Distribution function of the endpoint fluctuations of one-dimensional directed polymers in a random potential

0.3 MB·English

by  Victor Dotsenko

Distribution function of the endpoint fluctuations of one-dimensional directed
polymers in random potential
Victor Dotsenko
LPTMC, Universit´e Paris VI, 75252 Paris, France and
L.D. Landau Institute for Theoretical Physics, 119334 Moscow, Russia
(Dated: September28, 2012)
The explicit expression for the the probability distribution function of the endpoint of one-
2
dimensional directed polymers in random potential is derived in terms of the Bethe ansatz replica
1
techniquebymappingthereplicatedproblemtotheN-particlequantumbosonsystemwithattrac-
0
tiveinteractions.
2

p PACSnumbers: 05.20.-y75.10.Nr74.25.Qt61.41.+e
e
S

7
2
I. INTRODUCTION

]
h
c One-dimensional directed polymers in a quenched random potential and equivalent problem of the solutions of
e the KPZ-equation [1] describing the growth in time of an interface in the presence of noise have been the subject
m
of intense investigations during the past two decades (see e.g. [2–7]). The model of directed polymers describes an
- elastic string directed along the τ-axis within an interval [0,t]. Randomness enters the problem through a disorder
t
a potential V[φ(τ),τ], which competes against the elastic energy. The system is defined by the Hamiltonian
t
s
t. H[φ(τ),V]= tdτ 1 ∂ φ(τ) 2+V[φ(τ),τ] ; (1)
a 2 τ
m Z0 n (cid:2) (cid:3) o
- where the disorder potential V[φ,τ] is Gaussian distributed with a zero mean V(φ,τ)=0 and the δ-correlations:
d
n
V(φ,τ)V(φ′,τ′)=uδ(τ τ′)δ(φ φ′) (2)
o − −
c
[ Here the parameter u describes the strength of the disorder.
Inwhatfollowsweconsiderthe probleminwhichthe polymerisfixedatthe origin,φ(0)=0anditis freeatτ =t.
1
In other words, for a given realization of the random potential V the partition function of the considered system is:
v
6 +∞
6 Z = dxZ(x) = exp βF (3)
1 {− }
Z−∞
6
. where
9
0 φ(t)=x
2 Z(x)= φ(τ)e−βH[φ] (4)
1 Zφ(0)=0 D
:
v
is the partition function of the system with the fixed boundary condition, φ(t) = x and F is the total free energy.
i
X Besidestheusualextensivepartf t(wheref isthelinearfreeenergydensity),thetotalfreeenergyF ofsuchsystem
0 0
is known to contain the disorder dependent fluctuating contribution which in the limit of large t scales as t1/3 (see
r
a e.g. [4–7]). In other words, in the limit of large t the total (random) free energy of the system can be represented
as F = f t+ct1/3f, where c is a non-universal parameter, which depends on the temperature and the strength of
0
disorder,andf istherandomquantitywhichinthethermodynamiclimitt isdescribedbyanon-trivialuniversal
→∞
distribution function P(f). The trivial self-averaging contribution f t to the free energy can be eliminated from the
0
further study by the simple redefinition of the partition function, Z =exp βf t Z˜, so that Z˜ =exp λf , where
0
λ=βct1/3. Thus, to simplify notations the contribution f t will be just d{r−opped}out in the further ca{lc−ulat}ions.
0
For the problem with the zero boundary conditions, φ(0)=φ(t)=0, the distribution function P(f) was provedto
bedescribedbytheGaussianUnitaryEnsemble(GUE)Tracy-Widomdistribution[8–11]. Ontheotherhand,thefree
energydistribution function of the directedpolymers with the free boundary conditions, eqs.(1)-(4), was shownto be
given by the Gaussian Orthogonal Ensemble (GOE) Tracy-Widom distribution [12, 13] In the course of these proofs
ratherefficientBetheansatzreplicatechniquehasbeendeveloped[10–13]. Hereintermsofthistechniquewearegoing
tostudy onthe statisticalpropertiesofthe transversefluctuations ofthe directedpolymers. Thescalingpropertiesof
2
thetypicalvalueoftheendpointdeviations,φ(t), etlargetimesiswellknown: φ(t) 2 t4/3 (here ... denotesthe
h i ∝ h i
thermalaverageand (...) is the averageoverthe disorder potential, eq.(2)) [4–7]. Much more interesting objectis the
probabilitydistributionfunctionP(x)forthe rescaledquantityx=φ(t)/t2/3 whichisexpectedtobecomeauniversal
function in the limit t . Recently this function has been derived in terms of the so called maximal point of the
→ ∞
Airy process minus a parabola [14–16], which is believed to descibe the scaling limit of the endpoint of the directed
2
polymers in a random potential. The obtained explicit expression for P(x) turned out to be rather complicated and
its analytic properties is not so easy to analyze although the asymptotic behavior of this function is already known:
P(x ) exp x3/12 [15].
