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Probability distribution of the free energy of a directed polymer in a random medium

E´ ric Brunet*and Bernard Derrida

Laboratoire de Physique Statistique, E´ cole Normale Supe´rieure, 24 rue Lhomond, 75231 Paris Ce´dex 05, France 共Received 10 December 1999兲

We calculate exactly the first cumulants of the free energy of a directed polymer in a random medium for the geometry of a cylinder. By using the fact that the nth momentZn典of the partition function is given by the ground-state energy of a quantum problem of n interacting particles on a ring of length L, we write an integral equation allowing to expand these moments in powers of the strength of the disorder␥or in powers of n. For n small and n(L␥)1/2, the moments具Zn典take a scaling form which allows us to describe all the fluctuations of order 1/L of the free energy per unit length of the directed polymer. The distribution of these fluctuations is the same as the one found recently in the asymmetric exclusion process, indicating that it is characteristic of all the systems described by the Kardar-Parisi-Zhang equation in 1⫹1 dimensions.

PACS number共s兲: 64.60.Cn, 05.30.⫺d, 05.70.⫺a

I. INTRODUCTION

Directed polymers in a random medium is one of the sim- plest systems for which the effect of strong disorder can be studied 关1–3兴. At the mean-field level, it possesses a low- temperature phase, with a broken symmetry of replica 关4,5兴 similar to mean-field spin glasses关6兴. The problem is, how- ever, much better understood than spin glasses; in particular, one can write关4,5兴closed expressions of the mean-field free energy and one can predict the existence关7兴of phase transi- tions in all dimensions d⫹1⬎2⫹1. It is also an interesting system from the point of view of nonequilibrium phenom- ena: through the Kardar-Parisi-Zhang 共KPZ兲equation关8,9兴, it is related to ballistic growth models and, in 1⫹1 dimen- sions, to the asymmetric simple exclusion process 共ASEP兲 关3,9兴.

In the theory of disordered systems, the replica approach plays a very special role. On the one hand, it is one of the most powerful theoretical tools and often the only possible approach to study some strongly disordered systems. On the other hand, it is difficult to tell in advance whether the pre- dictions of the replica approach are correct or not. When it does not work, one can always try to break the symmetry of the replica 关6兴: this usually makes the calculations much more complicated without being certain that the results be- come correct. In the replica approach, the calculation usually starts with an integer number n of the replica. Then, as the limit of physical interest is the limit n→0, one has to extend to noninteger n results obtained for integer n. This is in fact the big difficulty of the replica approach, so it is useful to look at simple examples for which the n dependence can be studied in detail.

This is one of the motivations of the present work, where we show how to calculate integer and noninteger moments 具Znof the partition function Z of a directed polymer in 1

⫹1 dimensions. The geometry we consider is a cylinder in- finite in the t direction and periodic, of size L, in the x direc- tion 共i.e., xLx). The partition function Z(x,t) of a di-

rected polymer joining the points (0,0) and (x,t) on this cylinder is given by the path integral

Zx,t兲⫽

(0,0) (x,t)

Dys兲exp

0tds

12

d ydss

2

⫹␩„ys,s

册 冊

, 1

where the random medium is characterized by a Gaussian white noise␩(x,t),

具␩共x,t兲␩共x

,t

兲典⫽␥␦共xx

兲␦共tt

兲. 共2兲 One of the main goals of the present work is to calculate the cumulants limt→⬁lnkZ(t)c/t of the free energy per unit length of the directed polymer. These cumulants are the co- efficients of the small-n expansion of E(n,L,␥) defined as

En,L,␥兲⫽⫺lim

t→⬁

1

tln

ZZnx,tx,tn

. 3

This E(n,L,␥) was calculated exactly by Kardar 关10兴 for integer n and L⫽⬁. His closed expression E(n,⬁,␥)

⫽⫺n(n2⫺1)␥2/24 cannot, however, be continued to all values of n, in particular to negative n, as it would violate the fact that ⳵2E(n,L,)/n2 is negative. Therefore, one does not know the range of validity of this expression.

