Condensed Matter Physics, 2014, Vol. 17, No 3, 33003: 1–10
DOI: 10.5488/CMP.17.33003
http://www.icmp.lviv.ua/journal
N -point free energy distribution function
in one dimensional random directed polymers
V. Dotsenko1,2
1 LPTMC, Université Paris VI, 75252 Paris, France
2 L.D. Landau Institute for Theoretical Physics, 119334 Moscow, Russia
Received April 16, 2014, in final form May 14, 2014
Explicit expression for the N -point free energy distribution function in one dimensional directed polymers in a
random potential is derived in terms of the Bethe ansatz replica technique. The obtained result is equivalent to
the one derived earlier by Prolhac and Spohn [J. Stat. Mech., 2011, P03020].
Key words: directed polymers, random potential, replicas, fluctuations, distribution function
PACS: 05.20.-y, 75.10.Nr, 74.25.Qt, 61.41.+e
1. Introduction
In this paper we consider the model of one-dimensional directed polymers in a quenched random
potential. This model is defined in terms of an elastic string φ(τ) directed along the τ-axis within an
interval [0, t ] which passes through a random medium described by a random potential V (φ, τ). The
energy of a given polymer’s trajectory φ(τ) is
H [φ(τ),V ] =
Zt
0
dτ
½
¾
¤2
1£
∂τ φ(τ) + V [φ(τ), τ] ,
2
(1)
where the disorder potential V [φ, τ] is described by the Gaussian distribution with a zero mean V (φ, τ) =
0 and the δ-correlations: V (φ, τ)V (φ′ , τ′ ) = uδ(τ − τ′ )δ(φ − φ′ ) The parameter u describes the strength of
the disorder.
The system of this type as well as the equivalent problem of the KPZ-equation [1] describing the
growth of an interface with time in the presence of noise have been the subject of intense investigations
for about the last three decades (see e.g. [2–13]). Such a system exhibits numerous non-trivial features
due to the interplay between elasticity and disorder. In particular, in the limit t → ∞, the polymer mean
squared displacement exhibits a universal scaling form 〈φ2 〉 ∝ t 4/3 (where 〈. . . 〉 and (. . . ) denote the thermal and the disorder averages) while the typical value of the free energy fluctuations scales as t 1/3 . Note
that in the corresponding pure system (with V (φ, τ) ≡ 0) 〈φ2 〉 ∝ t while the free energy is proportional to
ln(t ).
A few years ago, an exact solution for the free energy probability distribution function (PDF) has
been found [14–27]. It was shown that depending on the boundary conditions, this PDF is given by the
Tracy-Widom (TW) distribution [28] either of the Gaussian Unitary Ensemble (GUE) or of the Gaussian
Orthogonal Ensemble (GOE) or of the Gaussian Simplectic Ensemble (GSE). Besides, recently the two-point
free energy distribution function which describes the joint statistics of the free energies of the directed
polymers coming to two different endpoints has been derived in [29–31].
© V. Dotsenko, 2014
33003-1
V. Dotsenko
For fixed boundary conditions, φ(0) = 0; φ(t ) = x , the partition function of the model (1) is
Z t (x) =
φ(t
Z)=x
φ(0)=0
£
¤
Dφ(τ) e−βH [φ] = exp −βF t (x) ,
(2)
where β is the inverse temperature and F t (x) is the free energy. In the limit t → ∞, the free energy scales
as
βF t (x) = β f 0 t + βx 2 /2t + λt f (x) ,
(3)
where f 0 is the selfaveraging free energy density and
λt =
1 ¡ 5 2 ¢1/3
β u t
∝ t 1/3 .
2
(4)
It is the statistics of rescaled free energy fluctuations f (x) which in the limit t → ∞ is expected to be
described by a non-trivial universal distribution W ( f ). In fact, the first two trivial terms of this free
energy can be easily eliminated by simple redefinition of the partition function:
©
ª
Z t (x) → exp −β f 0 t − βx 2 /2t Z̃ t (x)
(5)
©
ª
Z̃ t (x) = exp −λt f (x) .
(6)
£
¤
W ( f 1 , . . . , f N ; x1 , . . . , x N ) ≡ W (f; x) = lim Prob f (x1 ) > f 1 , . . . , f (x N ) > f N ,
(7)
so that
The aim of the present work is to study the N -point free energy probability distribution function
t →∞
which describes the joint statistics of the free energies of N directed polymers coming to N different
endpoints. Some time ago the result for this function has been derived in terms of the Bethe ansatz replica
technique under a particular decoupling assumption [32]. Here, I am going to recompute this function
using somewhat different computational tricks which do not require any supplementary assumptions
and which permit to represent the final result in somewhat more explicit form.
2. N -point distribution function
The probability distribution function, equation (7) can be defined as follows:
¸µ N
¶
N · (−1)L k
Y
¡
¢ Y
exp λL k f k
Z̃ t (xk ) ,
λ→∞ L ,...,L =0 k=1
Lk !
k=1
1
N
W (f; x) = lim
∞
X
(8)
where (. . .) denotes the average over random potentials. Indeed, substituting here equation (6) we get
W (f; x) = lim
µ
N
Y
λ→∞ k=1
¶ ·N
¸
n
h ¡
Y ¡
¢io
¢
exp − exp λt f k − f (xk )
=
θ f (xk ) − f k
(9)
k=1
which coincides with the definition (7).
Performing the standard averaging over random potentials in equation (8) one obtains (for details
see e.g. [20])
¸
N · (−1)L k
Y
¢
¡
¢
¡
W (f; x) = lim
exp λL k f k Ψ x1 , . . . , x1 , x2 , . . . , x2 , . . . , x N , . . . , x N ; t ,
|
{z
}
|
{z
}
|
{z
}
λ→∞ L ,...,L =0 k=1
L
!
k
1
N
∞
X
L1
where the time dependent n -point wave function Ψ(x1 , . . . , xn ; t ) (n =
inary time Schrödinger equation
β ∂t Ψ(x; t ) =
33003-2
"
L2
(10)
LN
PN
L ) is the solution of the imagk=1 k
#
n
n
1 X
1 X
2
δ(x a − xb ) Ψ(x; t )
∂ + κ
2 a=1 x a 2 a ,b
(11)
N -point free energy distribution function in one dimensional random directed polymers
with κ = β3 u and the initial condition
Ψ(x; t = 0) =
n
Y
δ(x a ) .
(12)
a=1
A generic eigenstate of such a system is characterized by n momenta {Q a } (a = 1, . . . , n) which split into
M (1 É M É n ) clusters described by continuous real momenta q α (α = 1, . . . , M) and having nα discrete
imaginary parts
Q a ≡ q rα = q α −
iκ
(nα + 1 − 2r ),
2
(r = 1, . . . , nα ),
(13)
with the global constraint
M
X
nα = n .
α=1
(14)
The time dependent solution Ψ(x, t ) of the Schrödinger equation (11) with the initial conditions, equation
(M)
(12), can be represented in the form of a linear combination of eigenfunctions ΨQ (x):
 +∞
 Ã
!
Z
N
M
M
∞
X
X
X
©
ª
κN |C M (q, n)|2 (M)
1 Y
dq
∗
α