→∞ ∼ {−| | }
In this work the explicit form of the distribution function of the directed polymer’s endpoint fluctuations will be
derived in terms of the Bethe ansatz replica technique. The distribution function we are going to consider is defined
as follows:
∞
W(x) = lim Prob φ(t)t−2/3 > x = dx′ P(x′) (5)
t→∞ Zx
(cid:2) (cid:3)
This function gives the probability that the rescaled value of the polymer’s right endpoint φ(t)/t2/3 is bigger than a
given value x. In this paper it will be shown that (see eqs.(79)-(84) below)
+∞ +∞ −1
W(x) = df F1( f) dωdω′ ˆ1 Bˆ−f Φω′ω(f,x) (6)
Z−∞ − Z Z0 (cid:16) − (cid:17)ωω′
where F ( f) = det ˆ1 Bˆ is the GOE Tracy-Widom distribution with the kernel B (ω,ω′) = Ai(ω +ω′
1 −f −f
− − −
f), (ω,ω′ > 0) and
(cid:2) (cid:3)
1 +∞ ∂ ∂ 1 1
Φωω′(f,x)=−2Z0 dy " (cid:16)∂ω − ∂ω′(cid:17)Ψ(cid:0)ω− 2f +y; x(cid:1)Ψ(cid:0)ω′− 2f +y; −x(cid:1)+
∂ ∂ 1 1
+ + Ψ ω f y; x Ψ ω′ f +y; x (7)
∂ω ∂ω′ − 2 − − 2 − #
(cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1)
where the function of two variables Ψ(ω;x) is the simple generalization of the Airy function:
dz 1 1
Ψ(ω;x) = exp z3 ωz xz2 (8)
2πi 6 − − 4
ZC n o
Heretheintegrationcontour inthe complexplanestartsatinfinity withthe argument π/3andendsupatinfinity
C −
with the argument +π/3. The above result looks quite similar to the one obtained in [14], although at the moment I
am not able to provide the proof that these results are indeed the same.
The paper is organized as follows. In Section II we define the distribution function W(x) via the two-point free
energydistributionfunctionV (f ,f )whichgivetheprobabilitythatthefreeenergyofthepolymerwiththeendpoint
x 1 2
located above a position x is bigger than a given value f , while the free energy of the polymer with the endpoint
1
locatedbelowthepositionxisbiggerthanagivenvaluef . InSectionIIIthefunctionV (f ,f )isdefinedbymapping
2 x 1 2
the considered problem to the one-dimensional N-particle system of quantum bosons with attractive δ-interactions.
In Section IV the explicit expression for the probability function V (f ,f ) is obtained in terms of the Bethe ansatz
x 1 2
replica technique. Finally, in Section V the result eqs.(6)-(8) is derived. Conclusions and future perspectives are
discussed in Section VI.
II. THE ENDPOINT PROBABILITY DISTRIBUTION FUNCTION
In terms of the partition function Z(x), eqs.(4), the probability distribution function of the polymer’s endpoint
W(x), eq.(5), can be defined as follows:
+∞ Z(x′)
W(x) = lim dx′
t→∞ Zx −+∞∞dx′ Z(x′)!
R
Z(+)(x)
= lim (9)
t→∞ Z(+)(x)+Z(−)(x)!