The second motivation of the present work is to test the universality class of the KPZ equation. The problem共1兲of a directed polymer in a random medium is described by the KPZ equation as several other problems such as growing interfaces or exclusion processes 关3兴. For certain models of this class, the asymmetric exclusion processes, the distribu- tion of the total current Yt integrated over time t, has been calculated exactly关11–15兴in the long-time limit. For large t, the generating function of this integrated current Yton a ring of L sites takes the form 关11,12兴

ln具e␣Yt⬃⌳max共␣兲t, 共4兲

*Electronic address: Eric.Brunet@physique.ens.fr

Electronic address: Bernard.Derrida@physique.ens.fr

PRE 61

1063-651X/2000/61共6兲/6789共13兲/$15.00 6789 ©2000 The American Physical Society

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and it was shown 关11–14兴, when L is large and when the parameter␣ in Eq.共4兲is of order L3/2that ⌳max(␣) takes the following scaling form:

max共␣兲⫺␣K1K2G共␣K3兲, 共5兲 where K1, K2, and K3 are three constants which depend on the system size L, the density of particles, and the asymme- try.

The interesting aspect of Eq.共5兲is that the function G() is universal关12,14,16兴in the sense that it does not depend on any of the microscopic parameters which define the model. It is given 共in a parametric form兲by

␤⫽⫺p

⫹⬁1 p3/2p , 6

G共␤兲⫽⫺p

⫹⬁1 p5/2p . 7

In the correspondence 关3兴 between the directed polymer problem and the asymmetric exclusion process through the KPZ equation, the role played by ln„Z(t)is the ratio Yt/L.

Comparing 具exp(Yt)典 andZn(t)in Eqs. 共3兲 and 共4兲, we see that n corresponds toL and E(n,L,) tomax(␣). If the function G(␤) is characteristic of systems described by the KPZ equation, we expect in the scaling regime 共large L and nL1/2) a relation similar to Eq. 共5兲 between E(n,L,) 关defined by Eq.共3兲兴 and n. This is indeed one of the main results of the present work: when L is large and n

L1/2, we find

En,L,␥兲⫽n2

24 ⫺

2

2L3/2Gn

2L␥兲. 共8兲 It is clear that in order to establish this relation we have to calculate noninteger moments of the partition function.

The paper is organized as follows. In Sec. II, we recall how the replica approach of Eq. 共1兲 can be formulated as a quantum problem with n particles on a ring and how this problem can be solved by the Bethe ansatz when the noise is

␦ correlated as in Eq.共2兲. In Sec. III, we write an integral equation 共26兲 which, together with some symmetry condi- tions共27兲and共28兲, allows us to solve the Bethe equations of Sec. II. The main advantage of Eq.共26兲is that the strength c of the disorder 共where c⫽␥L/2) and the number of the rep- lica appear as continuous parameters. We show how expan- sions in powers of c or in powers of the number n of replica can be obtained from this integral equation. In the expansion of the energy E(n,L,) in powers of c, all the coefficients are polynomials in n. This allows us to define E(n,L,) for a noninteger n at least perturbatively in c. At the end of Sec.

III, we show how to generate a small-n expansion which solves the integral equation 共26兲. We also give explicit ex- pressions up to order n3 and we notice that in this small-n expansion of the energy, we have to deal with coefficients that are functions of c with a zero radius of convergence. The content of Secs. II and III is essentially a recall of a method developed in our previous work 关17兴. In Sec. IV, we show that the recursion of Sec. III, which generates all the terms of

the small-n expansion, simplifies greatly in the scaling re- gime (c large and nc1/2), allowing us to calculate all the terms of the expansion and to establish Eq.共8兲.

II. A QUANTUM SYSTEM OF n PARTICLES WITHINTERACTIONS

Let us start with a case slightly more general than Eq.共2兲 where the noise ␩(x,t) in Eq. 共1兲 is a Gaussian noise

␦-correlated in time but with some given correlation v in space,

具␩共x,t兲␩共x

,t

兲典vxx

兲␦共tt

兲. 共9兲 If we consider the correlation function 具Z(x1,t) Z(x2,t)

•••Z(xn,t)of the partition function Z(x,t) at points x1, x2, . . . ,xn, one can check 关3兴from Eqs.共1兲and共9兲 that it satisfies

d

dtZx1,tZx2,t兲•••Zxn,t兲典

⫽⫺H˜Zx1,tZx2,t兲•••Zxn,t兲典, 10 where the HamiltonianH˜ is given by

⫽⫺1

2

x22

⬍␤ vxx兲⫺n2v共0兲, 共11兲 and where, because of the cylinder geometry in the directed polymer problem, we have xxL for 1⭐␣⭐n.

This implies that in the long-time limit,

Zx1,tZx2,t兲•••Zxn,t兲典⬃etE˜ (n,L,), 共12兲 where E˜ (n,L,␥) is the ground-state energy of Eq.共11兲.