(0) exp −E M (q, n)t .
Ψ(x; t ) =
nα , n
δ
¢ ΨQ (x)Ψ(M)
QM ¡
Q
M!
2π
N ! α=1 κnα
α=1
α=1
n α =1
M=1
−∞
(15)
Here, δ(k, m) is the Kronecker symbol, the normalization factor
¯
¯
M ¯ q − q − iκ (n − n )¯2
Y
α
α
β
β
2
|C M (q, n)| =
¯2
¯
iκ
α<β ¯ q α − q β − 2 (n α + n β )¯
2
and the eigenvalues:
E M (q, n) =
M
X
α=1
µ
(16)
¶
1
κ2 3
nα q α2 −
nα .
2β
24β
(17)
(M)
For a given set of integers {M; n1 , . . . ., n M }, the eigenfunctions ΨQ (x) can be represented as follows (for
details see [33–37]):
¶
¸
µ
·
Ψ(M)
q (x) =
n
X Y
P a<b
1 + iκ
n
X
sgn(x a − xb )
Q P a xa ,
exp i
QP a − QPb
a=1
(18)
where the summation goes over n! permutations P of n momenta Q a , equation (13), over n particles x a .
Substituting equations (15)–(18) into equation (10) we get
W (f; x) =1 + lim
λ→∞
×
×
½
¸
N · (−1)L k
Y
¡
¢
exp λL k f k
Lk !
L 1 +...+L N Ê1 k=1
L 1 +...+L
X N
M=1
X
∞
X