3
where
x
Z(−)(x) dx′ Z(x′) = exp λf (10)
(−)
≡ {− }
Z−∞
+∞
Z(+)(x) dx′ Z(x′) = exp λf (11)
(+)
≡ {− }
Zx
where the parameter λ t1/3 and f are the free energies of the polymers with the endpoint φ(t) located corre-
(±)
∝
spondingly above and below a given position x. According to these definitions we find
0, for f <f
exp λf (−) (+)
(+)
W(x) = lim {− } = (12)
λ→∞ exp{−λf(−)} + exp{−λf(+)}  1, for f(−) >f(+)
Let us introduce the joint probability density function f ;f . By definition the quantity
x (+) (−)
P
f ;f df df gives the probability that the free energy of the polymer with the endpoint located be-
Px (+) (−) (+) (−) (cid:2) (cid:3)
lowxisequaltof (withintheintervaldf ),whilethefreeenergyofthepolymerwiththeendpointlocatedabove
(−) (−)
(cid:2) (cid:3)
x is equal to f (within the interval df ). Thus, according to eq.(12),
(+) (+)
+∞ +∞
W(x) = df df f ;f (13)
(+) (−) x (+) (−)
P
Z−∞ Zf(+)
(cid:2) (cid:3)
Let us introduce one more joint probability distribution function:
+∞ +∞
V (f ,f ) = Prob f >f ; f >f = df df f ;f (14)
x 1 2 (+) 1 (−) 2 (+) (−) x (+) (−)
P
Zf1 Zf2
(cid:2) (cid:3) (cid:2) (cid:3)
This two-point free energy distribution function gives the probability that the free energy of the polymer with the
endpoint located above the position x is bigger than a given value f , while the free energy of the polymer with the
1
endpoint located below the position x is bigger than a given value f . According to this definition,
2
∂ ∂
f ;f = V (f ,f ) (15)
x 1 2 x 1 2
P ∂f ∂f
1 2
(cid:2) (cid:3)
Substituting this relation into eq.(13) we find
+∞ +∞ ∂ ∂
W(x) = df df V (f ,f ) (16)
1 2 x 1 2
∂f ∂f
Z−∞ Zf1 1 2
Integrating by parts over f and taking into account that V (f ,f ) =0 we get
2 x 1 2 f2=+∞
(cid:12)
+∞ ∂ (cid:12)
W(x) = df V (f ,f ) (17)
1 x 1 2
−Z−∞ (cid:16)∂f1 (cid:17)(cid:12)f2=f1+0
(cid:12)
Thus, to get the distribution function W(x) for the polymer’s endpoint fl(cid:12)uctuations we have to derive the two-point
free energy distribution function V (f ,f ) first. Note that this function is different from the two-point free energy
x 1 2
distributionfunctionderivedin[17]whichdescribesjointstatisticsofthefreeenergiesofthedirectedpolymerscoming
to two different endpoints
III. MAPPING TO QUANTUM BOSONS
According to the definition, eq.(14), the probability distribution function V (f ,f ) can be defined as follows:
x 1 2
∞ ∞ ( 1)L( 1)R L R
V (f ,f )= lim − − exp λLf +λRf Z(+)(x) Z(−)(x) (18)
x 1 2 1 2
λ→∞ L! R!
L=0R=0
X X (cid:0) (cid:1)(cid:2) (cid:3) (cid:2) (cid:3)
4
Indeed, substituting here the definitions, eqs.(10)-(11), we find:
+∞ +∞ ∞ ∞ ( 1)L( 1)R
V (f ,f ) = lim df df f ;f − − exp λL(f f ) exp λR(f f )
x 1 2 λ→∞Z−∞ (+)Z−∞ (−)Px (+) (−) L=0R=0 L! R! 1− (+) 2− (−)
(cid:2) (cid:3)X X (cid:2) (cid:3) (cid:2) (cid:3)
+∞ +∞
= lim df(+) df(−) x f(+);f(−) exp eλ(f1−f(+)) eλ(f2−f(−))
λ→∞Z−∞ Z−∞ P (cid:2) (cid:3) h− − i
+∞ +∞
= df df f ;f θ f f θ f f (19)
(+) (−) x (+) (−) (+) 1 (−) 2
P − −
Z−∞ Z−∞
(cid:2) (cid:3) (cid:0) (cid:1) (cid:0) (cid:1)
which coincides with the definition, eq.(14).
Further calculations of the two-point distribution function V (f ,f ) to a large extent repeats the procedure de-
x 1 2
scribed in detail in the previous paper [13] for the one-point free energy distribution function. Using the definitions,
eqs.(10)-(11), the distribution function, eq.(18), can be represented as follows:
∞ ( 1)L+R x +∞
V (f ,f )= lim − exp λLf +λRf dx ...dx dy ...dy Ψ(x ,...,x ,y ,...,y ;t) (20)
x 1 2 1 2 1 L 1 R 1 L R 1
λ→∞L,R=0 L!R! Z−∞ Zx
X (cid:0) (cid:1)
where
N φa(t)=xa
Ψ(x ,...,x ;t) Z(x )Z(x )...Z(x )= φ (τ) exp βH [φ ,φ ,...,φ ] (21)
1 N 1 2 N a N 1 2 N
≡ a=1"Zφa(0)=0 D # −
Y (cid:0) (cid:1)
with the replica Hamiltonian
1 t N 2 N
H [φ ,φ ,...,φ ] = dτ ∂ φ (τ) βu δ φ (τ) φ (τ) (22)
N 1 2 N τ a a b
2Z0 a=1 − a6=b − !
