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_ continuous time tasep . _ the totally asymmetric simple exclusion process ( tasep ) is the simplest non - reversible interacting stochastic particle system . in tasep , particles are on the lattice of integers , @xmath0 , with at most one particle at each site ( exclusion principle ) . the dynamics is defined as follows . particles jump to the neighboring right site with rate @xmath5 provided that the site is empty . jumps are independent of each other and take place after an exponential waiting time with mean @xmath5 , which is counted from the time instant when the right neighbor site is empty . it is known that the only translation invariant stationary measures are bernoulli product measures with a given density @xmath6 $ ] ( see  @xcite ) . in the sequel we fix a @xmath7 to avoid the trivial cases @xmath8 ( no particles ) and @xmath9 ( all sites occupied ) . one quantity of interest is the fluctuations of the currents of particles during a large time @xmath10 . consider the so - called _ characteristic line _ given by @xmath11 . then , the current fluctuation seen from the characteristic are @xmath12 ( in agreement with the scaling for the two - point function established in  @xcite ) , and the limit distribution function has been determined ( conjectured in  @xcite and proved in  @xcite , see also  @xcite ) . the limiting distribution was first discovered in the context of a directed last passage percolation model  @xcite .    on the other hand , if one looks at the current along a line different from the characteristic line , say @xmath13 with @xmath14 , then for large time @xmath10 one just sees the gaussian fluctuations from the initial condition on a @xmath15 scale , since the dynamics generates fluctuations only @xmath12 , thus irrelevant  @xcite . a non - trivial interplay between the dynamically generated fluctuations and the one in the initial condition occurs in a region of order @xmath4 around the characteristic line . one of the main results of this paper is the determination of multi - point limits of the current fluctuations around the characteristics @xmath16 ( see theorem  [ theoremtasep ] and theorem  [ theoremtasepb ] ) . there has been a great deal of work pertaining to the limiting fluctuation of tasep over the last ten years since the well - known work of johansson  @xcite . see for example , the review paper  @xcite for the limit processes which arise from deterministic initial conditions . a recent paper  @xcite , building on the earlier work  @xcite , is concerned on the situation where random and deterministic initial condition are both present , but not stationary .    _ directed percolation model . _ it is well - known that the currents of tasep ( of arbitrary initial condition ) can be expressible in terms of the last passage time of an associated directed last passage percolation model ( see for example , @xcite ) . we first discuss the asymptotic result of the following directed percolation model associated to stationary tasep @xcite : see the paragraphs preceding theorem  [ theoremtasep ] below for the exact relation to stationary tasep . let @xmath17 , @xmath18 , @xmath19 , be independent random variables with the following distributions @xmath20 here the notation @xmath21 means that @xmath22 is a random variable exponentially distributed with expectation @xmath23 . an _ up - right _ path @xmath24 from @xmath25 to @xmath26 is a sequence of points , starting from the origin , @xmath27 , ending at @xmath28 , @xmath29 , and satisfying @xmath30 . denote by @xmath31 . then , the last passage time is defined by @xmath32 we are interested in the limit distribution of the properly rescaled last passage time of @xmath33 , [ yn])$ ] in the @xmath34 limit .    when there are no ` borders ' i.e. @xmath35 when @xmath36 or @xmath37 , the limiting distribution of @xmath38 was first obtained in @xcite . the case of ` single - border ' when @xmath39 but @xmath40 was studied in @xcite . the above ` double - border ' case was considered in @xcite . ( the poisson and geometric variations were studied earlier in @xcite . ) one can also consider a more general model when @xmath41 for @xmath42 and @xmath43 for @xmath44 , where @xmath45 . such model was considered in @xcite and a conjecture on the limiting distribution left in that paper was recently confirmed in @xcite . for the poisson and geometric variations , this generalization was considered in @xcite earlier when @xmath46 . an interesting characteristic behavior in such bordered models is the transition phenomenon . one can imagine that the last passage time comes from the competition between the contributions from the ` bulk ' @xmath47 , and the edges @xmath36 or @xmath37 . see for example , section 6 of @xcite for an illustration of such heuristic ideas . for the model  , a crucial role is played by the critical direction ( which corresponds to the characteristic line of tasep ) , @xmath48 it is easy to check that along a direction other than the critical direction , the fluctuations of @xmath33 , [ yn])$ ] is given by gaussian distribution on the @xmath49 scale , see appendix  [ appgaussian ] . this situation corresponds to the gaussian fluctuations along a non - characteristic line in a stationary tasep @xcite mentioned earlier .    along the critical direction and also in a @xmath50 neighborhood of the critical direction , the fluctuations of @xmath33 , [ yn])$ ] are on the @xmath51 scale . precisely , the following , among other things , is proven in  @xcite . let us set @xmath52 then set the parameter @xmath53 , where @xmath54 is considered to be large . consider the scaling @xmath55 the parameter @xmath56 measures the displacement of the focus with respect to the critical line ( on a @xmath57 scale ) . in particular , for @xmath58 one looks exactly along the critical direction . @xmath59 is the macroscopic value of the last passage time for large @xmath54 , while the parameter @xmath60 in @xmath61 measures the amount of the fluctuations ( on a @xmath62 scale ) of the last passage time . then for any fixed @xmath63 , @xmath64 for an explicit distribution function @xmath65 , which satisfies @xmath66 . an analogous result was first obtained in  @xcite for a poissonized version of the problem in which @xmath65 was obtained in terms of a solution to the painlev ii equation ( see ( 3.22 ) of  @xcite : @xmath67 ) . based on this result , prhofer and spohn conjectured in  @xcite that   holds . this conjecture was proven in  @xcite whose analysis was somewhat different from  @xcite . this results in a different formula of @xmath65 , expressed in terms of the airy function ( see ( 1.20 ) in  @xcite ) . it can be checked that these two formulas do agree . one of the main objects of study in this paper is the limit of multi - point distribution @xmath68 , @xmath69 . the limit of the process for the no - border case was first obtained in @xcite following the earlier work of @xcite on the poissonized version of the model . the limit of the process for the double - border case was considered in @xcite for the model when @xmath70 and @xmath41 for @xmath42 , @xmath43 for @xmath44 when @xmath71 and @xmath72 . note that when @xmath73 , the random variable @xmath74 becomes singular in this model : the restriction @xmath75 is significant in connection to stationary tasep . the geometric counterpart of the model   was studied earlier by imamura and sasamoto  @xcite . denoting by @xmath76 a random variable with the probability mass function @xmath77 , @xmath78 , the authors of @xcite considered the model   where @xmath75 , @xmath79 , @xmath80 and @xmath81 for @xmath82 . since the exponential model   can be obtained as a limiting case of @xmath83 , the analysis of @xcite can in principle be used to yield the corresponding result for the model  . nevertheless , this paper differs from @xcite on the following aspects . ( a ) the authors of @xcite obtained explicit limiting distribution functions for the case that the situation @xmath72 ( and @xmath84 ) in  . however , they left the case corresponding to @xmath85 as the limit of the @xmath72 case and did not compute the limiting distribution