 Ã
!
+∞
µ
¶
2
N
∞ Z
M
M
X
X
X
κn
t
κ
t
1 Y
α
2
3 

dq α
L k |C M (q, n)|2
nα ,
δ
exp − nα q α +
n
M! α=1 nα =1
2πκnα
2β
24β α
α=1
k=1
−∞
N
Y
"
X
#
Ll
Lk Y
N Y
Y
P (L 1 ,...,L N ) k=1 P (L k ) k<l a k =1 a l =1


Q
(L k )
Pa
Q
k
−Q
(L k )
Pa
k
(L l )
Pa
−Q
l
− iκ
(L l )
Pa
l

Ã
 exp i
N
X
k=1
xk
Lk
X
a k =1
Q
(L k )
Pa
k
!¾
.
(19)
In the above expression, the summation over permutations of n = L 1 + . . . + L N momenta Q a split
into the internal permutations P (L k ) of L k momenta [taken at random out of the total list {Q a } (a =
1, . . . , n)] and the permutations P (L 1 ,...,L N ) of the momenta among the groups L k . It is evident that due to
the symmetry of the expression in equation (19), the summations over P (L k ) give just the factor L 1 !. . . L N !.
On the other hand, the structure of the Bethe ansatz wave functions, equation (18), is such that for the
positions of ordered particles in the summation over permutations, the momenta Q a belonging to the
same cluster also remain ordered (for details see e.g. [37]). Thus, in order to perform the summation over
33003-3
V. Dotsenko
the permutations P (L 1 ,...,L N ) in equation (19) it is sufficient to split the momenta of each cluster into N
parts:
α
α
α
α
α
},
, . . . , qP
{q 1α , . . . , q m
1 ; q m 1 +1 , . . . , q m 1 +m 2 ; . . . ; q PN −1 k
N
m α +1
mk
k=1
k=1 α
{z α α}
|
{z α} | α
|
{z
}
2
mα
1
mα
(20)
N
mα
k
where the integers m α
= 0, 1, . . . , nα are constrained by the conditions
N
X
k
mα
=
nα ,
(21)
k
mα
=
Lk ,
(22)
k=1
M
X
α=1
and the momenta of every group
½
α
qP
k−1
are all equal to xk . Let us redefine:
α
qP
k−1
l =1
l
+r
mα
l =1
≡
l
+1
mα
α
q k,r
, ...,
α
qP
k
l =1
l
mα
¾
all belong to the particles whose coordinates
!
Ã
k−1
X l
iκ
m α − 2r .
nα + 1 − 2
= qα +
2
l =1
(23)
k
In this way, the summation over P (L 1 ,...,L N ) is changed by the summation over the integers {m α
}. Substituting equations (20)–(23) into equation (19) after simple algebra, we find

+∞
Z
∞ (−1)M Y
M 
PN k
X
X
dq α
W (f; x) =1 + lim 
(−1) k mα −1
¡ PN k ¢
λ→∞ M=1 M! α=1 P N k
2πκ
m Ê1
k mα