X(cid:2) (cid:3) X (cid:2) (cid:3)
The propagator Ψ(x;t), eq.(21), describes N trajectories φ (τ) all starting at zero (φ (0) = 0), and coming to N
a a
different points x ,...,x at τ = t. One can easily show that Ψ(x;t) can be obtained as the solution of the the
1 N
{ }
imaginary-time Schr¨odinger equation
β∂ Ψ(x;t)=HˆΨ(x;t) (23)
t
−
with the initial condition
Ψ(x;0)=ΠN δ(x ) (24)
a=1 a
Here the Hamiltonian is
N N
1 1
Hˆ = ∂2 κ δ(x x ) (25)
−2 xa − 2 a− b
a=1 a6=b
X X
and the interaction parameter κ = β3u. This Hamiltonian describes N bose-particles interacting via the attractive
two-body potential κδ(x).
−
A generic eigenstate of such system is characterized by N momenta q (a = 1,...,N) which are splitted into M
a
{ }
(1 M N) ”clusters” described by continuous real momenta q (α = 1,...,M) and having n discrete imaginary
α α
≤ ≤
”components” (for details see [10, 20–24]):
iκ
q qα = q (n +1 2r) ; (r =1,...,n ) (26)
a ≡ r α− 2 α − α
with the global constraint
M
n =N (27)
α
α=1
X
5
A generic solution Ψ(x,t) of the Schr¨odinger equation (23) with the initial conditions, Eq.(24), can be representedin
the form of the linear combination of the eigenfunctions Ψ(M)(x):
q
N
Ψ(x ,...,x ;t)= 1 (M)(q,n) C (q,n)2 Ψ(M)(x)Ψ(M)∗(0) exp E (q)t (28)
1 N M!" D #| M | q q − M
M=1 Z
X (cid:8) (cid:9)
where we have introduced the notation
M +∞ dq ∞ M
(M)(q,n) α δ n , N (29)
α
Z D ≡αY=1"Z−∞ 2π nXα=1# (cid:16)αX=1 (cid:17)
and δ(k,m) is the Kronecker symbol; note that the presence of this Kronecker symbol in the above equation allows
to extend the summations over n ’s to infinity. Here (non-normalized) eigenfunctions are [10, 24]
α
N N
sgn(x x )
Ψ(M)(x)= 1+iκ a− b exp i q x (30)
q q q Pa a
XP aY<b" Pa − Pb # h Xa=1 i
wherethe summationgoesoverN!permutations ofN momentaq ,eq.(26), overN particlesx ; the normalization
a a
P
factor
κN M q q iκ(n n ) 2
C (q,n)2 = α− β − 2 α− β (31)
| M | N! Mα=1 κnα αY<β (cid:12)(cid:12)qα−qβ − i2κ(nα+nβ)(cid:12)(cid:12)2
and the eigenvalues: Q (cid:0) (cid:1) (cid:12)(cid:12) (cid:12)(cid:12)
1 N 1 M κ2 M
E (q) = q2 = n q2 (n3 n )
M 2β a 2β α α− 24β α− α
α=1 α=1 α=1
X X X
M 1 κ2 κ2
= n q2 n3 + N (32)
2β α α− 24β α 24β
αX=1h i
Thelasttermintheaboveexpressionprovidesjustthetrivialcontributiontothe selfaveragingpartofthefreeenergy
(discussed in the Introduction) and therefore it will be dropped out of the further calculations.
Using the definition, eq.(30), one can easily prove that
Ψ(M)(0)=N! (33)
q
In this way the problemof the calculationofthe probabilitydistribution function, eq.(20), reduces to the summation
over all the spectrum of the eigenstates of the N-particle bosonic problem, which is parametrized by the set of both
continuous, q ,...,q , and discrete n ,...,n ; (M =1,...,N); (N =1,..., ) degrees of freedom.