If one takes the limit v(xx

)(xx

), the energy E˜ (n,L,␥) becomes infinite because of the constant part nv(0)/2 in Eq.共11兲. This divergence disappears, however, if we consider the ratio 具Z(x1,t)Z(x2,t)•••Z(xn,t)/Z(x,t)典, and one can see that in the long-time limit,

Zx1,tZx2,t兲•••Zxn,t兲典

Zx1,t兲典具Zx2,t兲典•••具Zxn,t兲典etE(n,L,), 13 where E(n,L,␥) is the ground-state energy of the Hamil- tonian

H⫽⫺1

2

x22⫺␥

⬍␤ xx, 14

where the positions x of the n particles are on a ring of length L.

Lieb and Liniger have shown that the Bethe ansatz allows us to calculate the ground-state energy E(n,L,␥) of this one- dimensional quantum Hamiltonian exactly关18–24兴. The Be- the ansatz consists in looking for a ground-state wave func- tion ⌿(x1, . . . ,xn) of Eq.共14兲of the form

⌿共x1, . . . ,xn兲⫽

P aPe2(q1xP(1)• • •qnxP(n))/L 15

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in the region 0⭐x1⭐. . .⭐xnL. The sum in Eq.共15兲runs over all the permutations P of1, . . . ,n其 and the value of⌿ in other regions can be deduced from Eq. 共15兲 by symme- tries. One can show关22–24,17兴that Eq.共15兲is the ground- state wave function of Eq.共14兲at energy

En,L,␥兲⫽⫺ 2 L2 1⭐␣⭐

n

q2, 共16兲 if the q are the solutions of the n coupled equations

e2q␤⫽␣

qqqqcc, 共17兲 obtained by continuity from the solution兵q0at c0, where

c⫽␥L

2 . 共18兲

Moreover, the q are all different and the ground state is symmetric (兵qq). See, for instance, 关22兴. Note that ikj and c in 关22兴 are here (2/L)qj and ⫺␥; so our c defined by Eq. 共18兲and the c in关22兴are different.兴

If we introduce the polynomial P(X), PX兲⫽

q

Xq兲, 共19兲

the system of equations共17兲becomes

eqPqc兲⫹eqPqc兲⫽0 共20兲 for any 1⭐␣⭐n, and we have from the symmetry of the ground state

P共⫺X兲⫽共⫺1兲nPX兲. 共21兲 The knowledge of the polynomial P(X) determines the en- ergy共16兲as

PX兲⫽Xn⫺1

2

1⭐␣⭐

n q2

Xn2••• 22

关using Eq.共19兲and the fact that兺q⫽0].

For small c, it is possible to solve directly Eq.共20兲and to determine the q共see Appendix D兲. This leads to the follow- ing expression of the ground-state energy 共16兲:

En,L,␥兲⫽⫺ 2

L2nn⫺1兲

2c12c2nc1803Oc4

.

共23兲 We see that the first coefficients of the small-c expansion are polynomial in n. In fact, following the approach of Appendix D, one can see that each coefficient of the small-c expansion of E(n,L,) is polynomial in n, allowing us to define, at least perturbatively in c, the ground-state energy E(n,L,) for noninteger n. The approach of Appendix D becomes, however, quickly complicated. This is why in the next sec- tion we develop a different approach关17兴based on the inte- gral equation共26兲.

III. SOLUTION OF THE BETHE ANSATZ USING AN INTEGRAL EQUATION

In this section we recall the approach developed in our previous work 关17兴, which consists in writing an integral equation where c and n appear as continuous parameters and which allows us to expand the energy in powers of c as well as in powers of n.

Let us introduce the following function of兵q: Bu兲⫽1

nec(u21)/4

q qeq(u1), 24

where the parameters ␳(q) are defined by

␳共q兲⫽q

q qqqqc. 共25兲 If the 兵q其 are given by the solution of Eq. 共17兲, which corresponds to the ground state, one can show共see Appendix A兲that the function B(u) satisfies the integral equation

B共1⫹u兲⫺B共1⫺u兲⫽nc

0 u

dvec(v2uv)/2

B共1⫺v兲B共1⫹uv兲 共26兲 and the following two conditions:

B共1兲⫽1, 共27兲

Bu兲⫽B共⫺u兲. 共28兲 Moreover, the energy共16兲can be extracted from the knowl- edge of B(u) through

En,L,␥兲⫽ 2

L2

n36c2nc122nc2nB

1

. 29

The derivation of Eqs.共26兲–共29兲is given in Appendix A.

We are now going to see how one can find perturbatively in c or in n the solution of Eqs. 共26兲–共28兲 and, consequently, the ground-state energy共29兲.