× exp λ
N
X
k=1
k
mα
fk
+i
N
X
k=1
−∞
α
k
"
k
mα
xk q α −

¯
¡
¢¯
¡
¢
k ¯2
k 
× ¯C M q; {m α
} G M q; {m α
} ,
¯
!3 #)
Ã
N
N
N
¯
¯ t 2X
1 X
κ2 t X
k l ¯
k
k
¯
m m xk − xl −
m +
m
κ
q
4 k,l =1 α α
2β α k=1 α 24β k=1 α
(24)
¡
¢¯
PN
k ¯2
k
where the normalization constant ¯C M q; {m α
} is given in equation (16) (with nα = k=1
mα
) and
k
l µ α
¶ M N N mαk mαl µ q α − q α − iκ ¶
α
M Y
N m
Y Y Y Y k,r
Y
Yα m
Yα q k,r − q l ,r ′ − iκ Y
¡
l ,r ′
k ¢
.
G M q; {m α
} =
α
α
α
q k,r − q l ,r ′
q k,r − q lα,r ′
α=1 k<l r =1 r ′ =1
α<β k=1 l =1 r =1 r ′ =1
(25)
α
Substituting the expressions for q k,r
, equation (23), one can find an explicit formula for the above
factor G M which is rather cumbersome: it contains the products of all kinds of the Gamma functions of
£
¤
¢
¡ PN
k PN
the type Γ 1+ 12
(±)m α
+ l (±)m βl ± κ1 (q α − q β ) [the example of this kind of the product is given in
k
[38], equation (A17)]. We do not reproduce it here as it turns out to be irrelevant in the limit t → ∞ (see
below).
After rescaling
qα
→
xk
→
κ
qα ,
2λ
2λ2
xk ,
κ
(26)
(27)
with
µ
¶1/3
1 ¡ 5 2 ¢1/3
1 κ2 t
=
β u t
λ=
2 β
2
33003-4
(28)
N -point free energy distribution function in one dimensional random directed polymers
¯2
¯
k ¯
}) , equation (16) (with nα =
the normalization factor ¯C M (q; {m α
k 2
|C M (q; {m α
})|
PN
k
k
mα
), can be represented as follows:
¯ P
¯2
P
k
M ¯λ N m α
− λ kN m βk − iq α + iq β ¯
Y
k
¯ PN k
¯2
PN k
α<β ¯λ k m α + λ k m β − iq α + iq β ¯
=
"
=
M
Y
α=1
Ã
2λ
N
X
k
mα
k
!#
·
det ¡PN
k
1
¢ ¡P N
¢
k
λm α − iq α + k λm βk + iq β
¸
.
(29)
α,β=1,...,M
Substituting equation (25)–(28) into equation (23) and using the Airy function relation
! #
!3 #
" Ã
Ã
+∞
Z
N
N
X
1 3 X
k
k
mα y
mα
=
dy Ai(y) exp λ
exp λ
3
k
k
"
(30)
−∞
we get
W (f; x)
=
 +∞
∞ (−1)M Y
M Z Z dq dy
X
¡
¢
α
α
1 + lim 
Ai y α + q α2
λ→∞ M=1 M! α=1 
2π

×
PN
k
X
−∞
(−1)
k
Ê1
mα
× det K̂
"Ã
N
X
PN
k
mα
−1
k
"
exp λ
! Ã
k
λm α
, qα ;
k
k=1
N
X
k
N
X
¡
k
m α y α + f k + ixk q α
λm βk , q β
!#
GM
α,β=1,...,M
¢
#

N
1 2 X
k l
− λ
m α ∆kl
mα

2 k,l =1
³ κq
2λ
´

k 
; {m α
} ,
(31)
where
and
K̂
"Ã
¯
¯
∆kl = ¯xk − xl ¯
N
X
k
λm α
,
k
! Ã
!#
N
X
k
λm β , q β = ¡P
qα ;
N
k
k
(32)
1
³P
´.
¢
N
k − iq
λm α
λm βk + iq β
α +
k
(33)
k
The quadratic in m α
term in the exponential of equation (31) can be linearized as follows:
(
N
X
1
exp − λ2
m k m l ∆kl
2 k,l =1 α α
)
=
=
)
´2 X
³
´2 1 X
N
N
N ³
1 2 X
k
k
l
2
m
∆kl
∆kl m α + m α + λ
exp − λ
4 k,l =1
2 k=1 α l =1
 +∞
 +∞


