4 12)

'~2

i _-2IoRiIi RoIi

This

inhomogeneous

term increases at

infinity

like e~~.

Consequently,

_any solution of

equation (4.7)

for _{n =} 2 also increases _at

infinity

like e~~ Since the solutions of the

homogeneous equation

increase at most like e~~, ^{the dominant} terms in the

asymptotic expansion

of

#2

are

independent

^{of the} initial condition at the

origin.

^Since we are interested

only

in the

exponentially

dominant terms, _no new unknown _{constant appears}

here,

and _one obtains

j~(~)

₌ ~~y2

~~~ l~e 1~2(T)

~ ~~)

@

^Im

52(r)

'

where

52(r)

is

given by equation (C3).

The _same

phenomenon happens

at each further step _{n =}

3,4,..

of the

procedure.

This

readily

leads to the

expansion (4.I),

and the

expressions given

in

equations (C2)

to

(C5).

The calculation of the

asymptotic expansions (4.2), (4.3)

and those in

appendix

C is then

straightforward,

_as well _as that of the various

probabilities

defined in the introduction. _Note that all the

integration

constants that _appear when

calculating G+n,

for _n >

I,

with the _use of

equations (1.10')

and

(I. II'),

_are irrelevant sub-dominant _terms.

To find the unknown constant _{a we}

proceed

_as follows

[14].

^{It is known} [15] ^that

,j+i

I

lj

_cs

j2~~+~P+~'~e~~~~, (4.14)

J.

for

j

finite and t » I.

Substituting

it in the dominant terms

E+

_(n>

t)

~

fl

_~2>

,

E- (n>

t)

~

jf

_~2>+1

(~ i~~

E+(0, t)

~

l 12J

E-(0,t)

~

l

12j+1

>= >=

gives

~+(~'~)

c~

l(2j)jl,-n~2-n(3n-1)t-n(2n-1)/2~2«ni

_{~~ ~~~}

E+(0,t)

J"

~-(~,l)

~

fl(~j

_{~ ~)j}

,-n(n+1)~-n(3n+2)~-n(2n+1)/2 ~2«nt

_{~ ~~)}

E-(0,t)

~

J*

The

expansions (C16), (C17)-(C24)

and

equation (1.8)

will all agree with the above if _we choose the constant _{a as}

" "

~ (4.18)

With this

value,

all _our

expansions

_agree with those of

Basor,

lYacy ^{and Widom} [7].

From the

expansions

of

Sj (r)

listed in

appendix C,

_we observe that the first _{terms are}

given by

'~~~

~

~l6r~

^~

~~~~il~)2

_~~~

~ ~~~~~ ^~

~~~)~~(~~~~

~~~~

~

+

A _(i

+

~llil~

+

~~~~~~lllll~~

^~°°

+

(4.19)

This formula

gives

the _correct coefficient of r-k

up ^to k ₌

j

in the real part, and _up to k ₌

2j

in the

imaginary

part.

We observe also that

A>

(r)

_{+ B>}

(r)

₌ 2 Re

S>(r)

+ O

(r~~>)

,

(4.20) Jtj(r) Bj (r)

₌ 2 Im

Sj(r)

+ O

(r~~i)

,

(4.21)

We checked

equation (4.20)

for

j

< 8 and

equation (4.21)

for

j

< 5.

Formulae similar to

equation (4.19)

_can be written for the coefficient of

r~k

_in

G+j(r),

£+j(r)

and

£j(r).

5. Conclusion.

Our

objective

_was to derive the

large

t

asymptotic

behaviour of the

probabilities Ep(n,t).

So

far,

_we did not

explicitely complete

this _program. To

proceed,

_one should _use the

following

equations

El(o,I)

"

E+(°,1), E2(o,I)

"

E+(°>~)E-(°>1), (5'i)

E4(0, t)

=

(E+(0, 2t)

+ E-

(0, 2t))

,

(5.2)

E+(°,t)

₌

F+(I,t)

₌

?+(2/K, r), (5.3)

which first determine the

asymptotic

behaviours of

Ep(0,t)

for

fl

₌ 1, 2 and

4,

in terms of those of

F+(2/x,r).

These latter _are

given by equation (1.24)

with the cm's tabulated in

appendix

D. Then the

asymptotic

^behaviours of

Ep(n,t)

_are

given by Eqs.(1.5)-(1.7), (1.24)

and

(C25)-(C38);

recall that _r

= xi and

(

=

2z/x.

As

compared

with

previous methods,

_{ours seems} to

provide

_more

easily

successive _terms in

large

t

asymptotic expansions.

We

just

need

algebraic

calculations which _can be

performed routinely by

software systems of

symbolic programming

like MATHEMATICA _or AMP. This enables _{us to}

investigate numerically

for

large

_n, the behaviour of the

n'~

coefficient of various

asymptotic expansions

in t. As

typical examples,

let _us write the

asymptotic expansions (Cl), (C6)

and

(Cll)

_as follows

So(r)

_"

)

^~

_~(Pn

+

i~n) j

,

(5.4)

~0

co

Ao(r)

₌

)

₊

~ @

,

(5.5)

n=o

80(r)

₌

~j ^~ _{~() (5-6)}

r

Then,

for

large

_n

pn =

iii

^"

exp

j-c

+

) j

+

°(j)1, (5.7)

~"

