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directed polymers in a quenched random potential have been the subject of intense investigations during the past two decades ( see e.g. @xcite ) . in the one - dimensional case we deal with an elastic string directed along the @xmath1-axis within an interval @xmath2 $ ] . randomness enters the problem through a disorder potential @xmath3 $ ] , which competes against the elastic energy . the problem is defined by the hamiltonian @xmath4 = \int_{0}^{t } d\tau     \bigl\{\frac{1}{2 } \bigl[\partial_\tau \phi(\tau)\bigr]^2     + v[\phi(\tau),\tau]\bigr\};\ ] ] where the disorder potential @xmath5 $ ] is gaussian distributed with a zero mean @xmath6 and the @xmath7-correlations : @xmath8 here the parameter @xmath9 describes the strength of the disorder . note that such system is equivalent to the problem of the kardar - parisi - zang ( kpz ) equation describing the growth in time of an interface in the presence of noise @xcite .    in what follows we consider the problem in which the polymer is fixed at the origin , @xmath10 and it is free at @xmath11 . in other words , for a given realization of the random potential @xmath12 the partition function of the considered system is : @xmath13 where @xmath14}\ ] ] is the partition function of the system with the fixed boundary conditions , @xmath10 and @xmath15 and where @xmath16 is the total free energy . besides the usual extensive part @xmath17 ( where @xmath18 is the linear free energy density ) , the total free energy @xmath16 of such system is known to contain the disorder dependent fluctuating contribution @xmath19 . in the limit of large @xmath20 the typical value of the free energy fluctuations scales with @xmath20 as @xmath21 ( see e.g. @xcite ) . in other words , the total free energy of the system can be represented as @xmath22 where @xmath23 is a non - universal parameter , which depends on the temperature and the strength of disorder , and @xmath24 is the random quantity which in the thermodynamic limit @xmath25 is described by a non - trivial universal distribution function @xmath26 . note that according to eqs.([3])-([5 ] ) , the trivial self - averaging contribution @xmath27 to the free energy can be eliminated by a simple redefinition of the partition function : @xmath28 so that @xmath29 where @xmath30 for the similar problem with the zero boundary conditions , @xmath31 , the corresponding distribution function was proved to be described by the gaussian unitary ensemble ( gue ) tracy - widom distribution @xcite . in the course of this proof rather efficient bethe ansatz replica technique has been developed @xcite . in particular , in terms of this technique the corresponding multi - point free energy distribution functions have been derived @xcite . recently , the free energy distribution function for the directed polymer problem with the free boundary conditions , eqs.([1])-([4 ] ) , has been obtained @xcite . it was shown that the function @xmath26 is the gaussian orthogonal ensemble ( goe ) tracy - widom distribution . in this paper i would like to present sufficiently simple alternative way of derivation of the same result which does not require rather complicated technique of the fredholm pfaffian described in @xcite . let us introduce the function @xmath32 which gives the probability that the random free energy is bigger that a given value @xmath24 . it will be shown that in the thermodynamic limit , @xmath33 , this function is equal to the fredholm determinant @xmath34 with the kernel @xmath35 which is the goe tracy - widom distribution @xcite . explicitly , @xmath36\ ] ] where @xmath37 is the solution of the panlev ii differential equation , @xmath38 , with the boundary condition @xmath39 .    