·
·
¸
¸
N  Z dξα
N  Z dηα
Y
1 ¡ α ¢2  Y
1 ¡ α ¢2 
kl
k
p exp − ξkl
p exp − ηk

 k=1 

2
2
2π
2π
k,l =1 −∞
−∞
#
)
(
"
N p
N
X
¡
¢ p
i X
α
α
k
∆kl ξα
(34)
× exp λ
p
kl + ξl k − γk η k m α ,
2 l =1
k
(
where
γk =
N
X
l =1
∆kl =
N ¯
X
¯
¯ xk − xl ¯ .
(35)
l =1
Substituting the representation (34) into equation (31) and redefining the integration parameters
ηα
k
→
ηα
k+
N
X
i
p q α xk + i
γk
l =1
s
¢
∆kl ¡ α
ξkl + ξα
lk
2γk
(36)
33003-5
V. Dotsenko
we get
 +∞


 +∞
 +∞
Z dηα
Z dξα Y
N
N
∞ (−1)M Y
M Z Z dq dy
Y
X
α
α
k
kl
2



Ai(y α + q α )
W (f; x) =1 +
p 
p 
2π
2π k=1
2π
M=1 M! α=1
k,l
=1
−∞
−∞
−∞

s
"
#2  
 1 X

N
N ¡
N
X
X
¡
¡
¢
¢
¢
i
∆
1
kl
2
 S f, y, q, {ηk } ,
× exp −
ξα
+ ξα
ξα
ηα
+ p q α xk + i
−
kl
l
k
kl
k
 2 k,l =1

2 k=1
γk
2γ
k
l =1
(37)
where
¡
¢
S f, y, q, {ηk }
lim
=

M 
Y
λ→∞ α=1 PN
× det K̂
"Ã
k
N
X
X
(−1)
k
Ê1
mα
k
λm α
,
k
PN
k
k
mα
−1
! Ã
qα ;
N
X
k
"
exp λ
λm βk ,
qβ
N
X
k
mα
k=1
!#
¡
#
¢
p
y α + f k − γk η k
GM
α,β=1,...,M
³ κq
2λ
;
k
{m α
}
)
´
.
(38)
k
The summations over m α
in the above expression can be performed as follows:





h
∞
∞
N
N  X
M
X
Y
Y
Y
¢i
¡
¡
¢
k
p
m
−1
k
N



δm k ,0  − (−1)
(−1) α exp λm α y α + f k − γk ηk
S f, y, q, {ηk } = lim
α

 k
λ→∞ α=1 k=1
k
k=1 m α
=0
mα
=0
! Ã
"Ã
!#
³ κq
´
N
N
X
X
k
k
k
λm β , q β
λm α , q α ;
; {m α
}
× det K̂
× GM
2λ
k
k
α,β=1,...,M





Z
k
i
h
N Z
N
M
Y
Y
Y
¢
¡
dz
p
α
 dz k δ(z k ) − (−1)N


= lim
exp λzαk y α + f k − γk ηk
α
α
 2i sin(πzαk )

λ→∞ α=1 k=1
k=1
C
C
× det K̂
"Ã
N
X
k
! Ã
λzαk , q α ;
N
X
k
λzβk , q β
!#
α,β=1,...,M
GM
³ κq
2λ
´
; {zαk } ,
(39)
k
where the integration goes over the contour C shown in figure 1. Redefining z α
→ zαk /λ, in the limit
Figure 1. The contours of integration in the complex plane used for summing the series equation (39).
33003-6
N -point free energy distribution function in one dimensional random directed polymers
λ → ∞, we get
¡
S f, y, q, {ηk }
¢
=




Z
h ¡
N Z dz k
N
Y
Y
¢i
p
α
k
k
k
N
 dzα δ(zα ) − (−1)


exp zα y α + f k − γk ηk
 2πi zαk

α=1 k=1
k=1
M
Y

C
C
× det K̂
"Ã
N
X
zαk ,
k
! Ã
qα ;
N
X
k
zβk ,
qβ
!#
lim G M
α,β=1,...,M
λ→∞
Ã
!
κq zαk
;{ } .
2λ λ
(40)
Taking into account the Gamma function property lim|z|→0 Γ(1 + z) = 1, one can easily demonstrate (see
e.g. [38]) that
lim G M
λ→∞
Ã
κq
;
2λ
(
zαk
λ
)!
= 1.
(41)
Thus, in the limit λ → ∞, the expression (37) takes the form of the Fredholm determinant
W (f; x)
=
 +∞
 +∞
 +∞