~~~~

^{~~~ ~}

In ^~2

^~

~/3~~

'

~~'~~

~"

~~e

~

~~~ ~

In ^~2

^~

^~~ /3

'

~~'~~

bn =

iii

^~

exp

j-c

+

)

+

o(j)), _(5.io)

where the constant _c is

c = 0.451582705289.

(5.ll)

The coefficients _cn of

equation (1.24)

have two similar

large

_n

expansions,

_one for _{even n} and another _one for odd _n

~~

~~

^~ ~~~

~~

^~~'~~~

k=0

The constant term in the exponential does not

depend

_on the

parity

of _n

~)

= -2.144729885849.

(5.13)

while

~)

= -2.763930691.

,

~)

= -4.994470.

,

(n even),

~ ~

~/

= +0.930597358.

,

Kp = -2.394586.

,

(n odd).

It is not clear

numerically

whether these

~)

are rational.

It is to be noticed that the _same dominant factor

(n/2e)~(l/n)

_appears

everywhere.

The above formulae suggest that

(at least)

the functions

So, Jto,

So and F+ _are Borel summable at r = cc. If it is _so, their Borel transforms _are

analytic

for

(r(

> 2.

The

problem

of

connecting

the behaviour at _r

= 0 and

r = cc of _any of the fifth _or the third Painlevd transcendents encountered in section 3 remains _open.

Appendix

A.

A. Solution of

equation (1.18)

for

large

t.

By

_a

comparison

with the continuum

model,

Dyson showed [8] ^{that for t} » I and

ii

_z-j « I,

B(t)

is

nearly

_a Jacobian

elliptic

function. We _{can recover} this result _as follows. Let _us write

Eq.(1.18)

_as

B"

+

4x~B

₊

~~')

~

=

16B~ (B'

^{+ ~}

~

+

x~B~)

(Al)

t t

If _we

ignore

the terms in

Ill,

then

~

= +48,

(A2)

or

d

(~)

= +d

(28~) (A3)

On

integration

this

gives

B'~

₊ 4x~B~

= 4

(B~

+

c)~

,

(A4)

with _{c an}

integration

constant. As B'

equals

the _square root of _a fourth

degree polynomial

in

B,

B is _a Jacobian

elliptic

function. Let _us look for _a solution of the form _a

sn(u, I),

where _sn is the

elliptic

sine

function,

_a and I _are constants, ^and u is _a function oft to be determined.

Then

B'~

=

a~u'~(I sn~B)(I ^l~ sn~B)

j~2 j~2

=

a~u'~

_I

m I

l~m

,

(A5)

a a

or

B'~

₊ 4x~B~

4(B~

+ c)~ =

a~u'~

4c~ ₊ B~

(4x~

8c

u'~(

I +

l~))

~2 12

+B~

_-4

(A6)

a

Choose _{a, u} and I _so that the coefficients _on the

right

hand side of

equation (A6)

_{are zero;}

This

gives

u'

=

)

=

~, u'~(i

₊

i~)

= 4K~ 8C

(A7)

or

'~'

l~l'

~

~ l~l'

_~~~~

Hence

~~~~

l~l

~~

l~~~l

~

'~~

~~~~

is the

general

solution of

equation (A2);

b and I _are

integration

constants. This

gives

the solution of

equation (Al)

to the doiuinant order.

JOURNAL DE PHYSIQUE T 3 N' 3 MARCH '993 25

To calculate the corrections in

Ill, I/t~,

etc. the

procedure

would be to substitute

B ₌

j

sn

()

+

b,1)

⁺

^~)~~

⁺

~~~

^+..