it should be noted that the present paper is rather technical . the main message of this work is not the final result itself ( which is well known anyway ) but the presentation of the general method and new technical tricks used in the derivation . section ii is devoted to the standard reformulation of the considered problem in terms of one - dimensional @xmath0-particle system of quantum bosons with attractive @xmath7-interactions @xcite . here it is shown that the calculation of the the free energy probability distribution function , eq.([9 ] ) , reduces to the summation over all the spectrum of eigenstates of this @xmath0-particle problem . this summation is performed in section iii , where in the thermodynamic limit , @xmath25 , the result , eqs.([10])-([11 ] ) is derived . the concluding remarks and as well as the key points of the calculations are listed in the final section iv . in terms of the partition function @xmath40 , eq.([7 ] ) , the function @xmath41 , eq.([9 ] ) , can be defined as follows : @xmath42 where @xmath43 denotes the averaging over quenched disorder . indeed , substituting here eq.([7 ] ) , we have @xmath44 \\ \nonumber \\ \nonumber & = & \int_{-\infty}^{+\infty } \ ; df ' \ ; p(f ' ) \ ; \theta\bigl(f - f'\bigr)\end{aligned}\ ] ] which coincides with the definition , eq.([9 ] ) . later on we will see that the integration over @xmath45 in the definition of the partition function , eq.([3 ] ) , requires proper regularization at both limits @xmath46 . for that reason it is convenient to represent it in the form of two contributions : @xmath47 thus , taking into account the definition eq.([6 ] ) , we get @xmath48 where @xmath49 using the relations , eqs.([1 ] ) , ( [ 2 ] ) and ( [ 4 ] ) , after simple gaussian averaging we obtain @xmath50    \ ;   \exp\bigl(-\beta   h_{n } [ \phi_{1 } , \phi_{2 } , ... , \phi_{n } ] \bigr)\ ] ] where @xmath51 \ ; = \ ;     \frac{1}{2 } \int_{0}^{t } d\tau \biggl (     \sum_{a=1}^{n } \bigl[\partial_\tau\phi_{a}(\tau)\bigr]^2     - \beta u \sum_{a\not= b}^{n }     \delta\bigl[\phi_{a}(\tau)-\phi_{b}(\tau)\bigr ] \biggr)\ ] ] the propagator @xmath52 , eq.([17 ] ) , describes @xmath0 trajectories @xmath53 all starting at zero ( @xmath54 ) , and coming to @xmath0 different points @xmath55 at @xmath11 . one can easily show that @xmath52 can be obtained as the solution of the linear differential equation @xmath56 with the initial condition @xmath57 and the interaction parameter @xmath58 . one can easily see that eq.([19 ] ) is the imaginary - time schrdinger equation @xmath59 with the hamiltonian @xmath60 which describes @xmath0 bose - particles interacting via the _ attractive _ two - body potential @xmath61 . a generic eigenstate of such system is characterized by @xmath0 momenta @xmath62 which are splitted into @xmath63 ( @xmath64 ) `` clusters '' described by continuous real momenta @xmath65 @xmath66 and having @xmath67 discrete imaginary `` components '' ( for details see @xcite ) : @xmath68 with the constraint @xmath69 a generic solution @xmath70 of the schrdinger equation ( [ 19 ] ) with the initial conditions , eq.([20 ] ) , can be represented in the form of the linear combination of the eigenfunctions @xmath71 : @xmath72 \ ;    \psi^{(m)}_{{\bf q}}({\bf x } ) { \psi^{(m)}_{{\bf q}}}^{*}({\bf 0 } ) \ ; \exp\bigl\{-e_{m}({\bf q } ) t \bigr\}\ ] ] where we have introduced the notation @xmath73 { \boldsymbol \delta}\bigl(\sum_{\alpha=1}^{m } n_{\alpha } \ ; , \ ;   n\bigr)\ ] ] and @xmath74 is the kronecker symbol ; note that the presence of this kronecker symbol in the above equation allows to extend the summations over @xmath67 s to infinity . here ( non - normalized ) eigenfunctions are @xcite @xmath75 \ ; \exp\bigl[i \sum_{a=1}^{n } q_{{\cal p}_{a } } x_{a } \bigr]\ ] ] where the summation goes over @xmath76 permutations @xmath77 of @xmath0 momenta @xmath78 , eq.([23 ] ) , over @xmath0 particles @xmath79 ; the normalization factor @xmath80 and the eigenvalues : @xmath81 note that the eigenfunctions , eq.