explicitly ( see remarks after theorem  5.1 in  @xcite ) . this critical case is the most interesting ( and the most difficult ) case for our situation and we give an explicit formula in theorem  [ mainthm ] below . ( b ) the justification of the limits of fredholm determinant and other quantities appearing in the analysis requires proper conjugations of corresponding operators in order to make sense of the fredholm determinant and trace class limit . these issues were not discussed in @xcite ( the main issue was to determine the possible limit regimes and not specifically the @xmath85 case ) . ( c ) in addition to the multi - point distribution on the line @xmath86 considered in @xcite , we also obtain limit of the process result for points @xmath87 not necessarily on the same line ( see theorem  [ theoremdppext ] ) . ( d ) the limit of the process for points at more general positions than on a line mentioned in ( c ) is used to prove the limit of the process ( in the sense of finite distribution ) of stationary tasep ( see theorem  [ theoremtasep ] and theorem  [ theoremtasepb ] below ) . we first state the result extending   to the joint distributions at points on the line @xmath88 at and near the critical direction . we first need some definitions . [ defin1 ] fix @xmath89 . for real numbers @xmath90 and @xmath91 , set @xmath92}\frac{e^{-\frac23\tau_i^3-\tau_i x}}{\sqrt{4\pi(\tau_i-\tau_1)}}\int_{-\infty}^{s_1-x}dy\ , e^{-\frac{y^2}{4(\tau_i-\tau_1 ) } }   -\int_{0}^\infty dy\ , \operatorname*{ai}(y+x+\tau_i^2)e^{\tau_i y}. \end{aligned}\ ] ] for @xmath93 , where @xmath94 denotes the airy function . the first result is the following . [ mainthm ] fix @xmath89 . for real numbers @xmath90 and @xmath91 , with the scaling given in  , @xmath95 here @xmath96 is equipped with the standard measure @xmath97 where @xmath98 is the counting measure on @xmath99 . @xmath100 denotes the projection operator @xmath101}$ ] , and @xmath102 is the so - called extended airy kernel  @xcite with shifted entries defined by the kernel @xmath103_{i , j}(x , y)\\ & = \begin{cases}\displaystyle { \int_0^\infty d\lambda \operatorname*{ai}(x+\lambda+\tau_i^2)\operatorname*{ai}(y+\lambda+\tau_j^2 ) e^{-\lambda(\tau_j-\tau_i)}},&\textrm { if } \tau_i\leq \tau_j,\\ \displaystyle { -\int_{-\infty}^0 d\lambda \operatorname*{ai}(x+\lambda+\tau_i^2)\operatorname*{ai}(y+\lambda+\tau_j^2 ) e^{-\lambda(\tau_j-\tau_i)}},&\textrm { if } \tau_i > \tau_j . \end{cases } \end{aligned}\ ] ] the function @xmath104 is defined by @xmath105 where @xmath106 and @xmath107 , @xmath108 . finally the functions @xmath109 , @xmath110 , and @xmath111 are defined in definition  [ defin1 ] . observe that @xmath112 . when @xmath113 ,   agrees with the limiting function in ( 1.20 ) of @xcite , as one may expect . [ rem : inv ] the fact that @xmath114 is invertible follows from the fact that @xmath115 is trace class ( see @xcite ) and that @xmath116 for all given @xmath117 . see lemma  [ leminvertibility ] in appendix  [ appinvertibility ] . the shift in the integrand by @xmath118 is due to the fact that the last passage time is ( macroscopically ) a linear function along the line @xmath119 , in contrast with the non - border case ( i.e. , @xmath120 ) where the last passage time has a non - zero curvature . therefore , when @xmath121 , the contribution from @xmath122 will very close to one and the main contribution comes from @xmath104 . the second theorem is a generalization of theorem  [ mainthm ] to the case when the points @xmath87 are not necessarily on the same line @xmath123 , @xmath124 . we show that the fluctuation is unchanged even if some of the points are away from the line to the order smaller than @xmath125 . this is due to the slow de - correlation phenomena obtained in @xcite : along the critical direction the fluctuations decorrelate to order @xmath126 over a time scale @xmath125 ( instead of @xmath127 ) . more precisely , if we compare the last passage time @xmath128 at two points @xmath28 and @xmath129 with @xmath130 , their fluctuation will be @xmath131 ( see lemma  [ lemslowdecorrelation ] below for details ) . consider a @xmath132 and any fixed number @xmath133 and consider the scaling ( a generalization of ( [ eq6 ] ) ) given by @xmath134 see figure  [ figuredppext ] for an illustration . [ cb]@xmath135 [ cb]@xmath136 [ c]@xmath137 [ c]@xmath138 [ l]critical line   for some @xmath139 away from the line @xmath138 . then , the fluctuations of the passage time at the locations of the black dots are , on the @xmath62 scale , the same as the one of their projection along the critical direction to the line @xmath138 , the white dots.,title="fig:",height=188 ]    [ theoremdppext ] fix @xmath140 and @xmath132 . for real numbers @xmath90 , @xmath141 , and @xmath91 , with the scaling given in  , @xmath142 this generalization of theorem  [ mainthm ] is proven in section  [ sectproofmainthm ] . _ stationary tasep and directed percolation . _ we now discuss the result in terms of stationary tasep . the mapping between tasep and last passage percolation model is as follows . we assign label @xmath143 to the particle sitting at the smallest positive integer site initially . for the rest we use the right - to - left ordering so that @xmath144 . then @xmath145 for all @xmath146 , since the tasep preserves the ordering of the particles .    for @xmath147 such that @xmath148 , let @xmath149 be the waiting time of particle with label @xmath150 to jump from site @xmath151 to site @xmath152 ( the waiting time is counted from the instant where particle can jump , i.e. , particle is at @xmath151 and site @xmath152 is empty ) . the @xmath149 are iid @xmath153 random variables . let @xmath154 be the last passage time to @xmath26 along a directed path in the domain @xmath155 , starting any point in the domain . this is a curve - to - point optimization problem . an example of the non - trivial part of the boundary of this domain is illustrated in figure  [ figuretasep ] below as the curve at @xmath156 . then a straightforward generalization of the step - initial case of @xcite shows that @xmath157    when the initial condition is random , the domain of directed percolation is also random . note that ( see figure  [ figuretasep ] ) , the part of the domain @xmath158 in the first quadrant is the rectangle @xmath159 , but the parts in the second quadrant @xmath160 and the forth quadrant @xmath161 are of random shape . observe that @xmath158 does not intersect with the third quadrant @xmath162 . define @xmath163 to be the right - most empty site in @xmath164 in the initial particles configuration and @xmath165 to be the position of the left - most particle in @xmath166 . first let us focus on the case @xmath167 , @xmath168 ( as in figure  [ figuretasep ] ) . then , since the initial condition is stationary bernoulli , an interpretation of burke s theorem  @xcite , see also @xcite , shows that the @xmath169 is distributed as @xmath170 where @xmath171 s are iid @xmath172 distributed . similarly , by considering holes instead of particles one finds that @xmath173 is distributed as @xmath174 where @xmath175 s are iid @xmath176 random variables . hence @xmath154 has the same distribution as @xmath38 defined from  . this argument was sketched in section 2 of @xcite . consequently , we have for @xmath177 , @xmath178 . @xmath179 for stationary tasep when initially the position @xmath143 is empty and the position  @xmath5 is occupied .    