Z dξα Y
ZZ
Z dηα
N
M
N
∞ (−1)M Y
Y
X
dq
dy
α
α
kl
k



Ai(y α + q α2 )
1+
p 
p 
2π
2π k=1
2π
M=1 M! α=1
k,l
=1
−∞
−∞
−∞

s
"
#2 
 1 X

N
N
N
X
X
¡
¢
1
i
∆
kl
α
α
q
x
+
i
ξ2kl −
ξ
+
ξ
× exp −
ηα
+
p
α
k
lk
 2 k,l =1

2 k=1 k
γk
2γk kl
l =1

(
)
Z
h ¡
N
N
N
Y
Y
Y
¢i
1
p
k
k
k
N

 dz 
×
exp zα y α + f k − γk ηk
δ(zα ) − (−1)
α
k
k=1
k=1 2πiz α
k=1
C
! Ã
"Ã
!#
N
N
X
X
k
k
zβ , qβ
zα , qα ;
× det K̂
k
k
≡
(
∞ 1
X
det 1̂ − Â = exp −
Tr  M
M
M=1
£
¤
(42)
α,β=1,...,M
)
,
(43)
where  is the integral operator with the kernel
A
"Ã
N
X
k
k
! Ã
z ,q ;
N
X
k
k
z̃ , q̃
!#
 +∞
 +∞


Z
Z
N
N
Y
Y
dξ
dη
dy
kl
k
2


Ai(y + q )
=
p 
p 
2π
2π k=1
2π
k,l
=1
−∞
−∞
−∞

s
#2 
"

 1 X
N
N
N
X
X
1
i
∆
kl
ξ2kl −
× exp −
ηk + p q α xk + i
(ξkl + ξl k )

 2 k,l =1
2 k=1
γk
2γk
l =1
)
(
h ¡
N
N
Y
Y
¢i
1
p
exp z k y + f k − γk ηk
δ(z k ) − (−1)N
×
k
k=1 2πiz
k=1
+∞
Z
× PN
k
1
z k − iq +
PN
k
z̃ k + iq̃
.
(44)
Correspondingly, for the trace of this operator in the M -th power [in the exponential representation of
33003-7
V. Dotsenko
the Fredholm determinant, equation (43)] we get
Tr Â
M
=
 +∞
 +∞


 +∞
Z
Z
ZZ
N
N
M
Y
Y
Y
dη
dξ
dydq
α
k
kl
2



Ai(y + q α )
p 
p 
2π
2π k=1
2π
α=1
k,l =1 −∞
−∞
−∞

s
#2 
"
 1 X

N
N
N
X
1X
i
∆kl
2
× exp −
ξkl −
ηk + p q α xk + i
(ξkl + ξl k )
 2 k,l =1

2 k=1
γk
2γk
l =1
(

)
Z
i
h ¡
N
N
N
Y
Y
Y
¢
1
p
 dz k 

δ(zαk ) − (−1)N
×
exp zαk y + f k − γk ηk
α
k
k=1 2πiz α
k=1
k=1
C
Ã
!
M
Y
1
×
,
PN k
P k
+ iq α+1
z − iq α + kN zα+1
α=1
k α
(45)
k
where, by definition, z M+1
≡ z1k and q M+1 ≡ q 1 .
Substituting
PN
k
zαk − iq α +
1
PN
k
k
zα+1
+ iq α+1
(
!)
Ã
∞
Z
N
N
X
X
k
k
zα+1 + iq α+1
zα − iq α +
= dωα exp −ωα
k
0
(46)
k
into equation (45) we obtain
Tr Â
M
∞ Z
∞
Z
M
Y
A (ωα ; ωα+1 ) ,
= . . . dω1 . . . dωM
0
0
(47)
α=1
where
¡
A ω; ω
¢
′
=
 +∞


 +∞
Z
Z
N
N
Y
Y
dη
dydq
dξ
k
kl


Ai(y + q 2 + ω + ω′ )
p 
p 
2π
2π k=1
2π
k,l
=1
−∞
−∞
−∞


s
#2
"
 1 X

N
N
N
X
∆kl
1X
i
′
2
× exp −
ξkl −
ηk + p q xk + i
(ξkl + ξl k ) − iq(ω − ω )
 2 k,l =1