([28 ] ) , are symmetric with respect to permutations of all its arguments @xmath82 , and @xmath83 in this way the problem of the calculation of the free energy probability distribution function , eq.([15 ] ) , reduces to the summation over all the spectrum of the eigenstates of the @xmath0-particle bosonic problem , which is parametrized by the set of both the continuous , @xmath84 , and the discrete @xmath85 degrees of freedom . substituting eqs.([26])-([31 ] ) into eq.([15 ] ) , ( defining @xmath86 , the factor @xmath18 drops out of the further calculations ) we get : @xmath87 \ ; { \boldsymbol \delta}\bigl(\sum_{\alpha=1}^{m } n_{\alpha } \ ; , \ ;   n\bigr ) \ ; |\tilde{c}_{m}({\bf q } , { \bf n})|^{2 } \ ; i_{k , l } ( { \bf q } , { \bf n } ) \label{32}\end{aligned}\ ] ] where @xmath88 and @xmath89 \times \prod_{a < b}^{k}\biggl[\frac{q_{{\cal p}_a^{(k ) } } - q_{{\cal p}_b^{(k ) } }   - i \kappa } { q_{{\cal p}_a^{(k ) } } - q_{{\cal p}_b^{(k)}}}\biggr ] \times \prod_{c < d}^{l}\biggl[\frac{q_{{\cal p}_c^{(l ) } } - q_{{\cal p}_d^{(l ) } }   + i \kappa } { q_{{\cal p}_c^{(l ) } } - q_{{\cal p}_d^{(l)}}}\biggr ] \times \\ \nonumber \\ \nonumber & \times & \int_{-\infty < x_{1 } \leq ... \leq x_{k}\leq 0 } dx_{1 } ... dx_{k } \ ; \exp\bigl[i \sum_{a=1}^{k } ( q_{{\cal p}_{a}^{(k)}}-i\epsilon ) x_{a } \bigr ] \\ \nonumber \\ & \times & \int_{0 \leq y_{l } \leq ... \leq y_{1 } < + \infty } dy_{l } ... dy_{1 } \ ; \exp\bigl[i \sum_{c=1}^{l } ( q_{{\cal p}_{c}^{(l)}}+i\epsilon ) y_{c } \bigr ] \label{33}\end{aligned}\ ] ] here the summation over all permutations @xmath77 of @xmath90 momenta @xmath91 over @xmath92 `` negative '' particles @xmath93 and @xmath94 `` positive '' particles @xmath95 are divided into three parts : the permutations @xmath96 of @xmath92 momenta ( taken at random out of the total list @xmath91 ) over @xmath92 `` negative '' particles , the permutations @xmath97 of the remaining @xmath94 momenta over @xmath94 `` positive '' particles , and finally the permutations @xmath98 ( or the exchange ) of the momenta between the group @xmath99 and the group @xmath100 . note also that the integrations both over @xmath79 s and over @xmath101 s in eq.([33 ] ) require proper regularization at @xmath102 and @xmath103 correspondingly . this is done in the standard way by introducing a supplementary parameter @xmath104 which will be set to zero in final results . the result of the integrations can be represented as follows : @xmath105 \times \\ \nonumber \\ \nonumber & \times & \sum_{{\cal p}^{(k ) } } \ ; \ ; \frac{1}{q^{(-)}_{{\cal p}_{1}^{(k ) } } \bigl(q^{(-)}_{{\cal p}_{1}^{(k ) } } + q^{(-)}_{{\cal p}_{2}^{(k)}}\bigr ) ... \bigl(q^{(-)}_{{\cal p}_{1}^{(k ) } } + ... + q^{(-)}_{{\cal p}_{k}^{(k)}}\bigr ) } \prod_{a < b}^{k}\biggl[\frac{q^{(-)}_{{\cal p}_a^{(k ) } } - q^{(-)}_{{\cal p}_b^{(k ) } }   - i \kappa } { q^{(-)}_{{\cal p}_a^{(k ) } } - q^{(-)}_{{\cal p}_b^{(k)}}}\biggr ] \times \\ \nonumber \\ & \times & \sum_{{\cal p}^{(l ) } } \ ; \ ; \frac{(-1)^{l}}{q^{(+)}_{{\cal p}_{1}^{(l ) } } \bigl(q^{(+)}_{{\cal p}_{1}^{(l ) } } + q^{(+)}_{{\cal p}_{2}^{(l)}}\bigr ) ... \bigl(q^{(+)}_{{\cal p}_{1}^{(l ) } } + ... + q^{(+)}_{{\cal p}_{l}^{(l)}}\bigr ) } \prod_{c < d}^{l}\biggl[\frac{q^{(+)}_{{\cal p}_c^{(l ) } } - q^{(+)}_{{\cal p}_d^{(l ) } }   + i \kappa } { q^{(+)}_{{\cal p}_c^{(l ) } } - q^{(+)}_{{\cal p}_d^{(l)}}}\biggr ] \label{34}\end{aligned}\ ] ] where @xmath106 using the `` magic '' bethe ansatz combinatorial identity @xcite , @xmath107 \ ; = \ ; \frac{1}{\prod_{a=1}^{n } q_{a } } \ ; \prod_{a < b}^{n}\biggl[\frac{q_{a } + q_{b }   + i \kappa } { q_{a } + q_{b}}\biggr]\ ] ] ( where the summation goes over all permutations @xmath108 of @xmath0 momenta @xmath109 ) we get : @xmath105 \times \\ \nonumber \\ & \times & \frac{1}{\prod_{a=1}^{k } q^{(-)}_{{\cal p}_{a}^{(k ) } } } \prod_{a < b}^{k}\biggl[\frac{q^{(-)}_{{\cal p}_a^{(k ) } } + q^{(-)}_{{\cal p}_b^{(k ) } }   + i \kappa } { q^{(-)}_{{\cal p}_a^{(k ) } } + q^{(-)}_{{\cal p}_b^{(k)}}}\biggr ] \times \frac{(-1)^{l}}{\prod_{c=1}^{l } q^{(+)}_{{\cal p}_{c}^{(l ) } } } \prod_{c < d}^{l}\biggl[\frac{q^{(+)}_{{\cal p}_c^{(l ) } } + q^{(+)}_{{\cal p}_d^{(l ) } }   - i \kappa } { q^{(+)}_{{\cal p}_c^{(l ) } } + q^{(+)}_{{\cal p}_d^{(l)}}}\biggr ] \label{36}\end{aligned}\ ] ] further simplification comes from one important property of the bethe ansatz wave function , eq.