for generic @xmath180 and @xmath181 , @xmath154 is distributed as @xmath38 where now we assume in   that @xmath182 where @xmath183 and @xmath184 . but as shown in proposition 2.2 in  @xcite , this change does not affect the asymptotics . geometrically , the tasep and directed percolation can be thought of as two different cuts of the three - dimensional object @xmath185 ( a ) the directed percolation problem we analyze is the cut at @xmath186 , + ( b ) the particles configuration of the tasep at time @xmath10 is the cut at @xmath187 . + we have shown in theorem  [ theoremdppext ] that the limit process does not depend on the cut chosen for the analysis as long as we avoid the cut along the characteristic , @xmath188 . thus , to get the fluctuations around the characteristic line it is enough to project on @xmath189 ( see figure  [ figuretasep ] for an illustration ) , for which the limit theorem was proven in theorem  [ mainthm ] . [ c]@xmath150 [ l]@xmath190 [ cb]@xmath136 [ c]@xmath137 [ bc]@xmath156 [ bl]@xmath10 [ c]@xmath138 [ l]characteristics   and some later time @xmath10 . the position of particle with label @xmath191 is its projection on the @xmath192-axis . interpolating between particles as in this example , one gets a line configuration , which is interpreted as height function @xmath193 above position  @xmath150.,title="fig:",height=226 ]    the following two theorems are proven in section  [ sectproofmainthm ] . define the functions @xmath194    [ theoremtasep ] fix @xmath140 . for real numbers @xmath90 and @xmath91 , @xmath195    an equivalent but geometrically slightly different way of representing the tasep is via a height function ( see e.g.  @xcite ) . let us define the occupation variable , @xmath196 if there is a particle at site @xmath197 at time @xmath10 and @xmath198 otherwise . then , define the height function @xmath199 where @xmath200 is the number of particles which jumped from site @xmath143 to site @xmath5 during the time - span @xmath201 $ ] . then the link between the height functions and the locations of particles is @xmath202 let us consider the scaling @xmath203    [ theoremtasepb ] fix @xmath140 . for real numbers @xmath90 and @xmath91 . then , @xmath204    for simplicity , in theorems  [ theoremtasep ] and  [ theoremtasepb ] we stated the result only for fixed time . however , the statements can be extended to different times in a similar manner of the extension from theorem  [ mainthm ] to theorem  [ theoremdppext ] .    _ queues in tandem . _ there is a direct relation between queues in tandem and tasep . suppose that there are infinitely many servers , and we assume that the service time of a customer at each server is independent and distributed as @xmath153 . once a customer is served at the server @xmath197 , then the customer joins at the @xmath205th queue , and so on . in other words , the departure process from the @xmath197th queue is the arrival process at the @xmath205th queue . suppose that the system is in equilibrium with parameter @xmath1 : the arrival process at each queue is independent poisson process of rate @xmath1 . then the departure process at each queue is also independent poisson process of rate @xmath1 due to burke s theorem ( see for example , @xcite ; see also @xcite ) . this also implies that at each time , the number of customers in each queue is distributed as @xmath206 . ( here @xmath207 means that @xmath208 , @xmath78 . ) now consider a fixed time @xmath156 and arbitrary select one customer and assign label @xmath143 to that customer . for convenience , we call the queue in which that customer is in at time @xmath143 as the @xmath143th queue . we assign labels to the other customers so that the labels decreases for the customers ahead in the queues ( see figure  [ figurequeues ] ) . [ b]@xmath209 [ b]@xmath210 [ b]@xmath211 [ b]@xmath212 [ b]@xmath213 [ b]@xmath143 [ b]@xmath5 [ b]@xmath214 [ b]@xmath215 [ b]@xmath216 [ cb]c@xmath217 [ cb]c@xmath218 [ cb]c@xmath219 [ cb]c@xmath220 [ cb]c@xmath221 [ cb]c@xmath222 [ cb]server @xmath212 [ cb]server @xmath213 [ cb]server @xmath143 [ cb]server @xmath5 [ cb]server @xmath214     let @xmath223 denote the label of the queue in which the @xmath150th customer is in at time @xmath10 . the mapping from the queueing model to the tasep is obtained by setting @xmath224 ( see figure  [ figurequeues ] ) . the equilibrium condition implies that the initial condition for the corresponding tasep is stationary . hence if we define @xmath225 be the time the @xmath150th customer exits the queue @xmath197 , then we find that @xmath226 hence using   and theorem  [ theoremdppext ] , we immediately obtain the following result . [ theoremqueue ] fix @xmath140 and @xmath132 . for real numbers @xmath90 , @xmath141 , and @xmath91 , with the scaling given in  , @xmath227      to prove theorem  [ mainthm ] , which is the basis for theorems [ theoremtasep ] and [ theoremtasepb ] , we first consider a slightly different directed percolation model which is known to be determinantal . this model is then related to ( [ eqdp ] ) by a shift argument ( section  [ sectshift ] ) followed by an analytic continuation ( section  [ sectanalytic ] ) , resulting in an explicit formula of the joint distribution of the last passage times of a more general version of   when @xmath41 for @xmath42 and @xmath43 for @xmath44 , for @xmath228 ( see theorem  [ thm : pab ] ) . the formula for the model   is obtained by setting @xmath229 . these arguments are a generalization of the arguments given in @xcite and @xcite for one - point distribution . in section  [ sectasympt ] we carry out the asymptotic analysis of the formula obtained in theorem  [ thm : pab ] . the proofs of theorems  [ mainthm ] ,  [ theoremtasep ] and  [ theoremtasepb ] are given in section  [ sectproofmainthm ] . some technical computations are given in appendices : various expressions of integrals involving airy functions are given in appendix  [ sectairyfunctions ] , the invertibility of an operator appearing in theorem  [ mainthm ] is discussed in appendix  [ appinvertibility ] , certain operator are shown to be trace - class in appendix  [ secttraceclass ] , and finally we explain the gaussian fluctuation along non - critical directions in appendix  [ appgaussian ] . as mentioned above , theorem  [ thm : pab ] below contains a formula for the model with @xmath41 for @xmath42 and @xmath43 for @xmath44 , for @xmath228 , which is more general than  . this would correspond to other random initial data which possibly allows a shock . essentially all the ingredients for the asymptotic analysis for this more general case are in this paper , and one can obtain the limit laws for such models . however , for the sake of simplicity , we do not pursue this direction here . the work of jinho baik was supported in part by nsf grant dms075709 . patrik l. ferrari started the work while being at wias - berlin . finally , we would like to thank the referee for reading the manuscript carefully and suggesting many small changes which helped improving the presentation . we first consider a slightly different directed percolation model . let @xmath230 where the parameters @xmath231 and @xmath232 satisfy @xmath233 denote by @xmath234 to be the last passage time from @xmath25 to @xmath28 for this modified model . this model is well - studied and has nice mathematical structure . in particular , the correlation functions and joint distribution functions on @xmath138 are determinantal , which is well - suited for an asymptotic analysis . note that the original model given in   corresponds to the case when @xmath85 and @xmath235 . we will show in sections  [ sectshift ] and  [ sectanalytic ] how to obtain a joint distribution formula for the original model   from this modified model  . let @xmath236 be the last passage time for the model   with @xmath237 replaced by @xmath143 . we proceed as follows : + * ( 1 ) shift argument * : for @xmath238 with @xmath72 , we relate the distribution of @xmath239 with the one of @xmath240 . + * ( 2 ) analytic continuation * : we determine an expression for @xmath239 which can be analytically continued in all @xmath238 . + * ( 3 ) choice of parameter * : finally we set @xmath241 and @xmath242 . + the step ( 1 ) is done in section  [ sectshift ] and step ( 2 ) is presented in section  [ sectanalytic ] . [ propshift ] let @xmath238 with @xmath72 . let @xmath243 be the a set distinct of points in @xmath244 and define @xmath245 then , @xmath246    proposition  [ propshift ] let us only consider @xmath247 : the proof for general @xmath248 is a straightforward generalization and we leave it to the reader . set @xmath249 . then @xmath250 consider the laplace transform @xmath251 since @xmath252 when either @xmath253 or @xmath254 , we can restrict the integral from @xmath255 to @xmath256 . then the above equals @xmath257 multiplying by @xmath258 and integrating by parts leads to @xmath259 since this holds for all @xmath260 , by inverting the laplace transform , we obtain @xmath261    as mentioned earlier , the probability @xmath262 has an explicit determinantal expression . in fact , the joint distribution function at points on @xmath138 for the directed percolation model with , , has a determinantal structure ( it is a limit of geometric random variables studied in  @xcite or of the schur measure  @xcite ; see  @xcite for a direct approach to exponential random variables ) . specifically , this follows , for example , from theorem 3.14 and ( 1.3 ) of @xcite after substituting @xmath263 and @xmath264 in ( 1.3 ) and taking the limit @xmath265 . the kernel in this limit is explicitly derived in theorem 3 of  @xcite . for good survey papers on the topic , see  @xcite . by specializing to the case with weights as in , we have the following result . consider a set of distinct points @xmath243 on the line , which can be ordered and parametrized by @xmath266 $ ] , @xmath267 set @xmath268 and  , the notation @xmath269 means that the integration path goes anticlockwise around the points in @xmath270 but does not include any other poles . ] @xmath271 define the kernels @xmath272 where @xmath273}}{2\pi{{\rm i}}}\int_{{{\rm i}}{\mathbb{r } } } d\widetilde z\ , e^{\widetilde z ( y - x)}\frac{\phi_i(\widetilde z)}{\phi_j(\widetilde z)},\\ \widetilde k_{i , j}(x , y ) & = \frac{-1}{(2\pi{{\rm i}})^2}\oint_{\gamma_{-1/2}}dz \oint_{\gamma_{1/2}}dw\ , e^{zy - wx}\frac{\phi_i(w)}{\phi_j(z)}\frac{1}{w - z}. \end{aligned}\ ] ] then , @xmath274 where @xmath275}$ ] and the operator @xmath276 is defined by the kernel @xmath277    in  , we have used a slight abuse of notation as @xmath278 is not a trace - class operator : one can indeed observe that for @xmath279 , @xmath280 as @xmath281 . nevertheless with a suitable multiplication operator @xmath282 , the conjugate operator @xmath283 becomes trace - class so that the fredholm determinant is well - defined , and the identity becomes valid analytically . more concretely , if we take @xmath282 as in   below , then it is shown in appendix  [ secttraceclass ] below that @xmath283 is trace - class operator for @xmath284 . thus , after conjugation , the fredholm determinant is well - defined , and the identity becomes valid analytically , as proved in the following . [ propstartingkernel ] let @xmath285 . fix constants @xmath286 satisfying @xmath287 define the conjugated operator @xmath288 by the kernel @xmath289 then , @xmath290 is a trace - class operator on @xmath291 and @xmath292    proposition  [ propstartingkernel ] and give the standard extension from one - point kernel to the multi - point ( or extended ) kernel . all the ingredients are included in appendix b and section 3 of  @xcite , and also in  @xcite . the extended kernel can be found in the more recent work  @xcite . there the setting is more general allowing several lines / columns to have waiting times different from @xmath5 . observe @xmath293 is independent of @xmath231 and @xmath232 , and the only dependence on @xmath231 and @xmath232 in @xmath276 is through the rank - one term @xmath294 . using this , we can find an expression of @xmath262 in which the condition   can be relaxed to  . this relaxation is important when we take the limit @xmath295 in the next section . [ propmodifiedkernel ] let @xmath285 . fix constants @xmath286 satisfying @xmath296 define the conjugated operator @xmath297 by the kernel @xmath298 then @xmath299 where the notation @xmath300 denotes the real inner product in @xmath301 .    one can check that the expression   can be analytically extended to @xmath302 with @xmath72 . nevertheless since we will discuss the issue of extending the domain analyticity of a formula of @xmath303 to @xmath302 ( with no restriction of @xmath72 ) in the next section , we do not discuss the detail here . proposition  [ propmodifiedkernel ] from  , @xmath304 where @xmath305 and @xmath306 are multiplication operators by the functions @xmath307 , @xmath308 . the functions @xmath309 and @xmath310 are @xmath311 : see the end of the proof of proposition  [ proptraceclass ] . for the convenience of notations , set @xmath312=p_u k^{\rm conj } p_u$ ] , @xmath313=p_u \overline k^{\rm conj } p_u$ ] , @xmath314=p_uf^{\rm conj}$ ] and @xmath315=g^{\rm conj}p_u$ ] . thus @xmath316\right )      & = \det\left({\mathbbm{1}}-(a+b)({\mathbbm{1}}-[\overline k])^{-1}[f]\otimes [ g]\right)\cdot \det\left({\mathbbm{1}}-[\overline k]\right),\\      & = \left(1-(a+b)\langle({\mathbbm{1}}-[\overline k])^{-1}[f],[g]\,\rangle\right ) \cdot \det\left({\mathbbm{1}}-[\overline k]\right ) . \end{aligned}\ ] ] the above step holds assuming that @xmath317={\mathbbm{1}}-p_u\overline k^{\rm conj } p_u$ ] is invertible . the invertibility can be verified as follows . consider the modification of the directed percolation model where @xmath318 if @xmath197 and/or @xmath150 are equal to @xmath143 , and @xmath319 for @xmath82 ( i.e. , model without sources ) . denote by @xmath320 the new measure . this is the model with @xmath321 in  , and hence @xmath322 for any given @xmath323 . it is easy to obtain a lower bound @xmath324 , with @xmath325 . indeed , it is enough to take the configurations with @xmath326 for @xmath327 such that @xmath328 . this implies that @xmath329 , and hence @xmath330 is invertible . finally , since @xmath331 is a bounded operator in @xmath311 , and , @xmath332)^{-1}[f],[g]\,\rangle$ ] equals @xmath333 by conjugating back @xmath282 . now the condition   for the @xmath334 s can be relaxed to the condition   since @xmath297 is trace - class under this assumption : see proposition  [ proptraceclass ] . we now find a formula extending   to the case when @xmath228 without the restriction that @xmath72 . for this purpose , we show that both sides of   are analytic in the parameter @xmath335 . analyticity of the left - hand - side of   i.e. , of @xmath336 is a straightforward generalization of proposition 5.1 in  @xcite . this section is devoted to finding the analytic continuation of the right - hand - side of  .    by using ( [ eq25 ] ) , the right - hand - side of   becomes @xmath337 the last determinant is independent of @xmath338 . it is enough show that @xmath339 is analytically continued in @xmath238 . note that by changing the contour , @xmath340 where @xmath341    [ propanalyticcont ] for @xmath284 , @xmath342 where @xmath343 and @xmath344 with @xmath345 } \int_{-\infty}^{u_1 } dy\ , v_{i,1}(x , y ) f_1^{(a)}(y).\ ] ] the term @xmath346 is analytic in @xmath302 , and @xmath347 is convergent and is analytic in @xmath302 . hence the right - hand - side of   is an analytic continuation of   to @xmath348 . combining  ,   and , we finally obtain the following representation of @xmath349 defined in  . [ thm : pab ] recall the conditions and definitions from   through   above . for @xmath228 , @xmath350 , \end{aligned}\ ] ] for @xmath351 , where @xmath346 and @xmath352 are defined in proposition  [ propanalyticcont ] . proposition  [ propanalyticcont ] _ part i : decomposition_. first we decompose the contribution coming from the pole at @xmath353 and @xmath231 of @xmath354 , namely @xmath355 from lemma  [ lemma1piuv ] below , @xmath356 , where @xmath357 is defined in ( [ eq3.18 ] ) . here , @xmath358 is well - defined pointwise and is in @xmath311 as we can check as in the proof of lemma  [ lemma1piuv ] . hence using the identity ( recall that @xmath359 in  ) @xmath360 the last term in ( [ eq34 ] ) becomes @xmath361 observe that the function @xmath362 is precisely the last two terms in ( [ eq31 ] ) . hence from  , we obtain @xmath363    now a direct computation shows that @xmath364 here in the third equality , we used the fact that the contour @xmath365 can be made to be on the left of the point @xmath366 since @xmath367 . hence   and   imply  .    _ part ii : analyticity_. we now show that the functions on the right - hand - side of   are analytic in @xmath348 . clearly , @xmath346 is analytic in since both the poles @xmath366 and @xmath368 lie inside the integration contour . we need to show that @xmath369 is analytic . note that @xmath293 , @xmath370 and @xmath371 are independent of @xmath372 . as @xmath373 is analytic in @xmath374 , @xmath375 is analytic in @xmath231 in the same domain . also it is clear from the integral representation   that @xmath376 is analytic in @xmath377 . hence it it enough to show that @xmath369 is well - defined for @xmath378 . fix @xmath379 . let @xmath380 $ ] . using the identity @xmath381 and the estimates   and   in lemma  [ lemmaanalytic ] below , we see that @xmath382 for some constant @xmath383 . on the other hand , from  , @xmath384 for some constant @xmath383 . therefore , the inner product is convergent , and the proposition is obtained . [ lemma1piuv ] let @xmath385 . define the function @xmath357 in @xmath386 by @xmath387 then @xmath388 defined by @xmath389 is well - defined for each @xmath390 and is in @xmath311 . moreover , @xmath391    lemma  [ lemma1piuv ] recall from   that @xmath392 . the estimate   shows that the integral in   is well - defined . @xmath393 is well - defined pointwise . note that @xmath394 for some function @xmath395 ( see  ) . hence the integral in   equals @xmath396 . hence the @xmath311 norm of the integral is bounded by @xmath397 . but as @xmath398 by  , we find that the integral in   is in @xmath311 .    now let @xmath399 and @xmath400 be fixed . in the integral formula of @xmath401 in  , we can change the contour @xmath402 to either @xmath403 or @xmath404 . we will use the contour @xmath403 when @xmath405 and the contour @xmath404 when @xmath406 . then for @xmath407 , @xmath408 where the integral is evaluated using cauchy s formula . taking into account the case when @xmath409 , we obtain  . [ lemmaanalytic ] for any @xmath410 $ ] , there exists a constant @xmath383 such that @xmath411},\qquad x , y\in { \mathbb{r}},\ ] ] and @xmath412 for any @xmath377 , there is a constant @xmath383 such that @xmath413 finally , suppose that @xmath374 is given . for any @xmath414 $ ] , there is a constant @xmath383 such that @xmath415 here the constants are uniform if the parameters @xmath416 are in compact sets . lemma  [ lemmaanalytic ] when @xmath417 , by deforming the contour to in  , we obtain for @xmath418 , @xmath419 since the integrand is absolutely convergent ( recall that @xmath420 for @xmath279 ) . we obtain the bound   similarly by using the contour @xmath421 when @xmath422 . the estimate   is easily obtained by taking the contours @xmath423 and @xmath424 as the circles of radii of @xmath425 centered at @xmath426 and @xmath353 , respectively , in  .    recalling @xmath427 ( from  ) we obtain the estimate   by evaluating the residue at @xmath428 and making the remaining contour @xmath423 small enough .    finally , in order to estimate @xmath375 , first note that @xmath429 by using the integral representation   with the contour given by the circle of radius @xmath425 centered at @xmath353 . also from   and  , . using these estimates and @xmath430 in the definition  , @xmath431 } \int_{-\infty}^{u_1 } dy\ , v_{i,1}(x , y ) f_1^{(a)}(y),\ ] ] we obtain   for @xmath136 bounded below . in particular , to have the integrals over @xmath137 bounded we need @xmath432 for the integral with @xmath433 and @xmath434 for the integral with @xmath435 . both conditions are satisfied for @xmath436 . by setting @xmath229 in theorem  [ thm : pab ] , we obtain @xmath437 . \end{aligned}\ ] ] we now begin asymptotic analysis of this formula .    to obtain our main theorem ( theorem  [ mainthm ] ) we need to consider the following scaling limit . fix @xmath438 . set @xmath439 , @xmath242 , and @xmath241 . for a large parameter @xmath54 , according to ( [ eq6 ] ) and ( [ eq : xycond ] ) , we consider @xmath440 where we order @xmath90 with @xmath441 $ ] for all @xmath197 . the convergence of the fredholm determinants is ensured only after ( yet another ) proper conjugation . for this purpose , set @xmath442    [ propasymptkbar ] consider the scaling ( [ eqscaling ] ) . then @xmath443_{i , j}(s_i , s_j),\ ] ] uniformly for @xmath444 in a bounded set . the operator @xmath102 is defined in ( [ eqkhat ] ) . proposition  [ propasymptkbar ] recall that @xmath445 , with @xmath446 for @xmath447 . the same structure holds for @xmath102 . indeed , using the identity ( [ eqairyf ] ) of lemma  [ lemairyexpressions ] , we can rewrite ( [ eqkhat ] ) as follows , @xmath448_{i , j}(s_i , s_j ) & = \int_0^\infty d\lambda \operatorname*{ai}(s_i+\lambda+\tau_i^2)\operatorname*{ai}(s_j+\lambda+\tau_j^2 ) e^{-\lambda(\tau_j-\tau_i)}\\ & -\frac{\exp\left(-\frac{(s_i - s_j)^2}{4(\tau_i-\tau_j)}+\frac23(\tau_j^3-\tau_i^3)+ ( \tau_js_j-\tau_is_i)\right)}{\sqrt{4\pi ( \tau_i-\tau_j ) } } { \mathbbm{1}}(\tau_i>\tau_j ) \end{aligned}\ ] ] the proof is divided into the convergence for @xmath449 in lemma  [ lemasymptv ] and of @xmath450 in lemma  [ lemasymptk ] below . [ lemasymptv ] consider the scaling ( [ eqscaling ] ) and @xmath279 . then @xmath451 uniform for @xmath452 in a bounded set . lemma  [ lemasymptv ] recall that @xmath453 . we derive the asymptotics by saddle point analysis . set @xmath454,\\ g_1(\widetilde z)&:=-\widetilde z ( s_i - s_j)\chi^{-1/3 } , \\ g(z ) & : = g_0(z)+t^{-1/3}g_1(z ) . \end{aligned}\ ] ] then by plugging   into  , @xmath455 there is a unique critical point for @xmath456 in the interval @xmath457 , namely @xmath458 a straightforward computation gives @xmath459 from which @xmath460 . also observe that @xmath461    for the saddle point analysis we use the contour @xmath462 . first let us show that the contribution coming from @xmath463 is negligible in the @xmath464 limit . denote by @xmath465 . then we have @xmath466\right ) \\ & = \left(\frac{t}{\chi}\right)^{1/3}\frac{1}{\pi } \int_{\delta}^\infty dt \frac{1}{(|1-{{\rm i}}t/(1/2-\widetilde z_c)| |1+{{\rm i}}t/(1/2+\widetilde z_c)|)^\eta } \\ & = \left(\frac{t}{\chi}\right)^{1/3}\frac{1}{\pi } \int_{\delta}^\infty dt \frac{1}{(1+t^2/\rho^2)^{\eta/2}(1+t^2/(1-\rho)^2)^{\eta/2}}\\ \end{aligned}\ ] ] with @xmath467 when @xmath54 is large enough . for @xmath468 , we have the bound @xmath469 hence using the linear lower bound @xmath470 for all @xmath471 , we have @xmath472 therefore there exist a constant @xmath473 and a constant @xmath474 such that @xmath475 when @xmath476 , we obtain the same estimate by just replacing in some of the bounds @xmath1 by @xmath477 . next we determine the contribution from a @xmath425-neighborhood of the critical point . noting the taylor series @xmath478 we find that @xmath479 thus by choosing @xmath425 small enough , we find that for @xmath480 , @xmath481 and also @xmath482 for some constant @xmath383 . using  ,   and the general identity @xmath483 we deduce that @xmath484    finally , since @xmath485 for some constant @xmath486 , which is uniform for @xmath452 in a bounded set , we obtain   from   and  .    before discussing the asymptotic of @xmath370 we obtain also an exponential bound of @xmath487 when @xmath488 , which will be used later . [ lemasymptvbound ] consider the scaling ( [ eqscaling ] ) and @xmath279 ( i.e. , @xmath453 ) . then , for any given @xmath489 , there exists a @xmath490 large enough such that @xmath491 for all @xmath492 and @xmath493 . the constant @xmath494 is uniform in @xmath444 and in @xmath495 . lemma  [ lemasymptvbound ] set @xmath496 a+\frac{{\varepsilon}\chi^{1/3}}{2(\tau_i-\tau_j)},&\textrm { if } s_i - s_j > { \varepsilon}t^{1/3},\\[0.5em ] a-\frac{{\varepsilon}\chi^{1/3}}{2(\tau_i-\tau_j)},&\textrm { if } s_i - s_j < { \varepsilon}t^{1/3 } , \end{array } \right.