1 M 1 M
{ } { } ∞
IV. TWO-POINT FREE ENERGY DISTRIBUTION FUNCTION
Substituting eqs.(28)-(33) into eq.(20), we get:
∞ e
V (f ,f ) = 1+ lim ( 1)L+R λLf1+λRf2 (34)
x 1 2
λ→∞ − ×
L+R≥1
X
L+R 1 M ∞ +∞ dqα κnαe−2tβnαqα2+2κ42βn3α δ M nα , L+R C˜M(q,n)2 IL,R(q,n)
× MX=1M!αY=1"nXα=1Z−∞ 2πκnα # (cid:16)αX=1 (cid:17)| |
where
M q q iκ(n n ) 2
C˜ (q,n)2 = α− β − 2 α− β (35)
| M | αY<β (cid:12)(cid:12)qα−qβ − i2κ(nα+nβ)(cid:12)(cid:12)2
(cid:12) (cid:12)
(cid:12) (cid:12)
6
and
I (q,n) = L R qPa(L) −qPc(R) −iκ L qPa(L) −qPb(L) −iκ R qPc(R) −qPd(R) +iκ
L,R
PX(L,R)PX(L)PX(R) aY=1cY=1" qPa(L) −qPc(R) #×aY<b" qPa(L) −qPb(L) #×cY<d" qPc(R) −qPd(R) #×
L
dx ...dx exp i (q iǫ)x
× Z−∞<x1≤...≤xL≤x 1 L h Xa=1 Pa(L) − ai
R
dy ...dy exp i (q +iǫ)y (36)
× Zx≤yR≤...≤y1<+∞ R 1 h Xc=1 Pc(R) ci
Herethesummationoverallpermutations of(L+R)momenta q ,...,q overL”left”particles x ,...,x and
1 L+R 1 L
R ”right” particles y ,...,y are dividedPinto three parts: the p{ermutations} (L) of L momenta (ta{ken at ran}dom
R 1
{ } P
outofthe totallist q ,...,q )overL”left” particles,the permutations (R) ofthe remainingRmomentaoverR
1 L+R
”right” particles, an{d finally the}permutations (L,R) (or the exchange) of tPhe momenta between the group ”L” and
P
the group”R”. Notealsothatthe integrationsbothoverx ’s andovery ’sineq.(36)requireproperregularizationat
a c
and + correspondingly. This is done in the standard way by introducing a supplementary parameter ǫ which
−∞ ∞
will be set to zero in final results. The result of the integrations can be represented as follows:
M L R q q iκ
I (q,n) = i−(L+R) exp ix n q Pa(L) − Pc(R) −
L,R α α
(cid:8) αX=1 (cid:9)PX(L,R) aY=1cY=1" qPa(L) −qPc(R) #×
L q(−) q(−) iκ
1 Pa(L) − Pb(L) −
× q(−) q(−) +q(−) ... q(−) +...+q(−) " q(−) q(−) #×
PX(L) P1(L) P1(L) P2(L) P1(L) PL(L) aY<b Pa(L) − Pb(L)
(cid:0) (cid:1) (cid:0) (cid:1)
(+) (+)
(−1)R R qPc(R) −qPd(R) +iκ (37)
× q(+) q(+) +q(+) ... q(+) +...+q(+) " q(+) q(+) #
PX(R) P1(R) P1(R) P2(R) P1(R) PR(R) cY<d Pc(R) − Pd(R)
(cid:0) (cid:1) (cid:0) (cid:1)
where
q(±) q iǫ (38)
a ≡ a±
and where we have used the fact that for any permutation of the momenta, eq.(26), one has:
L+R M
q = n q (39)
Pa α α
a=1 α=1
X X
Using the Bethe ansatz combinatorial identity [12],
N N
1 q q iκ 1 q +q +iκ
pa − pb − = a b (40)
P qp1(qp1 +qp2)...(qp1 +...+qpN)a<b" qpa −qpb # Na=1qa a<b" qa+qb #
X Y Y
Q
(where the summation goes over all permutations P of N momenta q ,...,q ) we get:
1 N
{ }
M L R q q iκ
I (q,n) = i−(L+R) exp ix n q Pa(L) − Pc(R) −
L,R α α
(cid:8) αX=1 (cid:9)PX(L,R) aY=1cY=1" qPa(L) −qPc(R) #×
1 L qP(−a(L)) +qP(−b(L)) +iκ (−1)R R qP(+c(R)) +qP(+d(R)) −iκ (41)
× L q(−) " q(−) +q(−) #× R q(+) " q(+) +q(+) #
a=1 Pa(L) aY<b Pa(L) Pb(L) c=1 Pc(R) cY<d Pc(R) Pd(R)
Q Q
Further simplification comes from the following important property of the Bethe ansatz wave function, eq.(30). It
has such structure that for ordered particles positions (e.g. x <x <...<x ) in the summation over permutations
1 2 N
the momenta q belonging to the same cluster also remain ordered. In other words, if we consider the momenta,
a
7
eq.(26), of a cluster α, qα,qα,...,qα , belonging correspondingly to the particles x < x < ... < x , the
permutation of any two{m1ome2nta qαnαa}nd qα of this ordered set gives zero contributi{oni.1 Thusi,2in order toinpαe}rform
r r′
the summation over the permutations (L,R) in eq.(41) it is sufficient to split the momenta of each cluster into two
P
parts: qα,...,qα qα ...,qα , where m =0,1,...,n and where the momenta qα,...,qα belong to the particles
of the s{ec1tor ”Lm”α,|w|hmilαe+t1he mnoαm}enta qα α...,qα belonαg to the particles of the sec1tor ”Rm”.α
mα+1 nα
Let us introduce the numbering of the momenta of the sector ”R” in the reversed order:
qα q∗α
nα → 1
qα q∗α
nα−1 → 2
........