([28 ] ) . it has such structure that for ordered particles positions ( e.g. @xmath110 ) in the summation over permutations the momenta @xmath78 belonging to the same cluster also remain ordered . in other words , if we consider the momenta , eq.([23 ] ) , of a cluster @xmath111 , @xmath112 , belonging correspondingly to the particles @xmath113 , the permutation of any two momenta @xmath114 and @xmath115 of this _ ordered _ set gives zero contribution . thus , in order to perform the summation over the permutations @xmath98 in eq.([36 ] ) it is sufficient to split the momenta of each cluster into two parts : @xmath116 , where @xmath117 and where the momenta @xmath118 belong to the particles of the sector @xmath99 , while the momenta @xmath119 belong to the particles of the sector @xmath100 . let us introduce the numbering of the momenta of the sector @xmath100 in the reversed order : @xmath120 where @xmath121 and ( s.f . eq.([23 ] ) ) @xmath122 by definition , the integer parameters @xmath123 and @xmath124 fulfill the global constrains @xmath125 in this way the summation over permutations @xmath98 in eq.([36 ] ) is changed by the summations over the integer parameters @xmath123 and @xmath124 : @xmath126 \ ; { \boldsymbol \delta}\bigl(\sum_{\alpha=1}^{m } m_{\alpha}\ ; , \ ; k\bigr ) \ ; { \boldsymbol \delta}\bigl(\sum_{\alpha=1}^{m } s_{\alpha}\ ; , \ ; l\bigr ) \ ; \bigl ( ... \bigr)\ ] ] which allows to lift the summations over @xmath92 , @xmath94 , and @xmath127 in eq.([32 ] ) . in terms of the parameters @xmath123 and @xmath124 the product factors in eq.([36 ] ) are expressed as follows : @xmath128 @xmath129 & = & \prod_{\alpha=1}^{m } \prod_{1\leq r < r'}^{m_{\alpha } } \biggl [ \frac { { q^{\alpha}_{r}}^{(-)}+{q^{\alpha}_{r'}}^{(-)}+i\kappa } { { q^{\alpha}_{r}}^{(-)}+{q^{\alpha}_{r'}}^{(- ) } } \biggr ] \times \prod_{1\leq\alpha<\beta}^{m } \prod_{r=1}^{m_{\alpha}}\prod_{r'=1}^{m_{\beta } } \biggl [ \frac { { q^{\alpha}_{r}}^{(-)}+{q^{\beta}_{r'}}^{(-)}+i\kappa } { { q^{\alpha}_{r}}^{(-)}+{q^{\beta}_{r'}}^{(- ) } } \biggr ] \\ \nonumber \\ \nonumber \\ \label{45 } \prod_{c < d}^{l } \biggl [ \frac { q^{(+)}_{{\cal p}_c^{(l ) } } + q^{(+)}_{{\cal p}_d^{(l)}}-i \kappa } { q^{(+)}_{{\cal p}_c^{(l ) } } + q^{(+)}_{{\cal p}_d^{(l)}}}\biggr ] & = & \prod_{\alpha=1}^{m } \prod_{1\leq r < r'}^{s_{\alpha } } \biggl [ \frac { { { q^{*}}^{\alpha}_{r}}^{(+)}+{{q^{*}}^{\alpha}_{r'}}^{(+)}-i\kappa } { { { q^{*}}^{\alpha}_{r}}^{(+)}+{{q^{*}}^{\alpha}_{r'}}^{(+ ) } } \biggr ] \times \prod_{1\leq\alpha<\beta}^{m } \prod_{r=1}^{s_{\alpha}}\prod_{r'=1}^{s_{\beta } } \biggl [ \frac { { { q^{*}}^{\alpha}_{r}}^{(+)}+{{q^{*}}^{\beta}_{r'}}^{(+)}-i\kappa } { { { q^{*}}^{\alpha}_{r}}^{(+)}+{{q^{*}}^{\beta}_{r'}}^{(+ ) } } \biggr ] \\ \nonumber \\ \nonumber \prod_{a=1}^{k } \prod_{c=1}^{l } \biggl [ \frac { q_{{\cal p}_a^{(k ) } } - q_{{\cal p}_c^{(l ) } } - i\kappa } { q_{{\cal p}_a^{(k ) } } - q_{{\cal p}_c^{(l ) } } } \biggr ] & = & \prod_{1\leq\alpha<\beta}^{m } \biggl\ { \prod_{r=1}^{m_{\alpha}}\prod_{r'=1}^{s_{\beta } } \biggl [ \frac { q^{\alpha}_{r } - { q^{*}}^{\beta}_{r ' } - i\kappa } { q^{\alpha}_{r } + { q^{*}}^{\beta}_{r ' }   } \biggr ] \times \prod_{r=1}^{s_{\alpha}}\prod_{r'=1}^{m_{\beta } } \biggl [ \frac { { q^{*}}^{\alpha}_{r } - q^{\beta}_{r ' } - i\kappa } { { q^{*}}^{\alpha}_{r } - q^{\beta}_{r ' }   } \biggr ] \biggr\ } \times \\ \nonumber \\ \nonumber \\ \label{46 } & \times & \prod_{\alpha=1}^{m } \prod_{r=1}^{m_{\alpha}}\prod_{r'=1}^{s_{\alpha } } \biggl [ \frac { q^{\alpha}_{r } - { q^{*}}^{\alpha}_{r ' } - i\kappa } { q^{\alpha}_{r } - { q^{*}}^{\alpha}_{r ' }   } \biggr]\end{aligned}\ ] ] substituting eqs.