^(A10)

+ +

and _compare coefficients of various _powers oft to find

equations

satisfied

by fi(t), f2(t),

etc.

Of _course, this cannot

give

_us the constants I and b in _terms of the initial conditions

B(0)

_{= z}

and

B'(0)

₌

2z~,

since

equation (A.10)

is not valid _{near t}

= 0.

Dyson

_[8] found

by

_a finer

study

of the continuum model that b ₌ 0 and I varies

slowly

with t.

B. Power series of

F+(z,t)

and

F(z,t)

for small t.

Eq.(1.22)

_can be deduced from reference 2 with

aj$~

^and ajn as follows:

aj$~

= 0 if

j

<

n(n I), ajj~

₌ ^{0 if}

j

< n~ and ajn = 0 if

j

<

n(n -1)/2.

For values of

j

_greater than _or

equal

to these

they

_are

given by

the

following expressions (in

what follows

b(j, k)

is

equal

to I if

j

₌ k and is _zero if

j # k)

~~~~ ~~"~ ^~~

~0 ~0 ^{~~ ~~~}

l

~'~~~~~

~~~

,l' (

~~~

2m,

2t, + 2tk +

Ii 'l~

2tk ^' ~~~~

~~"~ ^{~ ~~}

~0 ~0 ^{~~ ~~~}

l

~'~~~~

~~~

,/~ _,l~~

^~~~

^2m;

_2t;

+ 2tk +

Ii fl ^2tk

+

l~

' ~~~~

~~" ~~"

~0 ~0 ^{~~ 1 ~~}

^"

l

~'~~~~~

~' j~~

~~~

~~~

2m, _;~+

tk +

~ '~~

~~~~

~~~~

~~~~

In

particular,

ajf~

=

-)b(j, 0)

_~

[2(2j

+

1)(2j

+ 1)1]~~,

(84)

(+) _~

j

+ 2~i

~~~

(2j

+

2)! 2j

+

(j

+

1)(2j +1)

~

(2j

₂₎₁

j~ _(2i

₊

^)(2~

_{2i +}

1)

~~ ^~

^~ ' ~~~~

aji "

-b(I, 0), (86)

aj2 "

-2~J+~[(j

₊

1)(2j

+

1)(2j

+

2)1]~~, (87)

~~~

(j

+

)(~~~

~(2j

₊

1)1

~~~~3~~ j~

~~~~

The rational _or numerical values of

aj$~

^or of ajn can

readily

be calculated for Small value8 of

j

and _n.

C. Inver8e power series

expansions

of

Sn(r), An (r), 8n(r),

etc..

For _r »

I,

the coefficient functions

Sn(r), An (r)

^and

Bn(r)

in

equations (4.1)-(4.3)

_can be

expanded

in powers of r~~ _As

explained

in section 4 _we get

(1 +

I) II

^{85 l1813 14121997 7374679967}

S°(~)

"

2

@ @ @

_{223r8 228rio} ⁺

~~

(~

₊

~

+

i

+

~j~~~

⁺

~~$~)~~~

⁺

)j ^(Cl)

3 5 469 2979 1227029 4411073 4184326389

Si(~)

_" l +

$

⁺

@

⁺

@

⁺

@

⁺ _223rs ⁺ _228r6 ⁺ _232r7 ⁺

7 213 4465 501667 17524881

+ ~

p

⁺

fi

⁺

@

⁺

@

⁺ _222rs ⁺ _226r6 ⁺

~~~

~ 5 63 771 43063 864555 37620327 1053966787

2 ^~ " +

p

⁺

fi

⁺

@

⁺ _21sr4 ⁺ _218rs ⁺ _222r6 ⁺ _22sr7 ~

13 387 6815 566999 13720419

+

@

⁺

@

⁺

@

⁺

@

⁺ _221r6 ⁺ _224r? ⁺

~~~

Ii 581 14439 1702083 65467383 5738450209

~~~~~ ~

24r ^~ 29r~ ^~ 213r3 ^~ 219r4 ^~ 2~3r5 ^~ 228r6 ^~

+ i

()

+

)

+

$

+

~/(((~

⁺

~~j()(~~

⁺ _,

^(C4)

3 63 823 24999 117585 1274445 33179409

54(~)