qα q∗α (42)
mα+1 → sα
where m +s =n and (s.f. eq.(26))
α α α
iκ iκ
q∗α = q + (n +1 2r) = q + (m +s +1 2r) (43)
r α 2 α − α 2 α α −
By definition, the integer parameters m and s fulfill the global constrains
α α
{ } { }
M
m = L (44)
α
α=1
X
M
s = R (45)
α
α=1
X
In this way the summation over permutations (L,R) in eq.(33) is changed by the summations over the integer
P
parameters m and s :
α α
{ } { }
M ∞ M M
... δ m +s , n δ m , L δ s , R ... (46)
α α α α α
→ " #
PX(L,R) (cid:0) (cid:1) αY=1 mαX+sα≥1 (cid:16) (cid:17) (cid:16)αX=1 (cid:17) (cid:16)αX=1 (cid:17)(cid:0) (cid:1)
which allows to lift the summations over L, R, and n in eq.(34). Straightforwardbut slightly painful calculations
α
{ }
result in the following expression (see Appendix):
V (f ,f ) = lim 1+ ∞ (−1)M M ∞ ( 1)mα+sα−1 +∞dq G qα,mα,sα
x 1 2 α
λ→∞( MX=1 M! αY=1"mαX+sα≥1 − Z−∞ 2π(cid:0)κ(mα+sα(cid:1)) ×
t κ2
exp (m +s )q2 + (m +s )3+λm f +λs f +ix(m +s )q
× −2β α α α 24β α α α 1 α 2 α α α #×
n o
C˜ (q,m+s)2 G q,m,s (47)
M M
× | | )
(cid:0) (cid:1)
where
M q q iκ(m +s m s ) 2
C˜ (q,m+s)2 = α− β − 2 α α− β − β (48)
| M | αY<β (cid:12)(cid:12)qα−qβ − i2κ(mα+sα+mβ +sβ)(cid:12)(cid:12)2
(cid:12) (cid:12)
and (cid:12) (cid:12)
Γ s + 2iq (−) Γ m 2iq (+) Γ 1+m +s
α κ α α− κ α α α
q ,m ,s = (49)
G(cid:0) α α α(cid:1) 2(mα+sα)Γ mα(cid:16)+sα+ 2κiqα((cid:17)−) (cid:16)Γ mα+sα−(cid:17)2κiq(cid:0)α(+) Γ 1+m(cid:1)α Γ 1+sα
(cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1)
The explicit expression for the factor G q,m,s is given in the Appendix, eq.(A.17).
M
(cid:0) (cid:1)
8
Redefining
κ
q = p (50)
α α
2λ
and
2λ2
x x (51)
→ κ
with
λ = 1 κ2t 1/3 = 1 β5u2t 1/3 (52)
2 β 2
(cid:16) (cid:17) (cid:0) (cid:1)
the normalization factor C˜ (q,m+s)2, eq.(48), can be represented as follows:
M
| |
M λ m +s λ m +s ip +ip 2
C˜ (q,m+s)2 = α α − β β − α β =
| M | αY<β (cid:12)(cid:12)λ(cid:0)mα+sα(cid:1)+λ(cid:0)mβ +sβ(cid:1)−ipα+ipβ(cid:12)(cid:12)2
M (cid:12) (cid:0) (cid:1) (cid:0) (cid:1) (cid:12)
(cid:12) (cid:12)1
= 2λ m +s det (53)
α α
αY=1(cid:2) (cid:0) (cid:1)(cid:3)× "λ mα+sα −ipα+λ mβ +sβ +ipβ#α,β=1,...,M
(cid:0) (cid:1) (cid:0) (cid:1)
where we have used the Cauchy double alternant identity
Mα<β(aα−aβ)(bα−bβ) = ( 1)M(M−1)/2det 1 (54)
Q Mα,β=1(aα−bβ) − haα−bβiα,β=1,...M
Q
with a =p iλ m +s and b =p +iλ m +s .