([41])-([46 ] ) into eq.([36 ] ) , and then substituting the resulting expression into eq.([32 ] ) we obtain @xmath130 \ ;    \ ; \prod_{1\leq\alpha<\beta}^{m } \ ; { \cal g}_{\alpha\beta } \bigl({\bf q } , { \bf m } , { \bf s}\bigr ) \biggr\ } \label{47}\end{aligned}\ ] ] where @xmath131 @xmath132 \prod_{r < r'}^{s_{\alpha } } \biggl [ \frac { { { q^{*}}^{\alpha}_{r}}^{(+)}+{{q^{*}}^{\alpha}_{r'}}^{(+)}-i\kappa } { { { q^{*}}^{\alpha}_{r}}^{(+)}+{{q^{*}}^{\alpha}_{r'}}^{(+ ) } } \biggr ] \prod_{r=1}^{m_{\alpha}}\prod_{r'=1}^{s_{\alpha } } \biggl [ \frac { q^{\alpha}_{r } - { q^{*}}^{\alpha}_{r ' } - i\kappa } { q^{\alpha}_{r } - { q^{*}}^{\alpha}_{r ' }   } \biggr]\ ] ] and @xmath133 \prod_{r=1}^{s_{\alpha}}\prod_{r'=1}^{s_{\beta } } \biggl [ \frac { { { q^{*}}^{\alpha}_{r}}^{(+)}+{{q^{*}}^{\beta}_{r'}}^{(+)}-i\kappa } { { { q^{*}}^{\alpha}_{r}}^{(+)}+{{q^{*}}^{\beta}_{r'}}^{(+ ) } } \biggr ] \prod_{r=1}^{m_{\alpha}}\prod_{r'=1}^{s_{\beta } } \biggl [ \frac { q^{\alpha}_{r } - { q^{*}}^{\beta}_{r ' } - i\kappa } { q^{\alpha}_{r } + { q^{*}}^{\beta}_{r ' }   } \biggr ] \times \prod_{r=1}^{s_{\alpha}}\prod_{r'=1}^{m_{\beta } } \biggl [ \frac { { q^{*}}^{\alpha}_{r } - q^{\beta}_{r ' } - i\kappa } { { q^{*}}^{\alpha}_{r } - q^{\beta}_{r ' }   } \biggr]\ ] ] the product factors in eq.([48 ] ) can be easily expressed it terms of the gamma functions : @xmath134 \ ; = \ ; ( i\kappa)^{m_{\alpha } } \frac { \gamma\bigl ( \frac{1}{2}-\frac{s_{\alpha}-m_{\alpha}}{2 } - \frac{i{q_{\alpha}}^{(-)}}{\kappa } \bigr ) } { \gamma\bigl ( \frac{1}{2}-\frac{s_{\alpha}+m_{\alpha}}{2 } - \frac{i{q_{\alpha}}^{(-)}}{\kappa } \bigr ) } \\ \nonumber \\ \nonumber \\ \prod_{r=1}^{s_{\alpha}}{{q^{*}}^{\alpha}_{r}}^{(+ ) } & = & \prod_{r=1}^{s_{\alpha } } \bigl [ { q_{\alpha}}^{(+ ) } + \frac{i\kappa}{2}(m_{\alpha}+s_{\alpha}+1 ) - i\kappa r \bigr ] \ ; = \ ; ( -i\kappa)^{s_{\alpha } } \frac { \gamma\bigl ( \frac{1}{2}-\frac{m_{\alpha}-s_{\alpha}}{2 } + \frac{i{q_{\alpha}}^{(+)}}{\kappa } \bigr ) } { \gamma\bigl ( \frac{1}{2}-\frac{m_{\alpha}+s_{\alpha}}{2 } + \frac{i{q_{\alpha}}^{(+)}}{\kappa } \bigr ) } \label{51}\end{aligned}\ ] ] @xmath135 & = & 2^{-(m_{\alpha}-1 ) } \frac { \gamma\bigl ( m_{\alpha}-s_{\alpha } - \frac{2i{q_{\alpha}}^{(-)}}{\kappa } \bigr ) \gamma\bigl ( 1-\frac{m_{\alpha}+s_{\alpha}}{2 } - \frac{i{q_{\alpha}}^{(-)}}{\kappa } \bigr ) } { \gamma\bigl ( \frac{m_{\alpha}-s_{\alpha}}{2 } - \frac{i{q_{\alpha}}^{(-)}}{\kappa } \bigr ) \gamma\bigl ( 1 - s_{\alpha } - \frac{2i{q_{\alpha}}^{(-)}}{\kappa } \bigr ) } \\ \nonumber \\ \nonumber \\ \label{53 } \prod_{r < r'}^{s_{\alpha } } \biggl [ \frac { { { q^{*}}^{\alpha}_{r}}^{(+)}+{{q^{*}}^{\alpha}_{r'}}^{(+)}-i\kappa } { { { q^{*}}^{\alpha}_{r}}^{(+)}+{{q^{*}}^{\alpha}_{r'}}^{(+ ) } } \biggr ] & = & 2^{-(s_{\alpha}-1 ) } \frac { \gamma\bigl ( s_{\alpha}-m_{\alpha } + \frac{2i{q_{\alpha}}^{(+)}}{\kappa } \bigr ) \gamma\bigl ( 1-\frac{m_{\alpha}+s_{\alpha}}{2 } + \frac{i{q_{\alpha}}^{(+)}}{\kappa } \bigr ) } { \gamma\bigl ( \frac{s_{\alpha}-m_{\alpha}}{2 } + \frac{i{q_{\alpha}}^{(+)}}{\kappa } \bigr ) \gamma\bigl ( 1 - m_{\alpha } + \frac{2i{q_{\alpha}}^{(+)}}{\kappa } \bigr ) } \\ \nonumber \\ \nonumber \\ \label{54 } \prod_{r=1}^{m_{\alpha}}\prod_{r'=1}^{s_{\alpha } } \biggl [ \frac { q^{\alpha}_{r } - { q^{*}}^{\alpha}_{r ' } - i\kappa } { q^{\alpha}_{r } - { q^{*}}^{\alpha}_{r ' }   } \biggr ] & = & \frac { \gamma\bigl(1 + m_{\alpha } + s_{\alpha}\bigr ) } { \gamma\bigl(1 + m_{\alpha}\bigr ) \gamma\bigl(1 + s_{\alpha}\bigr)}\end{aligned}\ ] ] substituting the above expressions into eq.([48 ] ) and using the standard relations for the gamma functions , @xmath136 for the factor @xmath137 , eq.