A. Expansion in powers of c

To obtain the small-c expansion of B(u) for arbitrary n, we write

Bu兲⫽B0u兲⫹cB1u兲⫹c2B2u兲⫹•••. 共30兲 Conditions 共27兲 and 共28兲 impose that B0(0)⫽1 and all Bk(1)⫽0 for k0, and that the Bk(u) are all even. More- over, as can be seen directly from Eq.共17兲, the q scale like

c when c is small. 共Appendix D shows how to obtain the small-c expansion of the q.) This implies from the defini- tion 共24兲of B(u) that all the Bk(u) are polynomials in u.

At zeroth order in c, Eq. 共26兲becomes

B0共1⫹u兲⫺B0共1⫺u兲⫽0. 共31兲 The only polynomial solution of Eq. 共31兲 consistent with Eqs. 共27兲and 共28兲, i.e., B0(u)B0(⫺u) and B0(1)⫽1, is simply

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B0u兲⫽1 共32兲 for any u. We put this back into Eq.共26兲and we get at first order in c

B1共1⫹u兲⫺B1共1⫺u兲⫽nu. 共33兲 Again, there is a unique polynomial solution which satisfies the facts that B1(u) is even and that B1(1)⫽0:

B1u兲⫽n

4共u2⫺1兲. 共34兲 It is easy to see from Eq.共26兲that at any order in c, we have to solve

Bk共1⫹u兲⫺Bk共1⫺u兲⫽␾ku兲, 共35兲 where ␾k(u) is a polynomial odd in u. There is a unique even polynomial Bk(u) solution of Eq. 共35兲 satisfying Bk(1)⫽0: it is one degree higher than ␾k(u) and can be determined by equating each power of u in both sides of Eq.

共35兲.„Alternatively, we found a way of writing the solution for any␾k(u):

Bku兲⫽

s0

1udvkv兲⫹s1k

u兲⫺k

1兲兴

s2关␾k⵮共u兲⫺␾k⵮共1兲兴⫹•••

sp关␾k

(2 p1)u兲⫺␾k

(2 p1)共1兲兴⫹•••

2,

共36兲 where the skare the coefficients of the expansion of x/sinh x in powers of xi.e., as x/sinh x⫽1⫺x2/6⫹7x4/360⫹•••, one has s01, s1⫽⫺1/6, s2⫽7/360, . . . ).…

This procedure gives for the first terms Bu兲⫽1⫹cnu2⫺1兲

4 ⫹c2n2n⫹1兲共u2⫺1兲2 96

c3nu2⫺1兲25n2u2⫺1兲⫹4n2u2⫺1兲

⫹2共u2⫺3兲…

5760 ⫹Oc4兲.

共37兲 The energy can then be deduced from Eq.共29兲:

En,L,␥兲⫽⫺2nn⫺1兲

L2

c212c2180n c3

1512n2 1260n

c4•••

. 38

关For Eq. 共38兲, we used more terms than given above in B(u).兴 Of course, this expression agrees with Eq. 共23兲 ob- tained directly by expanding the q.

B. Expansion in powers of n

The number of particles n is a priori an integer. However, when we look at the small-c expansion共37兲of B(u) or Eq.

共38兲 of the energy, we see that at any given order in c the expression is polynomial in n. Therefore, one can extend the

definition of the small-c expansion of B(u) or of E(n,L,) to noninteger n. We can also collect in the small-c expansion of B(u) all the terms proportional to n and call this series b1(u). From Eq.共37兲we see that

b1u兲⫽共u2⫺1兲

4 c⫹共u2⫺1兲2 96 c2

⫹共u2⫺1兲2u2⫺3兲

2880 c3Oc4兲. 共39兲 More generally, we can collect all the terms proportional to nk in the small-c expansion and call the series bk(u). This means that we can write B(u) as a power series in n,

Bu兲⫽1⫹nb1u兲⫹n2b2u兲⫹•••, 共40兲 where all the bk(u) are defined perturbatively in c. Condi- tions 共27兲 and 共28兲 impose that all the bk(u) are even and that bk(1)⫽0 for all k1. We define b0(u)⫽1 for consis- tency. 关It is easy to see in the small-c expansion that if n

0, then B(u)⫽1.兴

We are now going to describe the procedure we used关17兴 to determine the whole function b1(u) and eventually all the bk(u). If we insert Eq.共40兲into Eq.共26兲we get, at first order in n,

b1共1⫹u兲⫺b1共1⫺u兲⫽c

0 u

ec(v2uv)/2dv. 共41兲 It is easy to check that a solution of Eq.共41兲compatible with the conditions b1(1)⫽0 and b1(u)b1(⫺u) is

b1u兲⫽

c

0⫹⬁dcosh

u

c

2 ⫺cosh␭

c

2 sinh␭

c

2

e2/2.