α α α α α α β β
−
After rescaling, eqs.(50)-(52), for the exponential factor in eq.(47) we find
(cid:0) (cid:1) (cid:0) (cid:1)
t κ2
(m +s )q2 + (m +s )3+λm f +λs f +ix(m +s )q
−2β α α α 24β α α α 1 α 2 α α α →
1
λ(m +s )p2 + λ3(m +s )3+λm f +λs f +iλx(m +s )p (55)
→ − α α α 3 α α α 1 α 2 α α α
The cubic exponential term can be linearized using the Airy function relation
1 +∞
exp λ3(m +s )3 = dy Ai(y ) exp λ(m +s )y (56)
α α α α α α α
3
h i Z−∞ h i
Substituting eqs.(53),(55) and (56) into eq.(47), and redefining y y +p2 ixp , we get
α → α α− α
∞ ( 1)M M +∞ dy dp ∞
V (f ,f ) = lim 1 + − α α Ai y +p2 ixp ( 1)mα+sα−1
x 1 2 λ→∞( MX=1 M! αY=1"Z Z−∞ 2π (cid:0) α α− α(cid:1)mαX+sα≥1 − ×
p
α
exp λm (y +f )+λs (y +f ) , m , s
α α 1 α α 2 α α
× G λ #×
n o (cid:16) (cid:17)
p
detKˆ (λm , λs , p );(λm , λs , p ) G , m, s (57)
× α α α β β β α,β=1,...,M M λ )
(cid:2) (cid:3) (cid:16) (cid:17)
where
1
Kˆ (λm, λs, p);(λm′, λs′, p′) = (58)
λm+λs ip+λm′+λs′+ip′
−
(cid:2) (cid:3)
9
The crucialpointofthe further calculationsis the procedureoftakingthe thermodynamic limit λ . Inthis limit
→∞
thesummationsover m and s areperformedaccordingtothefollowingalgorithm. Letusconsidertheexample
α α
{ } { }
of the sum of a general type:
M ∞
p
R(y,p) = lim ( 1)nα−1exp λn y Φ , p, λn, n (59)
α α
λ→∞ " − { }# λ
αY=1 nXα=1 (cid:16) (cid:17)
where Φ is a function which depend on the factors λn , p /λ as well as on the parameters n and p (which do not
α α α α
contain λ). The summations in the above example can be representedin terms of the integrals in the complex plane:
M
1 dz p
R(y,p) = lim α exp λz y Φ , p, λn, n (60)
α α
λ→∞αY=1"2iZC sin(πzα) { }# (cid:16)λ (cid:17)
where the integration goes over the contour shown in Fig.1(a). Shifting the contour to the position ′ shown in
C C
Fig.1(b) (assuming that there is no contribution from infinity), and redefining z z/λ, in the limit λ we get:
→ →∞
M
1 dz p z
R(y,p) = α exp z y lim Φ , p, z, (61)
α α
αY=1"2πiZC′ zα { }# λ→∞ (cid:16)λ λ(cid:17)
where the parameters y , p and z remain finite in the limit λ .
α α α
→∞
FIG. 1: The contours of integration in the complex plane used for summing the series: (a) the original contour C; (b) the
deformed contour C′;
To perform the summations over m and s in eq.(57) it is convenient to represent it in the following way:
α α
∞ ( 1)M M +∞ dy dp
V (f ,f ) = 1+ − α α Ai y +p2 ixp p,y; f ,f (62)
x 1 2 M=1 M! α=1"Z Z−∞ 2π α α− α #SM 1 2
X Y (cid:0) (cid:1) (cid:0) (cid:1)
where
M ∞
p,y; f ,f = lim ( 1)mα+sα−1exp λm y +f +λs y +f
M 1 2 α α 1 α α 2
S λ→∞ " − #×
(cid:0) (cid:1) αY=1 mαX+sα≥1 n (cid:0) (cid:1) (cid:0) (cid:1)o
M
p p
α, m , s detKˆ (λm ,λs ,p );(λm ,λs ,p ) G , m, s (63)
α α α α α β β β M
× "G λ # λ
αY=1 (cid:16) (cid:17) (cid:2) (cid:3) (cid:16) (cid:17)
The summations over m and s in the above expression can be represented as follows