([48 ] ) , we get @xmath138 similar calculations for the factor @xmath139 yield the following expression @xmath140 \ , \gamma \bigl [ 1 - \frac{m_{\alpha } + m_{\beta } + s_{\alpha } + s_{\beta}}{2 } -\frac{i}{\kappa}\bigl({q_{\alpha}}^{(- ) } + { q_{\beta}}^{(-)}\bigr ) \bigr ] } { \gamma \bigl [ 1 - \frac{m_{\alpha } - m_{\beta } + s_{\alpha } + s_{\beta}}{2 } -\frac{i}{\kappa}\bigl({q_{\alpha}}^{(- ) } + { q_{\beta}}^{(-)}\bigr ) \bigr ] \ , \gamma \bigl [ 1 + \frac{m_{\alpha } - m_{\beta } - s_{\alpha } - s_{\beta}}{2 } -\frac{i}{\kappa}\bigl({q_{\alpha}}^{(- ) } + { q_{\beta}}^{(-)}\bigr ) \bigr ] } \times \\ \nonumber \\ \nonumber \\ \nonumber & \times & \frac { \gamma \bigl [ 1 - \frac{m_{\alpha } + m_{\beta } - s_{\alpha } - s_{\beta}}{2 } + \frac{i}{\kappa}\bigl({q_{\alpha}}^{(+ ) } + { q_{\beta}}^{(+)}\bigr ) \bigr ] \ , \gamma \bigl [ 1 - \frac{m_{\alpha } + m_{\beta } + s_{\alpha } + s_{\beta}}{2 } + \frac{i}{\kappa}\bigl({q_{\alpha}}^{(+ ) } + { q_{\beta}}^{(+)}\bigr ) \bigr ] } { \gamma \bigl [ 1 - \frac{m_{\alpha } + m_{\beta } + s_{\alpha } - s_{\beta}}{2 } + \frac{i}{\kappa}\bigl({q_{\alpha}}^{(+ ) } + { q_{\beta}}^{(+)}\bigr ) \bigr ] \ , \gamma \bigl [ 1 - \frac{m_{\alpha } + m_{\beta } - s_{\alpha } + s_{\beta}}{2 } + \frac{i}{\kappa}\bigl({q_{\alpha}}^{(+ ) } + { q_{\beta}}^{(+)}\bigr ) \bigr ] } \times \\ \nonumber \\ \nonumber \\ \nonumber & \times & \frac { \gamma \bigl [ 1 + \frac{m_{\alpha } + m_{\beta } + s_{\alpha } + s_{\beta}}{2 } + \frac{i}{\kappa}\bigl(q_{\alpha } - q_{\beta}\bigr ) \bigr ] \ , \gamma \bigl [ 1 + \frac{-m_{\alpha } + m_{\beta } + s_{\alpha } - s_{\beta}}{2 } + \frac{i}{\kappa}\bigl(q_{\alpha } - q_{\beta}\bigr ) \bigr ] } { \gamma \bigl [ 1 + \frac{-m_{\alpha } + m_{\beta } + s_{\alpha } + s_{\beta}}{2 } + \frac{i}{\kappa}\bigl(q_{\alpha } - q_{\beta}\bigr ) \bigr ] \ , \gamma \bigl [ 1 + \frac{m_{\alpha } + m_{\beta } + s_{\alpha } - s_{\beta}}{2 } + \frac{i}{\kappa}\bigl(q_{\alpha } - q_{\beta}\bigr ) \bigr ] } \times \\ \nonumber \\ \nonumber \\ \label{59 } & \times & \frac { \gamma \bigl [ 1 + \frac{m_{\alpha } + m_{\beta } + s_{\alpha } + s_{\beta}}{2 } -\frac{i}{\kappa}\bigl(q_{\alpha } - q_{\beta}\bigr ) \bigr ] \ , \gamma \bigl [ 1 + \frac{m_{\alpha } - m_{\beta } - s_{\alpha } + s_{\beta}}{2 } -\frac{i}{\kappa}\bigl(q_{\alpha } - q_{\beta}\bigr ) \bigr ] } { \gamma \bigl [ 1 + \frac{m_{\alpha } + m_{\beta } - s_{\alpha } + s_{\beta}}{2 } -\frac{i}{\kappa}\bigl(q_{\alpha } - q_{\beta}\bigr ) \bigr ] \ , \gamma \bigl [ 1 + \frac{m_{\alpha } - m_{\beta } + s_{\alpha } + s_{\beta}}{2 } -\frac{i}{\kappa}\bigl(q_{\alpha } - q_{\beta}\bigr ) \bigr]}\end{aligned}\ ] ]    redefining @xmath141 with @xmath142 the normalization factor @xmath143 , eq.([47a ] ) , can be represented as follows : @xmath144 \times \det \biggl [ \frac{1 } { \lambda\bigl(m_{\alpha}+s_{\alpha}\bigr ) - ip_{\alpha } + \lambda\bigl(m_{\beta}+s_{\beta}\bigr ) + ip_{\beta } } \biggr]_{\alpha,\beta=1, ... ,m}\end{aligned}\ ] ] where we have used the cauchy double alternant identity @xmath145_{\alpha,\beta=1, ... m}\ ] ] with @xmath146 and @xmath147 . after rescaling , eq.([60 ] ) , for the exponential factor in eq.([47 ] ) we find @xmath148 the cubic exponential term can be linearized using the airy function relation @xmath149 \ ; = \ ; \int_{-\infty}^{+\infty } dy_{\alpha } \ ; \operatorname{ai}(y_{\alpha } ) \ ; \exp\bigl[\lambda ( m_{\alpha}+s_{\alpha } ) \ , y_{\alpha } \bigr]\ ] ] substituting eqs.([65]),([64 ] ) and ( [ 62 ] ) into eq.([47 ] ) , and redefining @xmath150 , we get @xmath151 \times \\ \nonumber \\ & \times & \det \hat{k}\bigl[(\lambda m_{\alpha},\ , \lambda s_{\alpha } , \ , p_{\alpha } ) ; ( \lambda m_{\beta } , \ , \lambda s_{\beta } , \ , p_{\beta})\bigr]_{\alpha,\beta=1, ... ,m } \times \prod_{1\leq\alpha<\beta}^{m } \ ; { \cal g}_{\alpha\beta}\bigl(\frac{{\bf p}}{\lambda } , \ ; { \bf m } , \ ; { \bf s}\bigr ) \biggr\ } \label{66}\end{aligned}\ ] ] where @xmath152 \ ; = \ ; \frac{1 } { \lambda m + \lambda s - ip + \lambda m ' + \lambda s ' + ip'}\ ] ] the crucial point of the further calculations is the procedure of taking the thermodynamic limit @xmath153 . first of all one can easily note that according to eqs.