共42兲 There are, however, many other solutions of Eq. 共41兲, which can be obtained by adding to Eq. 共42兲 an arbitrary function F(u,c) even and periodic in u of period 2 and van- ishing at u1. If we require that each term in the small-c expansion of b1(u) is polynomial in u 共as justified in Sec.

III A兲, we see that all the terms of the small-c expansion of F(u,c) must be identically zero. This already shows that Eq.

共42兲 has the same small c expansion共39兲 as one would get by collecting all the terms proportional to n in the small-c expansion of Sec. III A.

If the solution 共42兲 of Eq.共41兲 had a nonzero radius of convergence in c, it would be natural to choose this solution and set F(u,c)⫽0. However, it is easy to see that Eq. 共42兲 has a zero radius of convergence in c: by making the change of variable␭2⫽2␯, it is easy to see that Eq.共42兲is the Borel sum of a divergent series关25兴.

Apart from being the Borel sum of its expansion in pow- ers of c, we did not find definitive reasons why Eq. 共42兲 is the solution of Eq. 共41兲we should select. However, we can notice that for integer n, all the q are real and B(u) defined by Eq. 共24兲 is analytic in u and remains bounded asIm u

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. The solution b1(u) given by Eq.共42兲is also analytic in u and grows as ln(u) asIm u兩→⬁. Adding any function F(u,c) periodic and analytic in u to Eq.共42兲would produce a much faster growth.

If we insert Eq. 共40兲 into Eq. 共26兲, we have to solve at order nk

bk共1⫹u兲⫺bk共1⫺u兲⫽␸ku兲, 共43兲 where␸k(u) is some function odd in u which can be calcu- lated if we know the previous orders b1(u), . . . ,bk1(u),

ku兲⫽cki

01

0 u

dvec(v2uv)/2bi共1⫺v兲

bki1共1⫹uv兲. 共44兲 We see that the difficulty of selecting a solution of a differ- ence equation appears at all orders in the expansion in pow- ers of n, and we are now going to explain the procedure we have used to select one solution.

If we write, as ␸k(u) is an odd function of u,

ku兲⫽2

0

⫹⬁

d␭sinh␭u

c

2 ak共␭兲, 共45兲 which is equivalent, by inverting when u is imaginary the Fourier transform in Eq. 共45兲, to define ak(␭) by

ak共␭兲⫽ 1 2i

0

⫹⬁

du sinu

2 ␸k

iuc

, 46

then the solution for bk(u) we select is given by

bku兲⫽

0

⫹⬁

d

cosh␭u

c

2 ⫺cosh␭

c

2 sinh␭

c

2

ak共␭兲. 共47兲

Indeed, bk(u) is an even function, vanishes at u⫽1, and one can check using Eq. 共45兲that Eq.共47兲solves Eq.共43兲.

The integrals in Eqs. 共45兲–共47兲 are convergent关17兴 and Eqs. 共44兲–共47兲 give an automatic way of calculating the bk(u) up to any desired order.

This procedure is the direct generalization of the choice 共42兲we did to solve Eq.共41兲. In fact, for k⫽1, Eqs.共44兲and 共46兲give共for␭⭓0) a1(␭)⫽

c exp(⫺␭2/2) and Eq.共47兲is identical to Eq.共42兲.

As for Eq.共42兲, the solution共47兲is not the only solution of Eq. 共43兲. At any order k, we could add an arbitrary even periodic function F(u,c) of period 2, the expansion of which vanishes to all orders in c. As for b1(u), we did not find an unquestionable justification of our choice. One can notice, nevertheless, that Eq.共47兲is the solution of Eq.共43兲analytic in u and with the slowest growth with u in the imaginary direction.

At order n2, the procedure共44兲and共46兲gives

a2共␭兲⫽ce⫺␭2/2

0de⫺␮2/22 coshtanh22

c 2

⫹⬁

de⫺␮2/2e

␭␮/2⫺2

tanh␮

c

2

, 48

with b2(u) given by Eq.共47兲. Writing down b3(u) or a3(u) would take here about half a column.