α α
∞ ∞ ∞ ∞ ∞
( 1)mα+sα−1 = ( 1)mα−1δ(s ,0)+ ( 1)sα−1δ(m ,0) ( 1)mα−1 ( 1)sα−1 (64)
α α
− − − − − −
mαX+sα≥1 mXα=1 sXα=1 mXα=1 sXα=1
Thus in the integral representation,eqs.(59)-(61), for the function p,y; f ,f , eq.(63), we get
M 1 2
S
M dz dz 2πi 2πi (cid:0) 1 (cid:1)
p,y; f ,f = 1α 2α δ(z )+ δ(z ) exp z y +f +z y +f
SM(cid:0) 1 2(cid:1) αY=1"Z ZC′ (2πi)2 (cid:16)z1α 2α z2α 1α − z1αz2α(cid:17) n 1α(cid:0) α 1(cid:1) 2α(cid:0) α 2(cid:1)o#×
M
p z z p z z
= lim α, 1α, 2α G , 1, 2 detKˆ (z ,z ,p );(z ,z ,p ) (65)
λ→∞( "G λ λ λ # M λ λ λ ) 1α 2α α 1β 2β β
αY=1 (cid:16) (cid:17) (cid:16) (cid:17) (cid:2) (cid:3)
10
Taking into account the Gamma function properties, Γ(z) = 1/z and Γ(1+z) = 1, for the factors ,
|z|→0 |z|→0
| | G
eq.(49), and G, eq.(A.17), we obtain
lim pα, z1α, z2α = z1α+z2α+ipα(−) z1α+z2α−ip(α+) (66)
λ→∞G(cid:16) λ λ λ (cid:17) (cid:0) z2α+ipα(−)(cid:1)(cid:0)z1α−ip(α+) (cid:1)
and (cid:0) (cid:1)(cid:0) (cid:1)
p z z
lim G , 1, 2 = 1 (67)
λ→∞ λ λ λ
(cid:16) (cid:17)
Thus, in the limit λ the expression for the probability distribution function, eq.(62), takes the form of the
→ ∞
Fredholm determinant
∞ ( 1)M M +∞ dy dp
V (f ,f ) = 1+ − α α Ai y +p2 ixp
x 1 2 M=1 M! α=1"Z Z−∞ 2π α α− α ×
X Y (cid:0) (cid:1)
dz dz 2πi 2πi 1 z z
1α 2α δ(z )+ δ(z ) 1+ 1α 1+ 2α
× Z ZC′ (2πi)2 (cid:16)z1α 2α z2α 1α − z1αz2α(cid:17)(cid:16) z2α+ipα(−)(cid:17)(cid:16) z1α−ip(α+)(cid:17)×
1
exp z y +f +z y +f det
× n 1α(cid:0) α 1(cid:1) 2α(cid:0) α 2(cid:1)o# hz1α+z2α−ipα+z1β +z2β +ipβi(α,β)=1,2,...,M
= det ˆ1 Aˆ (68)
−
with the kernel (cid:2) (cid:3)
+∞ dy 2πi 2πi 1
Aˆ (z , z , p);(z ′, z ′, p′) = Ai y+p2 ixp δ(z )+ δ(z )
1 2 1 2 2 1
2π − z z − z z ×
(cid:2) (cid:3) Z−∞ (cid:0) (cid:1)(cid:16) 1 2 1 2(cid:17)
z z
1 2
1+ 1+ exp z y+f +z y+f
× z +ip(−) z ip(+) 1 1 2 2 ×
2 1
(cid:16) (cid:17)(cid:16) − (cid:17) n (cid:0) (cid:1) (cid:0) (cid:1)o
1
(69)
× z +z ip+z ′+z ′+ip′
1 2 1 2
−
In the exponential representation of this determinant we get
∞
1
V (f ,f ) = exp TrAˆM (70)
x 1 2
− M
h MX=1 i
where
M +∞ dy dp
TrAˆM = α α Ai y +p2 ixp
α=1"Z Z−∞ 2π α α− α ×
Y (cid:0) (cid:1)
dz dz 2πi 2πi 1 z z
1α 2α δ(z )+ δ(z ) 1+ 1α 1+ 2α
× Z ZC′ (2πi)2 (cid:16)z1α 2α z2α 1α − z1αz2α(cid:17)(cid:16) z2α+ipα(−)(cid:17)(cid:16) z1α−ip(α+)(cid:17)×
M
1
exp z y +f +z y +f (71)
× n 1α(cid:0) α 1(cid:1) 2α(cid:0) α 2(cid:1)o# αY=1"z1α+z2α−ipα+z1α+1+z2α+1+ipα+1#
Here, by definition, it is assumed that z z (i=1,2) and p p . Substituting
iM+1 ≡ i1 M+1 ≡ 1
1 ∞
= dω exp z +z ip +z +z +ip ω (72)
z +z ip +z +z +ip α − 1α 2α− α 1α+1 2α+1 α+1 α
1α 2α− α 1α+1 2α+1 α+1 Z0 h (cid:0) (cid:1) i
into eq.(71), we obtain
∞ M +∞ dydp
TrAˆM = dω ...dω Ai y+p2+ω +ω ixp exp ip ω ω S p,y;f ,f (73)
1 M α α−1 α α−1 1 2
Z0 α=1"Z Z−∞ 2π − { − } #
Y (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)