2 k=1
γk
2γk
l =1



h ¡
N Z dz k
Y
¢i
p
k
N
.
(48)
γ
η
exp
z
y
+
f
−
× 1 − (−1)
k
k
k


2πiz k
k=1
+∞
ZZ
C
Integrating over z 1 , . . . , z N , we finally get
¡
A ω; ω
¢
′
=
 +∞
 +∞


Z
Z
N
N
Y
Y
dydq
dη
dξ
k
kl


Ai(y + q 2 + ω + ω′ )
p 
p 
2π
2π k=1
2π
k,l
=1
−∞
−∞
−∞


s
#2
"
 1 X

N
N
N
X
X
¡
¢
∆kl
1
i
× exp −
ξ2kl −
ηk + p q xk + i
(ξkl + ξl k ) − iq ω − ω′
 2 k,l =1

2 k=1
γk
2γk
l =1
#
"
N
Y
¡
p ¢
(49)
θ −y − f k + ηk γk ,
× 1 − (−1)N
+∞
ZZ
k=1
¯
¯
P
where ∆kl = ¯ xk − xl ¯ and γk = lN=1 ∆kl .
¡
¢
Thus, the N -point free energy distribution function W f 1 , . . . , f N ; x1 , . . . , x N , equation (7), is given by
the Fredholm determinant
£
¤
W (f; x) = det 1̂ − Â ,
(50)
¡
¢
where  is the integral operator with the kernel A ω; ω′ (with ω, ω′ Ê 0) represented in equation (49).
33003-8
N -point free energy distribution function in one dimensional random directed polymers
3. Conclusions
In this paper using the method developed in [30] we extended our result to the spatial N -point free
energy distribution function in the thermodynamic limit t → ∞. It should be noted that following the
ideas of the proof [31] for the two-point function, one can easily demonstrate that the result (49)–(50) obtained in this paper is equivalent to that derived earlier by Prolhac and Spohn [32]. It should be stressed,
¡
¢
however, that since the obtained result for the kernel A ω; ω′ , equation (49), has a rather complicated
structure, its analytic properties are at present completely unclear and their study would require special
efforts.
Acknowledgements
This work was supported in part by the grant IRSES DCPA PhysBio-269139.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
Kardar M., Parisi G., Zhang Y.-C., Phys. Rev. Lett., 1986, 56, 889; doi:10.1103/PhysRevLett.56.889.
Halpin-Healy T., Zhang Y.-C., Phys. Rep., 1995, 254, 215; doi:10.1016/0370-1573(94)00087-J.
Burgers J.M., The Nonlinear Diffusion Equation, Reidel, Dordrecht, 1974.
Kardar M., Statistical Physics of Fields, Cambridge University Press, Cambridge, 2007.
Huse D.A., Henley C.L., Fisher D.S., Phys. Rev. Lett., 1985, 55, 2924; doi:10.1103/PhysRevLett.55.2924.
Huse D.A., Henley C.L., Phys. Rev. Lett., 1985, 54, 2708; doi:10.1103/PhysRevLett.54.2708.
Kardar M., Zhang Y.-C., Phys. Rev. Lett., 1987, 58, 2087; doi:10.1103/PhysRevLett.58.2087.
Kardar M., Nucl. Phys. B, 1987, 290, 582; doi:10.1016/0550-3213(87)90203-3.
Bouchaud J.P., Orland H., J. Stat. Phys., 1990, 61, 877; doi:10.1007/BF01027306.
Brunet E., Derrida B., Phys. Rev. E, 2000, 61, 6789; doi:10.1103/PhysRevE.61.6789.
Johansson K., Commun. Math. Phys., 2000, 209, 437; doi:10.1007/s002200050027.
Prähofer M., Spohn H., J. Stat. Phys., 2002, 108, 1071; doi:10.1023/A:1019791415147.
Ferrari P.L., Spohn H., Commun. Math. Phys., 2006, 265, 1; doi:10.1007/s00220-006-1549-0.