We can now give the first terms in the small-n expansion of the energy. Using relation共29兲, we find

L2

2 En,L,␥兲⫽n

2c12c2

n2c43/2

0⫹⬁d 2

tanh␭

c

2

e⫺␭2/2n3 c2

4

0⫹⬁d 2

tanh␭

c

2

e⫺␭2/2

0de⫺␮2/22 coshtanh2

2c2

⫹⬁de⫺␮2/2tanhe␭␮2

22c

n36c2On4. 49

By making the change of variable␭2⫽2␯, the terms of order n2 and n3 appear as Borel transforms of series in c with a finite radius of convergence. We conclude that these terms both have a zero radius of convergence in c.

This small-n expansion gives quickly very complicated expressions of bk(u). It turns out, as we shall see in the next section, that for large c, the expressions of the bk(u) get simpler and the energy E(n,L,␥) can be calculated to all orders in powers of n.

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IV. EXPANSION IN POWERS OF n IN THE REGIME c\

In the preceding section, we have developed a procedure allowing to get the small-n expansion of the energy by solv- ing the problem 共26兲–共28兲. Here, we show how this proce- dure becomes greatly simplified for large c.

The expansion in powers of n of the preceding section can be summarized as follows: if we use Eq.共40兲and we write a共␭兲⫽na1共␭兲⫹n2a2共␭兲⫹•••, 共50兲 the bk(u) and ak(␭) can be obtained by expanding in powers of n the following two equations:

Bu兲⫽1⫹

0

⫹⬁

d

cosh␭u

c

2 ⫺cosh␭

c

2 sinh␭

c

2

a共␭兲 共51兲

关this is a rewriting of Eq.共47兲兴and

a共␭兲⫽ nc 2i

0

⫹⬁

du sinu 2

0

iu/c

dvec(v2iuv/c)/2

B共1⫺vB

1

iuc⫺v

. 52

关This is a rewriting of Eqs. 共44兲and共46兲.兴It will be conve- nient in the following to replace Eq.共52兲by its Fourier trans- form,

2

0⫹⬁dsinhu2

c a共␭兲

nc

0 u

dvec(v2uv)/2B共1⫺vB共1⫹uv兲. 共53兲 关This is a rewriting of Eqs.共44兲and共45兲.兴

We are going to see how one can simplify Eqs.共51兲–共53兲 when c is large. First we observe that for large c and u fixed of order 1, the expression b1(u) takes the scaling form

b1

1

uc

c

0⫹⬁eu/21e⫺␭2/2d. 54

One can check from Eqs.共44兲,共46兲, and共47兲that this scaling form is present at any order in the small-n expansion. In- deed, Eq. 共51兲becomes in the large-c limit

B

1

uc

1

0⫹⬁d␭ 共eu/21a共␭兲, 55

and using Eq. 共53兲we find

2

0⫹⬁dsinh2ua共␭兲⫽n

c

0udve⫺共v2uv兲/2B

1

vc

B

1u

⫺vc

. 56

It is apparent from Eqs.共55兲and共56兲that in the large-c limit the function B(1u/

c) depends only on u and n

c, and

a(␭) depends only on␭ and n

c. Let us introduce the con- stant K,

K⫽1⫺

0

⫹⬁

da共␭兲. 共57兲 Equation共55兲becomes

B

1

uc

K

0⫹⬁deu/2a共␭兲. 58

In Eq. 共56兲, if we write the integral from 0 to u as the dif- ference between an integral from 0 to ⫹⬁ and an integral from u to⫹⬁, and if we change the variable in the second integral to shift it to 0 to⫹⬁, we obtain

2

0

⫹⬁

d␭sinh␭u

2 a共␭兲⫽n

c

0

⫹⬁

dvev2/2B

1

vc

euv/2B

1u

⫺vc

euv/2

B

1u

cv

冊 册

. 59

If we replace B关1⫹(u⫺v)/

cand B关1⫺(u⫹v)/

c兴 by their expression 共58兲, we get after some rearrangements

2

0

⫹⬁

d␭sinh␭u

2 a共␭兲⫽n

c

0

⫹⬁

dvev2/2B

1

vc

2Ksinhu2v

0⫹⬁da

e␮v/22 sinh

uv⫹2

冊 册

. 60

Taking the Fourier transform of this expression for imagi- nary u, we get for␭⭓0

a共␭兲⫽n

c

0

⫹⬁

dvev2/2B

1

vc

K共␭⫺v兲⫹

0⫹⬁dae␮v/2共␭⫺v⫺

.

共61兲 This last expression can be used to calculate B(1u/

c)

using Eq.共58兲:

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B

1

uc

Kn

c

0⫹⬁dvev2/2B

1

vc

Kevu/2

0⫹⬁dae⫺␮v/2e(v⫹␮)u/2

.