_" l +

p

⁺

@

⁺

@

⁺

@

⁺ _213~s ⁺ _21sr6 ⁺ _218r7 ⁺

+ i

()

+

~i(~s

⁺

I

+

)

+

~j(((~

⁺

,

(C5) A°(T)

_" +

)

+

~

+

~

+

~

+

$

+

~)~$~

+

~i~(~~~

⁺ _'

^(~~)

~ 5 65 1823 163691 7266843 551317093

i(T)

_" 1+ ~~~~

$

⁺

@

⁺

@

⁺ _219~4 ⁺ _223~s ⁺ _228~6 ⁺ ,

A~(r)

₌ 1+

)

+

)

+

)

+

(((((

+

(((((~

⁺

~((((~~

⁺

,

(C8)

~ ^{Ii 581} 14455 1704963 65581207 5747809249

3(T)

_" +

$

⁺

@

⁺ _213r3 ⁺ _219r4 ⁺ _223rs ⁺ _228~6 ⁺ , ~~~~

~~~~~

~

IT

^~

~~2

~

~~3

~

~~~

~

~~$

_~

~~~~~~

^~ '

~~~~~

Bo(r)

₌

) ) /

~/()(~ ~((()((~ ~~((()((~~

_,

^{(Cl i)} Bi(r)

₌ 1+

)

+

)

+

$

+

((()(

+

~((()(~

⁺

~~/()i~

+

,

(C12)

5 61 745 41579 839407 36598265

~~~~~

82(T)

_" I +

p

⁺

fi

⁺

@

⁺

@

⁺ _218~5 ⁺ _222~6 ⁺

Ii 581 14423 1699203 65353559 5729123937

~~~~~

83(T)

_" +

$

⁺

@

⁺

@

⁺ _219~4 ⁺ _223~s ⁺ 228~6 ⁺

3 63 823 12499 117577 5097387

84(T)

_" I +

p

⁺

@

⁺

@

⁺ _211~4 ⁺ _213~s ⁺ 217~6 ⁺

(~l~)

Similarly,

let _us write

log F+((, r)

_as follows

log J~+(<, r)

₌

_i~ £ ^{a <} )) jj

^{~~ ~}

_{g+n(r), (c16)}

Then

first, -G+o(r)

coincide with the

right

^hand-side of

Eq.(1.24),

except ^{that the constant}

terms remain unknown. We list the coefficients _cn up ^to n = 25 in

appendix

D.

The

G+n(r)

for _{n =} 1, 2, 3, ^{have the}

expansions

~ ^{7 205 5305}

682963 28971265

~

2835864953

~~~~~

+~ ^~ ~

16r ^~ 29r2 ^~ 213r3 ^~ 219r4 ^~ 223r5 228r6 ^{~ '}

~ ^{l 7 l19 1485 90647 1773137} ~

80288703

~

~~~~

+2 T "

§

⁺

$

⁺

@

⁺

@

⁺

@

⁺ _{219~s 223~6}

7 287 22475 926961 35417889 9421927049

~~~~

~+~~~~

3 ~ 16r ~

29r2 ^~ 3 _x 213r3 ^~ 219r4 ^~ 223r5 ^~ 3 _x 228r6 ^'

~ ^{l 7 21 l159 36589 344937}

7512339

~~~~~

+4 ^{T "}

j

⁺

$

⁺

@

⁺

@

⁺

@

⁺ _216~s ⁺ _217~6 ⁺ '

~ ^l

~

19 1069 39293 7099315 378600725

~~~~~

~~ ~

8r ^~ 16r ^~ 29r2 ^~ 213r3 ^~ 2"r4 ^~ 223r5 ^{~ '}

~ ^l

(1

^{19 691 13725 1226375}

30927109

~~~~~

~~ ^~

(8~)2

~ ~ ifi~ ~

~8~2 ^~ ~ll~3 ^~ ~16~4 ^~ ~19~5 ^{~ '}

~^{~~ ~}

_(8r)3

^l

(1

₃ ^~ _16r^{19 ~ 1743}_29r2 ^~ ₃ ^{236615 15023537 773134357 ~~~~~}

x 213r3 ^~ 219r4 ^~ 223r5 ^{~ '}

~^{~~ ~}

_(8r)4

^l

(1

₄ ^~ _16r^{19 ~} _26r2^{263 ~ 13427}_21°r3 ^{~ 686667}_214r4 ^{~ 9145995}_216r5 ^{~ ~~~~~}

For _n > I _we write

Then

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