$N$-point free energy distribution function in one dimensional random directed polymers

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].


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 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][3][4][5][6][7][8][9][10][11][12][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 〈φ 2 〉 ∝ 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) 〈φ 2 〉 ∝ t while the free energy is proportional to ln(t ).
where β is the inverse temperature and F t (x) is the free energy. In the limit t → ∞, the free energy scales where f 0 is the selfaveraging free energy density and 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: so thatZ The aim of the present work is to study the N -point free energy probability distribution function 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.

N -point distribution function
The probability distribution function, equation (7) can be defined as follows: where (. . .) denotes the average over random potentials. Indeed, substituting here equation (6) we get which coincides with the definition (7). Performing the standard averaging over random potentials in equation (8) one obtains (for details see e.g. [20]) where the time dependent n-point wave function Ψ(x 1 , . . . , x n ; t ) (n = N k=1 L k ) is the solution of the imaginary time Schrödinger equation 33003-2 N -point free energy distribution function in one dimensional random directed polymers with κ = β 3 u and the initial condition A generic eigenstate of such a system is characterized by n momenta {Q a } (a = 1, . . . , n) which split into M (1 M n) clusters described by continuous real momenta q α (α = 1, . . . , M ) and having n α discrete imaginary parts with the global constraint The time dependent solution Ψ(x, t ) of the Schrödinger equation (11) with the initial conditions, equation (12), can be represented in the form of a linear combination of eigenfunctions Ψ (M ) Q (x): Here, δ(k, m) is the Kronecker symbol, the normalization factor and the eigenvalues: For a given set of integers {M ; n 1 , . . . ., n M }, the eigenfunctions Ψ (M ) Q (x) can be represented as follows (for details see [33][34][35][36][37]): 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 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 the permutations P (L 1 ,...,L N ) in equation (19) it is sufficient to split the momenta of each cluster into N parts: where the integers m k α = 0, 1, . . . , n α are constrained by the conditions and the momenta of every group q α , . . . , q α k l =1 m l α all belong to the particles whose coordinates are all equal to x k . Let us redefine: In this way, the summation over P (L 1 ,...,L N ) is changed by the summation over the integers {m k α }. Substituting equations (20)-(23) into equation (19) after simple algebra, we find where the normalization constant C M q; {m k α } 2 is given in equation (16) (with n α = N k=1 m k α ) and Substituting the expressions for q α 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 the type Γ 1 [the example of this kind of the product is given in [38], equation (A17)]. We do not reproduce it here as it turns out to be irrelevant in the limit t → ∞ (see below).

33003-4
N -point free energy distribution function in one dimensional random directed polymers the normalization factor C M (q; {m k α }) 2 , equation (16) (with n α = N k m k α ), can be represented as follows: Substituting equation (25) we get where The quadratic in m k α term in the exponential of equation (31) can be linearized as follows: Substituting the representation (34) into equation (31) and redefining the integration parameters

33003-5
we get where S f, y, q, The summations over m k α in the above expression can be performed as follows: where the integration goes over the contour C shown in figure 1. Redefining z k α → z k α /λ, in the limit

33003-6
N -point free energy distribution function in one dimensional random directed polymers λ → ∞, we get S f, y, q, Taking into account the Gamma function property lim |z|→0 Γ(1 + z) = 1, one can easily demonstrate (see e.g. [38]) that Thus, in the limit λ → ∞, the expression (37) takes the form of the Fredholm determinant whereÂ is the integral operator with the kernel where, by definition, z k M +1 ≡ z k 1 and q M +1 ≡ q 1 .

33003-8
N -point free energy distribution function in one dimensional random directed polymers

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.