\ ] ] where @xmath497 is a fixed constant chosen small enough so that @xmath498 lies in a compact subset of @xmath457 . taking the integration path as , a computation as in   yields ( the function @xmath499 is defined in  ) @xmath500 where @xmath501 when @xmath54 is large enough . hence @xmath502 the last integral satisfies @xmath503 as @xmath504 . therefore we find that there is a constant @xmath383 such that for @xmath54 large enough , @xmath505 now from   and   ( with @xmath506 and @xmath507 ) , if we have taken @xmath508 small enough , then @xmath509 now we plugging the value of @xmath498 in   for three difference cases . when @xmath510 , @xmath511 when @xmath512 , then @xmath513 when @xmath514 , then @xmath515 therefore , we obtain @xmath516 therefore , for any given @xmath517 , by taking @xmath497 small enough but fixed and then taking @xmath518 large enough , there is a constant @xmath383 such that for all @xmath493 , @xmath519 and hence @xmath520    finally , given any @xmath489 , by take @xmath521 . then when @xmath522 , then @xmath523 since @xmath524 . similarly , when @xmath525 , then @xmath526 this implies  . we now prove an asymptotic result for @xmath370 . [ lemasymptk ] consider the scaling ( [ eqscaling ] ) . then , @xmath527 uniformly for @xmath444 in a bounded set . lemma  [ lemasymptk ] by the definition  , @xmath528 the steepest - descent analysis of integrals very similar to this one with the same scaling   has been performed repeatedly in various places ( see for example , @xcite ) . the only difference here is that we have a double integral and we have to make sure that the two path do not touch . however this can be easily handled by locally modifying the steep(est)-decent contours near the critical point . except for this modification , the analysis of our case is similar to those in the literature . nevertheless we provide the proof for the completeness of the paper , and also since the analysis of this lemma and lemma  [ lemasymptkbound ] is a prototype of all the other lemmas in this section below ( except for lemma  [ lem : vlow ] ) . the first step is to determine a steep descent path for @xmath529 and @xmath530 . let us write @xmath531 where @xmath532,\\ h_{2,i}(w)&=-s_i w\chi^{-1/3}. \end{aligned}\ ] ] note that @xmath533 is independent of @xmath197 . for the steep descent analysis we need to determine a steep descent path for @xmath534 . the steep descent path will always be taken symmetric with respect to complex conjugation . thus we can restrict the discussion below to the part lying in the upper half plane . we have @xmath535 which both vanishes at the critical point @xmath536 , and @xmath537 . let @xmath538 with @xmath539 . then , for , @xmath540 is strictly decreasing in @xmath133 . indeed , @xmath541 since the last parenthesis is strictly positive : for @xmath542 , @xmath543 and @xmath544 , while @xmath545 . in the neighborhood of the critical point we consider a second path , @xmath546\}$ ] . along that path , we have @xmath547 the second degree term , @xmath548 , is strictly positive for all @xmath7 and @xmath549 $ ] . therefore also @xmath550 is a steep descent path , and close to the critical point will be steepest descent . now we can define the steep descent path used in the analysis . let @xmath551 . in a similar way , one obtains a steep descent path for @xmath552 , namely , @xmath553 is the path obtained by rotation around the origin of the steep descent path @xmath554 but with @xmath477 instead of @xmath1 . finally , as we shall discuss below , any local modification of the contours in a region of order @xmath555 around the critical point is allowed , see figure  [ figsteepk ] , so to have @xmath556 for a fixed @xmath497 . [ c]@xmath554 [ c]@xmath553 [ c]@xmath231 [ c]@xmath353 [ c]@xmath426   and @xmath553 are both anticlockwise oriented . the modification of @xmath553 close to the critical point @xmath231 is of order @xmath555.,title="fig:",height=188 ]    the path @xmath554 and @xmath553 are steep descent path for @xmath533 and @xmath552 . therefore , for any given small @xmath557 , the contribution to the double integral ( [ eq84 ] ) coming from @xmath558 is only of order @xmath559 for some @xmath473 ( with @xmath560 for small @xmath425 ) , which is smaller than @xmath561 . note that @xmath562 is the value of the integrand at the critical point . next we analyze the contribution coming from a @xmath425-neighborhood of the critical point @xmath563 . there we can use taylor series expansion of @xmath564 , which are given by @xmath565 therefore , the main contribution to ( [ eq84 ] ) is given by @xmath566 where @xmath567 and @xmath568 are the pieces of @xmath554 and @xmath553 which lies in a @xmath425-neighborhood of the critical point @xmath563 . more precisely , setting @xmath569 one has that @xmath570    assume that @xmath571 , then @xmath572 and @xmath573 thus , it is an easy computation to check that one can find @xmath574 small enough so that @xmath575 assume that @xmath576 . then it is easy to show as in ( [ borneot13 ] ) , lemma  [ lemasymptv ] that @xmath577    it thus remains us only to determine the asymptotic of ( [ eq90 ] ) . we make the change of variables @xmath578 and @xmath579 and obtain @xmath580 where the integration paths do not touch . we can now extend the integration path to infinity , since @xmath581 the conjugation by @xmath582 and the identity ( [ eqairya ] ) end the proof . now we give a bound which holds uniformly for @xmath444 bounded from below . let @xmath583 be given . [ lemasymptkbound ] consider the scaling ( [ eqscaling ] ) . then , for any given @xmath489 , there exists a @xmath490 large enough such that for all @xmath584 and @xmath585 @xmath586 the constant @xmath494 depends on @xmath587 only .    lemma  [ lemasymptkbound ] the upper bound of the integrals similar to   has also been obtained in various places ( see for example , @xcite ) . the analysis in our case is similar to those in the literature . however , again we provide the proof for the completeness of the paper ; also the analysis in this proof is going to be used and adapted in all the lemmas below in this section ( except for lemma  [ lem : vlow ] ) . first of all , we can rewrite @xmath588 set @xmath589 ( which corresponds to @xmath590 ) . then @xmath591 where @xmath592    we now show the exponential decay of @xmath593 for large positive @xmath594 . lemma  [ lemasymptk ] indeed implies the result when @xmath595 . it is thus enough to consider the case where @xmath596 . the analysis of @xmath597 is made in exactly the same way , up to a rotation around the origin and exchange of @xmath1 with @xmath477 .    to this aim we modify the contour given on figure  [ figsteepk ] as follows , defining a new contour @xmath598 . call @xmath599 the part of the contour lying in the upper half - plane @xmath600 . let @xmath489 be a fixed constant and @xmath601 then we set @xmath602 the contour @xmath598 is then completed by adjoining the conjugate of @xmath599 . this modification of the contour @xmath554 has no impact on the saddle point analysis made in the proof of lemma  [ lemasymptk ] . indeed , for @xmath54 large enough @xmath603 where @xmath604 is a uniformly bounded constant as @xmath605 is chosen in a compact interval of @xmath606    along @xmath598 it holds , @xmath607 thus , along @xmath598 @xmath608 using the saddle point argument used in the proof of lemma  [ lemasymptk ] and ( [ star ] ) , we easily deduce that there exists @xmath609 independent of @xmath594 such that for @xmath54 large enough @xmath610 thus , @xmath611 for some other constant @xmath494 . [ c]@xmath612 [ c]@xmath612 [ c]@xmath231 [ c](a ) [ c](b ) [ c](c ) [ c]@xmath353 [ c]@xmath426   that are only of order @xmath555.,title="fig:",height=188 ]    the saddle point analysis made for @xmath370 in the preceding lemmas will now be used , up to minor modifications , to consider the asymptotics of the remaining terms . [ lemr ] uniformly for @xmath613 in a bounded interval one has that @xmath614 moreover , for any given @xmath489 , there exists a @xmath490 large enough , such that for all @xmath495 it holds @xmath615 for any @xmath616 . the constant @xmath494 does not depend on @xmath617 or @xmath613 . lemma  [ lemr ] we have @xmath618 and also @xmath619\ ] ] with @xmath620 s defined in ( [ eq89 ] ) . as steep descent path we use almost the same contour @xmath553 as in figure  [ figsteepk ] , up to @xmath555-local deformation so that it passes to the right of @xmath231 , see figure  [ figsteeptwo ] ( a ) . then , using the same asymptotic argument as in the proof of lemma  [ lemasymptk ] , we deduce that @xmath621 where with @xmath622 we mean the path from @xmath623 to @xmath624 and passing on the right of @xmath143 . we can take it to its left up to adding a residue term , which is just @xmath613 . then the identity ( [ eqairyc ] ) gives the first formula of the lemma . + for the exponential decay , one first computes the residue at @xmath625 , yielding that @xmath626 the contour now lies to the left of @xmath231 , which ensures using the same argument as in lemma  [ lemasymptkbound ] , the exponential decay for large positive @xmath627    now we we turn to the asymptotics of the function @xmath628 defined in ( [ eq22 ] ) and the different terms of the function @xmath352 defined in ( [ eq31 ] ) . [ lemf12 ] consider the scaling ( [ eqscaling ] ) . then @xmath629 uniformly for @xmath594 in a bounded set . moreover , for any given @xmath489 , there exists a @xmath490 large enough , such that for all @xmath495 it holds @xmath630 for some constant @xmath494 independent of @xmath54 and uniform for @xmath594 bounded from below . lemma  [ lemf12 ] we have @xmath631 with the @xmath620 s defined in ( [ eq89 ] ) . as a steep descent path we use @xmath554 in figure  [ figsteepk ] with a local modification on the @xmath555 scale so that it passes on the right of @xmath231 , see figure  [ figsteeptwo ] ( b ) . then , the same asymptotic argument of the proofs of lemma  [ lemasymptk ] and lemma  [ lemasymptkbound ] still holds and gives us the claimed result . we also use the identity ( [ eqairyd ] ) to express the result in terms of airy functions . [ lemkfa ] consider the scaling ( [ eqscaling ] ) . then @xmath632 uniformly for @xmath633 in a bounded set . moreover , for any given @xmath489 , there exists a @xmath490 large enough , such that for all @xmath495 it holds @xmath634 for some constant @xmath494 independent of @xmath54 and uniform for @xmath635 bounded from below . lemma  [ lemkfa ] notice that @xmath636 . then , choosing the integration path for @xmath637 in ( [ eq24 ] ) so that @xmath638 , we can explicitly integrate the variable @xmath137 and get @xmath639 where the integration path for @xmath637 does not include @xmath231 and does not cross the integration path for @xmath640 . the rest follows along the same lines of lemma  [ lemasymptk ] and lemma  [ lemasymptkbound ] . finally one uses ( [ eqairye ] ) to rewrite the double integral expression in terms of airy functions . [ lemvfa][lem : vlow ] consider the scaling ( [ eqscaling ] ) and let @xmath641 . then @xmath642 uniformly for @xmath643 in a bounded set . moreover , for any given @xmath489 , there exists a @xmath490 large enough , such that for all @xmath495 it holds @xmath644}e^{-2\tau_i^3/3-\tau_is_i } \bigg)\ ] ] for some constant @xmath494 independent of @xmath54 and uniform in @xmath635 . lemma  [ lemvfa ] using @xmath645 and modifying the integration path in ( [ eq24 ] ) to @xmath646 for any @xmath647 , we can integrate out the @xmath137 variable with the result @xmath648 for the saddle point analysis we then deform back our integration path to @xmath649 with only a local deviation order @xmath555 around @xmath625 to make it passing on the right of @xmath231 . specifically , for a given small @xmath497 , we use the integration path @xmath650\},\ ] ] see figure  [ figsteeptwo ]  ( c ) . the asymptotic analysis for @xmath643 in a bounded set is the same as in lemma  [ lemasymptv ] except that the integral has an extra @xmath651 in the denominator . explicitly , we get @xmath652 for any @xmath497 . the function @xmath653 satisfies @xmath654 and the boundary condition @xmath655 therefore , @xmath656 is a gaussian distribution function , so that our limiting function is @xmath657    in order to prove  , we adapt the proof of lemma  [ lemasymptvbound ] . first , suppose that @xmath658 . for this case , we take the contour @xmath659 as in the proof of lemma  [ lemasymptvbound ] . note that @xmath660 . hence and the term @xmath661 in   is bounded by @xmath126 and an analysis as in the proof of lemma  [ lemasymptvbound ] as in the proof ( noting the presence of @xmath62 in  ) implies  . secondly , suppose that @xmath662 . we take the same contour . in this case , the residue at the simple pole @xmath231 should be taken into account . except for this term , the analysis on the contour @xmath554 is the same and we obtain  .    finally , when @xmath663 , we take the contour @xmath664 where @xmath550 is the part of the contour @xmath665 which lies outside the circle of radius @xmath666 centered at @xmath231 , and @xmath667 is the part of the circle of radius @xmath23 centered at @xmath231 whose real part is at least @xmath498 . this contour is same as the straight line @xmath665 , except that it goes around @xmath231 on the right in a @xmath668-neighborgood of @xmath231 . on @xmath550 , as @xmath669 , and hence we obtain , as in the proof of lemma  [ lemasymptvbound ] , @xmath670 on the other hand , on @xmath667 , observe that @xmath671 and the arc length of @xmath667 is also @xmath668 . by using the estimate   and  , which holds on @xmath667 if @xmath54 is large enough , we obtain that @xmath672 is bounded for @xmath673 ( where @xmath499 is in  ) . hence ( recall the first equation of  ) @xmath674 this , together with   implies  . the last term to be computed is the asymptotic of @xmath628 . [ lemg ] consider the scaling ( [ eqscaling ] ) . then @xmath675 uniformly for @xmath676 in a bounded set . moreover , for any given @xmath489 , there exists a @xmath490 large enough , such that for all @xmath495 it holds @xmath677 for some constant @xmath494 independent of @xmath54 and uniform for @xmath676 bounded from below . lemma  [ lemg ] we have @xmath678 with the @xmath620 s defined in ( [ eq89 ] ) . as steep descent path we use @xmath553 in figure  [ figsteepk ] with a local modification on the @xmath555 scale so that it passes on the right of @xmath231 , see figure  [ figsteeptwo ] ( a ) . then , the asymptotic argument of the proofs of lemma  [ lemasymptk ] and lemma  [ lemasymptkbound ] still holds and gives us the claimed result . the only small difference is that for the bound the integration path for large @xmath676 s has to be chosen to pass on the left of @xmath231 , which is fine up to adding the residue at @xmath231 . the factor @xmath679 comes from the multiplication by @xmath680 . to rewrite the result in terms of airy functions we used ( [ eqairyb ] ) . in this section we prove theorem  [ mainthm ] . recall   @xmath681 . \end{split}\ ] ] where @xmath682 and @xmath241 . we take the limit @xmath464 with the scaling  . note that the scaling of   is related to the scaling   by the identities @xmath683 and @xmath684 . the difference ( at most ) @xmath685 does not contribute to the asymptotics and only makes the notations complicated . for this reason , we will ignore this difference @xmath685 in the following presentation . [ propconvkai ] fix @xmath686 , and real numbers @xmath90 , @xmath687 . let @xmath688 , @xmath689 and @xmath690 be defined as in . then it holds @xmath691 where @xmath692}$ ] and @xmath101}$ ] . proposition  [ propconvkai ] proposition  [ propasymptkbar ] implies that for @xmath444 in a bounded interval @xmath693 @xmath694_{i , j}(s_i , s_j),\ ] ] and that the convergence is uniform on @xmath693 . now lemmas  [ lemasymptvbound ] and  [ lemasymptkbound ] then imply the convergence in trace class norm of the rescaled kernels @xmath695 and @xmath696 . this completes the proof . [ propconvscalarproduct ] fix @xmath686 real numbers @xmath90 , . let @xmath688 , @xmath689 and @xmath690 be defined as in  . then @xmath697 where @xmath698 is defined in  . proposition  [ propconvscalarproduct ] consider the term @xmath699 . we first scale and also conjugate by some operators . define the function @xmath700 set @xmath701 , and define the multiplication operator @xmath702 set @xmath703 and @xmath704 where one recalls the definitions of @xmath705 in ( [ eq31 ] ) and @xmath706 in ( [ eq22 ] ) . here the superscript @xmath54 indicates the dependence on @xmath54 .    