共62兲 Finally, using Eq.共58兲, we recognize the relation

B

1

uc

Kn

c

0⫹⬁dve⫺v2/2B

1

vc

evu/2B

1u

⫺vc

. 63

We see that, in the large-c limit, Eqs.共51兲and共52兲reduce to this single equation共63兲. We are now going to see that Eq.

共63兲can be solved to all orders in the parameter n

c. If we

introduce the function␤(u) and the parameterdefined by

␤共u兲⫽ 1 2K

e

u2/4B

1

uc

64

and

⑀⫽2nK

c, 共65兲 then Eq.共63兲simply becomes

␤共u兲⫽ 1 2

e

u2/4⫹⑀

0

⫹⬁

dv␤共uv兲␤共⫺v兲. 共66兲 Using Eqs. 共27兲, 共29兲, and 共64兲, we can express the ground-state energy E(n,L,␥) in terms of␤(u):

En,L,␥兲⫽ 2

L2

n36c2nc122nc

00

. 67

It is clear that relation共66兲alone determines␤(u), at least perturbatively in ⑀. So, from Eq.共67兲, we only need to ex- tract␤(0) and

(0) from Eq. 共66兲.

It is easy to do it for the first orders in⑀directly from Eq.

共66兲. Moreover, we have found a way of calculating ␤(0) and ␤

(0), and hence the energy, to all orders in ⑀. This calculation is technical and we present it in Appendix B. The final result can be written as

n

c⫽ 1

2

k

⫹⬁1 k3/2k , 68

En,L,␥兲⫽ 2

L2

nc1224

c k

⫹⬁1 k5/2k

. 69

We see that the energy is defined in an implicit way:

expression共68兲allows us to calculate⑀as a function of n

c,

and Eq.共69兲gives the energy as a function of⑀. If we sub- stitute c using Eq. 共18兲, we obtain the result announced in Eq. 共8兲.

For small n, one can eliminatefrom Eqs.共68兲and共69兲. We get

L2

2 En,L,␥兲⫺nc2 12 ⫽

c

4

2n

c

822n

c2

18227

3

2n

c3

O„共n

c4

. 70

V. CONCLUSION

In this paper, we have calculated, using the replica method, the first cumulants共13兲and共49兲of the free energy of a directed polymer in a random medium共1兲for a cylinder geometry. We used the integral equation 共26兲of关17兴which together with conditions共27兲and共28兲allowed us to expand the moments 具Zn典 of the partition function in powers of the strength c of the disorder or in powers of the number n of the replica. All the coefficients of the small-c expansion共38兲are polynomial in n, allowing us to define the expansions for noninteger n. On the other hand, the coefficients of the ex- pansion共49兲in powers of n are complicated functions of c, with in general a zero radius of convergence at c⫽0. As already mentioned in 关17兴, we think that weak disorder ex- pansions of the moments具Zn典 have generically a zero radius of convergence for noninteger n when the disorder is Gauss- ian; this is already the case for a single Ising spin in a Gauss- ian random field.

To obtain our small-n expansion, we solved a difference equation共26兲which at each order in powers of n has several solutions. We selected the particular solution which has the slowest growth in the imaginary u direction and has the right small-c expansion, but we could not exclude other solutions.

A different approach, with a direct calculation of the first cumulants of the free energy, and not based on the replica, would therefore be very useful to test the validity of our expressions 共49兲, which we have been able to derive only perturbatively to all orders in c.

Although our expansion in powers of n becomes quickly very complicated, it simplifies when c is large and we could write in this limiting case all the terms of the small-n expan- sion 共68兲and共69兲. The expression共8兲 we obtain of the en- ergy E(n,L,) 关that is, through Eq. 共3兲, the expression of 具Zn典] is given exactly by the same scaling function as found for the ASEP. The present work therefore gives additional evidence that the scaling function G(␤) given by Eqs. 共6兲 and共7兲is characteristic of the long-time behavior of the KPZ equation in 1⫹1 dimensions on a ring and that the probabil- ity distribution of the free energy for a very long directed polymer on a ring should have a universal shape in the range where the fluctuations per unit length of the free energy are of order 1/L. Other universal distributions for the free energy of a directed polymer have been found recently for different geometries关26–30兴. Our present approach, based on the Be- the ansatz, is, at the moment, unable to recover these other distributions. One can try, however, to extend it to open boundary conditions 共in this case too, the Bethe ansatz can be used 关24兴兲 instead of periodic boundary conditions and

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see how this change of boundary conditions affects the dis- tribution of ln Z. Of course, it would be very nice to find a simpler approach which would somehow unify all these re- sults and allow us to relate all these universal distributions corresponding to the possible geometries, in the spirit of critical phenomena in two dimensions where conformal in- variance关31兴allows us to connect the properties of different geometries.