Sasamoto T., Spohn H., Phys. Rev. Lett., 2010, 104, 230602; doi:10.1103/PhysRevLett.104.230602.
Sasamoto T., Spohn H., Nucl. Phys. B, 2010, 834, 523; doi:10.1016/j.nuclphysb.2010.03.026.
Sasamoto T., Spohn H., J. Stat. Phys., 2010, 140, 209; doi:10.1007/s10955-010-9990-z.
Amir G., Corwin I., Quastel J., Commun. Pure Appl. Math., 2011, 64, 466; doi:10.1002/cpa.20347.
Dotsenko V., Klumov B., J. Stat. Mech., 2010, P03022; doi:10.1088/1742-5468/2010/03/P03022.
Dotsenko V., Europhys. Lett., 2010, 90, 20003; doi:10.1209/0295-5075/90/20003.
Dotsenko V., J. Stat. Mech., 2010, P07010; doi:10.1088/1742-5468/2010/07/P07010.
Calabrese P., Le Doussal P., Rosso A., Europhys. Lett., 2010, 90, 20002; doi:10.1209/0295-5075/90/20002.
Calabrese P., Le Doussal P., Phys. Rev. Lett., 2011, 106, 250603; doi:10.1103/PhysRevLett.106.250603.
Le Doussal P., Calabrese P., J. Stat. Mech., 2012, P06001; doi:10.1088/1742-5468/2012/06/P06001;
Preprint arXiv:1204.2607, 2012.
Dotsenko V., J. Stat. Mech., 2012, P11014; doi:10.1088/1742-5468/2012/11/P11014.
Gueudré T., Le Doussal P., Europhys. Lett., 2012, 100, 26006; doi:10.1209/0295-5075/100/26006.
Corwin I., Random Matrices: Theory Appl., 2012, 1, 1130001; doi:10.1142/S2010326311300014;
Preprint arXiv:1106.1596, 2011.
Borodin A., Corwin I., Ferrari P., Preprint arXiv:1204.1024, 2012.
Tracy C.A., Widom H., Commun. Math. Phys., 1994, 159, 151; doi:10.1007/BF02100489.
Prolhac S., Spohn H., J. Stat. Mech., 2011, P01031; doi:10.1088/1742-5468/2011/01/P01031.
Dotsenko V., J. Phys. A: Math. Theor., 2013, 46, 355001; doi:10.1088/1751-8113/46/35/355001.
Imamura T., Sasamoto T., Spohn H., J. Phys. A: Math. Theor., 2013, 46, 355002;
doi:10.1088/1751-8113/46/35/355002.
Prolhac S., Spohn H., J. Stat. Mech., 2011, P03020; doi:10.1088/1742-5468/2011/03/P03020.
Lieb E.H., Liniger W., Phys. Rev., 1963, 130, 1605; doi:10.1103/PhysRev.130.1605.
McGuire J.B., J. Math. Phys., 1964, 5, 622; doi:10.1063/1.1704156.
Yang C.N., Phys. Rev., 1968, 168, 1920; doi:10.1103/PhysRev.168.1920.
Calabrese P., Caux J.-S., Phys. Rev. Lett., 2007, 98, 150403; doi:10.1103/PhysRevLett.98.150403.
Dotsenko V.S., Physics-Uspekhi, 2011, 54, No. 3, 259; doi:10.3367/UFNe.0181.201103b.0269.
Dotsenko V., J. Stat. Mech., 2013, P02012; doi:10.1088/1742-5468/2013/02/P02012.
33003-9
V. Dotsenko
N -точкова функцiя розподiлу вiльної енергiї
в одновимiрних хаотично напрямлених полiмерах
В. Доценко1,2
1 Унiверситет м. Париж VI, 75252 Париж, Францiя
2 Iнститут теоретичної фiзики iм. Л.Д. Ландау, 119334 Москва, РФ
Отримано явний вираз для N -точкової функцiї розподiлу вiльної енергiї в одновимiрному напрямленому
полiмерi в термiнах анзацу Бете в рамках методу реплiк. Отриманий результат еквiвалентний результату,
ранiше отриманому в роботi Пролака i Шпона [J. Stat. Mech., 2011, P03020].
Ключовi слова: напрямленi полiмери, хаотичний потенцiал, реплiки, флуктуацiї, функцiя розподiлу
33003-10