([60 ] ) and ( [ 66 ] ) , it is the parameters @xmath154 which remain finite in the limit @xmath153 . in other words , all the parameters @xmath65 which are not multiplied by @xmath155 ( e.g. in the expressions for @xmath156 and @xmath139 , eqs([58 ] ) and ( [ 59 ] ) ) have to be taken to zero in this limit . simultaneously , the summations over @xmath123 and @xmath124 have to be performed . the general algorithm of such summation is in the following . let us consider the example of the sum of a general type : @xmath157\ ; \phi\bigl [ { \bf p},\ ; \frac{{\bf p}}{\lambda } , \ ; \lambda { \bf n } ; \ ; { \bf n } \bigr]\ ] ] where @xmath158 is a function which depend both of @xmath159 s and @xmath67 s ( which are not multiplied by @xmath155 ) . the summations in the above example can be represented in terms of the integrals in the complex plane : @xmath160\ ; \phi\bigl [ { \bf p},\ ; \frac{{\bf p}}{\lambda } , \ ; \lambda { \bf z } ; \ ; { \bf z } \bigr]\ ] ] where the integration goes over the contour @xmath161 shown in fig.1(a ) . shifting the contour to the position @xmath162 shown in fig.1(b ) ( assuming that there is no contribution from @xmath163 ) , and redefining @xmath164 , in the limit @xmath153 we get : @xmath165\ ; \lim_{\lambda\to\infty } \phi\bigl [ { \bf p},\ ; \frac{{\bf p}}{\lambda } , \ ; { \bf z } ; \ ; \frac{{\bf z}}{\lambda } \bigr]\ ] ] where the parameters @xmath166 , @xmath167 and @xmath168 remain finite in the limit @xmath153 .    ; ( b ) the deformed contour @xmath162 ; , width=453 ]    [ figure1 ]    let us consider now the summations in eq.([66 ] ) . here the double sum can be represented as follows : @xmath169 thus , according to the above summation algorithm , we get @xmath170 \lim_{\lambda\to\infty } f\bigl(\frac{{\bf z}_{1}}{\lambda } ; \frac{{\bf z}_{2}}{\lambda}\bigr)\ ] ] where the rescaled integration parameters @xmath171 and @xmath172 remain finite in the limit @xmath153 . finally , taking into account the gamma function properties , @xmath173 and @xmath174 , for the factors @xmath137 and @xmath139 , eqs.([58])-([59 ] ) , we easily find @xmath175 and @xmath176 where @xmath177 thus , in the limit @xmath153 the expression for the probability distribution function , eq.([66 ] ) , takes the form of the fredholm determinant @xmath178 \bigl ( 1 + \frac{{z_{1}}_{\alpha}}{{z_{2}}_{\alpha } + i p_{\alpha}^{(- ) } } \bigr ) \bigl ( 1 + \frac{{z_{2}}_{\alpha}}{{z_{1}}_{\alpha } - i p_{\alpha}^{(+ ) } } \bigr ) \ ; \mbox{\large e}^{({z_{1}}_{\alpha } + { z_{2}}_{\alpha } ) y_{\alpha } } \biggr ] \\ \nonumber \\ \nonumber & \times & \det \hat{k}\bigl[({z_{1}}_{\alpha},{z_{2}}_{\alpha},p_{\alpha } ) ; ( { z_{1}}_{\beta},{z_{2}}_{\beta},p_{\beta})\bigr]_{\alpha,\beta=1, ... ,m } \\ \nonumber \\ & = & \det\bigl[1 \ , - \ , \hat{k } \bigr ] \label{76}\end{aligned}\ ] ] with the kernel @xmath179 \ ; = \ ; \frac{1 } { { z_{1 } } + { z_{2 } } - ip + { z_{1 } } ' + { z_{2 } } ' + ip'}\ ] ] in the exponential representation of this determinant we get @xmath180\ ] ] where @xmath181 \bigl ( 1 + \frac{{z_{1}}_{\alpha}}{{z_{2}}_{\alpha } + i p_{\alpha}^{(- ) } } \bigr ) \bigl ( 1 + \frac{{z_{2}}_{\alpha}}{{z_{1}}_{\alpha } - i p_{\alpha}^{(+ ) } } \bigr ) \ ; \mbox{\large e}^ { ( { z_{1}}_{\alpha } + { z_{2}}_{\alpha } ) y_{\alpha } } \biggr ] \times \\ \nonumber \\ & \times & \prod_{\alpha=1}^{m } \biggl [ \frac{1 } { { z_{1}}_{\alpha } + { z_{2}}_{\alpha } - i p_{\alpha } + { z_{1}}_{\alpha + 1 }   + { z_{2}}_{\alpha + 1 }   + i p_{\alpha + 1 } } \biggr ] \label{79}\end{aligned}\ ] ] here , by definition , it is assumed that @xmath182 ( @xmath183 ) and @xmath184 . substituting @xmath185\ ] ] into eq.