Technically, the approach followed in the present work is simply to try to find the q solution of Eq.共17兲and to cal- culate the energy共16兲, which is a symmetric function of the roots q, in such a way that n becomes a continuous vari- able. One could do the same in all kinds of situations. For example, in Appendix C, we show how to define and calcu- late symmetric functions of the roots of Hermite polynomials when the degree of the polynomial becomes noninteger.

Another interesting extension of the present work would be to consider more general correlations of the noise共9兲. The corresponding quantum problem becomes then the general problem of quantum particles interacting with an arbitrary pair potential. If the interactions are short ranged, one ex- pects the universality class of the KPZ equation to hold, so one could try to repeat our expansion in powers of c for a general potential共without the use of the Bethe ansatz兲simply by a standard perturbation theory in the strength of the po- tential. We believe that at any order in the strength of the potential, the ground-state energy is polynomial in n allow- ing us to define the perturbation expansion for noninteger n as we did here. If, with such an approach based on perturba- tion theory, one could recover the scaling function G of Eqs.

共6兲and 共7兲, one could try to extend the approach to higher dimension as the relation between the directed polymer prob- lem and the quantum Hamiltonian is valid in any dimension.

ACKNOWLEDGMENTS

We thank Franc¸ois David, Michel Gaudin, Vincent Pas- quier, Herbert Spohn, and Andre´ Voros for interesting dis- cussions.

APPENDIX A: DERIVATION OF EQS.2629… Let us first establish some useful properties of the num- bers␳(q) defined by Eq. 共25兲. If the q are the n roots of the polynomial P(X),

PX兲⫽

q Xq, A1

it is easy to see that the ␳(q) defined in Eq. 共25兲satisfy PXc

PX兲 ⫽1⫹c

q Xqq. 共A2兲 共The two sides have the same poles with the same residues and coincide at X→⬁.兲Expanding the right-hand side of Eq.

共A2兲for large X, we get PXc

PX兲 ⫽1⫹c

q Xq

1qXqX22

O

X14

.

共A3兲

On the other hand, using Eqs.共16兲and共A1兲and the symme- try兵qq, we have

PX兲⫽XnL2

4 En,L,␥兲Xn2OXn4兲, 共A4兲 so that

PXc

PX兲 ⫽1⫹nc X

c2

n2

X2

c3

n3

cEn,L,L2/2

X3

O

X14

. A5

Comparing Eqs.共A3兲and共A5兲, we get the relations

q

␳共q兲⫽n, 共A6兲

q

q␳共q兲⫽c

n2

, A7

q q2q兲⫽c2

n3

En,L,2L2. A8

Moreover, by letting X⫽⫾qc in Eq.共A2兲, we get for any q root of P(X)

1

c

q qqq c

q qqqc. 共A9兲 Lastly, using the symmetry兵qq其 and the definition 共25兲, the Bethe ansatz equations共17兲reduce to

eq␳共⫺q兲⫺eq␳共q兲⫽0. 共A10兲 From the definition共24兲of B(u) and the properties共A6兲– 共A10兲, it is straightforward to establish Eqs. 共26兲–共29兲: the integral equation 共26兲 is a direct consequence of Eqs. 共24兲 and共A9兲. Properties共27兲and共28兲follow from Eqs.共24兲and 共A6兲and Eqs.共24兲and共A10兲, respectively. Lastly, Eq.共29兲 is a consequence of Eqs.共24兲and共A6兲–共A8兲.

APPENDIX B: THE ENERGY IN THE SCALING REGIME In this appendix, we show how to calculate the energy from the integral equation共66兲. This equation is of the form

␤共u兲⫽Hu兲⫹⑀

0⫹⬁dvuvv, B1

where, in our case, H(u) is given by Hu兲⫽ 1

2

e

u2/4. 共B2兲

We are going to do our calculations for an arbitrary function H(u), even in u and decreasing fast enough共to make all the integrals converge兲when兩u兩→⬁.

To find the energy, we see from Eq.共67兲that we have to calculate from Eq. 共B1兲 the quantities ␤(0) and

(0) as functions of ⑀. We first show that Eq. 共B1兲is equivalent to

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