([79 ] ) , we obtain @xmath186\ ] ] where , by definition , @xmath187 , and @xmath188 \bigl ( 1 + \frac{z_{1}}{z_{2 } + i p^{(- ) } } \bigr ) \bigl ( 1 + \frac{z_{2}}{z_{1 } - i p^{(+ ) } } \bigr ) \ ; \mbox{\large e}^{(z_{1 } + z_{2 } ) y_{\alpha}}\ ] ] simple calculations yield : @xmath189 \ ; \delta(y)\end{aligned}\ ] ] taking the limit @xmath190 we find : @xmath191 substituting this result into eq.([81 ] ) we obtain @xmath192\ ] ] in other words , the free energy distribution function of our problem is given by the fredholm determinant , @xmath193\ ] ] with the kernel @xmath194 which is the goe tracy - widom distribution @xcite . in this paper we have presented sufficiently simple derivation of the goe tracy - widom distribution function for the free energy fluctuations in random directed polymers with free boundary conditions . the main message of this somewhat technical work is not the final result itself ( which is not new anyway ) , but the demonstration of the efficiency of the general method and new technical tricks used in the derivation . by mapping the original problem to the @xmath0-particle quantum boson system with attractive interactions the derivation is done in the framework of the integer replica series summations and the bethe ansatz formalism for the quantum boson system . the key technical tricks of presented calculations includes the following points . first of all , to make the integration over particle coordinates of the bethe ansatz propagator well defined one has to introduce proper regularization at @xmath46 which requires formal splitting the partition function into two parts : the one in the positive particles coordinates sector ( up to @xmath103 ) and another one in the negative particles coordinates sector ( down to @xmath102 ) , eqs.([14])-([15 ] ) . next is the `` magic '' bethe ansatz combinatorial identity , eq.([35 ] ) , which allows to perform the summation over the momenta permutations and `` disentangle '' sophisticated products containing in the bethe ansatz propagator . one more trick is the reformulation of the summation over permutations of the momenta between the positive and the negative particles positions sectors in terms of the series summations , eq.([41 ] ) , which allows to represent the probability distribution function in terms of the problem of the series summations , eq.([47 ] ) . finally , the crucial point of the considered derivation is the procedure of the series summations in the thermodynamic limit @xmath33 . in this limit , due to the integral representation of the series , eqs.([68])-([70 ] ) , one obtains dramatic simplifications of some factors , eqs.([73])-([74 ] ) , in the expression for the probability distribution function , which allows to represent it in the form of the fredholm determinant , eq.([76 ] ) . hopefully , the experience obtained in the presented calculations would help to solve more serious long standing problems of this scope , such as the distribution function of the directed polymer s end point fluctuations or the statistical properties of the free energy fluctuations at different times . an essential part of this work was done during my visit to the fields institute , toronto , in the spring of 2011 in the framework of the program `` dynamics and transport in disordered systems '' . i would like to thank jeremy quastel and kostya khanin for numerous illuminating discussions which were crucial for the progress in these somewhat complicated calculations .
the distribution function of the free energy fluctuations in one - dimensional directed polymers with free boundary conditions is derived by mapping the replicated problem to the @xmath0-particle quantum boson system with attractive interactions . 
 it is shown that in the thermodynamic limit this function is described by the universal tracy - widom distribution of the gaussian orthogonal ensemble .