by lemmas  [ lemf12 ] ,  [ lemkfa ] ,  [ lemvfa ] and  [ lemg ] , for any given @xmath489 , there exists a @xmath490 large enough , such that for all @xmath495 , @xmath707 and @xmath708 are @xmath709 for any given @xmath594 . then for @xmath54 large enough , @xmath710    let @xmath711 be the multiplication operator @xmath712 and set @xmath713 where the functions @xmath714 are defined in  . observe that the functions @xmath715 . since @xmath716 it is enough to prove @xmath717 where . clearly , @xmath718 first consider @xmath719 . as the operator - norm is bounded by the hilbert - schmidt norm @xmath720 , @xmath721 by taking @xmath722 in lemmas  [ lemasymptkbound ] and  [ lemasymptvbound ] , and using the dominated convergence theorem and proposition  [ propasymptkbar ] , the limit as @xmath464 of   becomes @xmath143 . by lemma  [ leminvertibility ] , the operator @xmath723 is invertible and it follows that @xmath724 as @xmath464 . now consider the term @xmath725 . from   of lemma  [ lemg ] , we have @xmath726 for @xmath727 for a fixed @xmath676 . the conjugation @xmath728 is introduced to make this function decay exponentially as @xmath729 . hence @xmath730 is bounded uniformly for large enough @xmath54 . also noting that the right - hand - side of   is precisely @xmath731 , it follows from dominated convergence theorem that @xmath732 as @xmath464 . it follows from lemmas  [ lemf12 ] ,  [ lemkfa ] and  [ lemvfa ] that @xmath733 for @xmath734 for a fixed @xmath676 . hence @xmath735 is uniformly bounded for large enough @xmath54 if we set for example @xmath736 . also it follows that @xmath737 as @xmath464 . hence we have shown that   converges to  . the remaining term in @xmath738 is @xmath739 . but lemma  [ lemr ] shows that this converges to @xmath740 ( after the changes of variables @xmath741 ) . this completes the proof . the above two propositions proves that @xmath742 converges to @xmath743 as @xmath464 . it remains to show that the derivatives with respect to @xmath744 s also converge . let us first consider @xmath745 of @xmath738 . as in  , this equals @xmath746 . note that the dependence on @xmath744 is only through the projection operator @xmath100 . hence by simple translations , @xmath747 where @xmath748 , @xmath749 and @xmath750 , and all the ( real ) inner product , the operator and the functions are all defined on @xmath751 . then @xmath752 hence we need to control the asymptotics of the derivatives of the kernels . but since all the asymptotic bounds for large @xmath136 in the lemmas in section  [ sectasympt ] are all exponential , while the derivatives of the kernels yields only polynomial terms , we can check that the dominated convergence theorem still applies and obtain the convergence of   to the corresponding derivatives of the limit . we omit the detail . the other terms are also similar . this completes the proof of theorem  [ mainthm ] .    to prove theorems  [ theoremdppext]-[theoremtasepb ] we use the following slow - decorrelation result , which is an extension of proposition 8 of  @xcite . [ lemslowdecorrelation ] let @xmath753 for some @xmath754 . let then with @xmath755 with @xmath756 . then , for any @xmath757 , it holds @xmath758    lemma  [ lemslowdecorrelation ] first of all , it is easy to check by proposition 2.2 of  @xcite that the difference between the model   and the last passage percolation model corresponding exactly to the stationary tasep becomes irrelevant in the order as @xmath464 . denote @xmath759 the measure for the last passage percolation corresponding to the stationary tasep . due to the stationarity in space and time of tasep , we observe that @xmath760 and @xmath761 have the same distribution . therefore , @xmath762 but since the distribution of @xmath763 converges to @xmath764 @xcite , and @xmath765 as @xmath464 , we find that   converges to @xmath5 as @xmath464 . this completes the proof . theorem  [ theoremdppext ] the projection of @xmath766 on the line along the critical direction @xmath767 is @xmath768 with @xmath769 defined in ( [ eq6 ] ) . we have @xmath770 with @xmath771 . in particular then , @xmath773 with @xmath774 by lemma  [ lemslowdecorrelation ] we have @xmath775 so that @xmath776 see the proof of theorem  1 of  @xcite for detailed steps . this is by theorem  [ mainthm ] the desired result . theorem  [ theoremtasep ] the goal is the express our setting into the one of theorem  [ theoremdppext ] . denote @xmath777 , which is given by @xmath778 and recall @xmath779 ( since @xmath780 and @xmath781 are integers , the above formulas are are exact up to the error of size @xmath214 at most : since this difference does not affect the asymptotics but only complicates the formulas , we drop this difference in the following presentation . ) by ( [ eqdptasep ] ) we have @xmath782 denote by @xmath783 the projection of @xmath784 on the line along the critical direction @xmath767 . we get @xmath785 where @xmath786 let us further denote @xmath787 then , @xmath788 where @xmath789 is defined in  . then , replacing ( [ eq5.22 ] ) into ( [ eq5.21 ] ) we get @xmath790 where @xmath791 are defined in ( [ eq6 ] ) .    with these notations we can rewrite ( [ eq5.20 ] ) as follows , @xmath792 where @xmath793 . then , by theorem  [ theoremdppext ] the result follows .    theorem  [ theoremtasepb ] the proof of theorem  [ theoremtasepb ] is very similar to the one of theorem  [ theoremtasep ] . the scaling ( [ eqscalingheight ] ) corresponds , in terms of @xmath769 , to the scaling @xmath794 the projections @xmath783 of @xmath28 on the line @xmath795 along the critical direction @xmath767 are given by @xmath796 , @xmath797 . explicitly we find @xmath798 setting @xmath799 we get @xmath800 then , setting @xmath801 we get ( [ eq5.25 ] ) and then by theorem  [ theoremdppext ] the result follows . in the asymptotic analysis we get two basic integral expressions which can be rewritten in terms of airy functions and exponential , namely @xmath802 starting from these two formulas we state some identities . [ lemairyexpressions ] @xmath803 @xmath804 @xmath805 @xmath806 @xmath807 @xmath808    lemma  [ lemairyexpressions ] ( a ) since we can choose the paths for @xmath809 and @xmath810 such that @xmath811 , using @xmath812 and ( [ eqairy ] ) we get ( [ eqairya ] ) . \(b ) let @xmath813 . then differentiating and using ( [ eqairy ] ) we get @xmath814 . together with the boundary condition @xmath815 , we get ( [ eqairyb ] ) . \(c ) let @xmath816 . differentiating twice and applying ( [ eqairy ] ) we obtain @xmath817 . the boundary conditions are @xmath818 . integrating twice and shifting the integration bounds to zero , we get ( [ eqairyc ] ) . \(d ) this can be derived by ( [ eqairyb ] ) by change of variable @xmath819 . \(e ) this identity can be found by first using ( [ eq141 ] ) and then the two complex integrals are decoupled . for the integral over @xmath809 one uses ( [ eqairy ] ) , while for the integral over @xmath810 one uses ( [ eqairyb ] ) . \(f ) one applies the identity ( 2.20 ) of  @xcite , which holds for @xmath820 , @xmath821 and then we set @xmath822 , @xmath823 . [ leminvertibility ] for any fixed real numbers @xmath91 , @xmath824 exists . lemma  [ leminvertibility ] first of all , notice that @xmath825 now , let @xmath826 . then , @xmath827 the equality with the goe tracy - widom distribution , @xmath828 , is proven in corollary 1.3 of  @xcite , while the strict inequality follows from the monotonicity of the distribution function and the large @xmath829 asymptotics ( see for example ,  @xcite ) .    moreover , as shown in section 2.2 of  @xcite , @xmath115 is trace - class ( the shift by @xmath830 is irrelevant for that property , since it holds for any @xmath744 s ) . the proof ends by applying a known result on fredholm determinant , see e.g.  theorem xiii.105 ( b ) in  @xcite : let @xmath831 be a trace - class operator , then @xmath832 in this section , we discuss the basic properties of the operators @xmath293 and @xmath276 in section  [ sectshift ] . [ proptraceclass]@xmath833    * the operators @xmath834 and @xmath278 are bounded in @xmath291 . * let @xmath835 . fix constants @xmath286 such that @xmath836 then @xmath837 defined in   is a trace - class operator on @xmath291 . on the other hand , @xmath838 defined in   is a trace - class in the same space if the constants satisfy more restricted condition @xmath839    proposition  [ proptraceclass ] we only prove ( ii ) . the proof of ( i ) follows easily by suitably modifying the analysis for part ( ii ) . we start with @xmath837 . since the set of trace - class operators is a linear space , it is enough to prove that @xmath840 is a trace - class operator in @xmath311 for each @xmath841 . as @xmath842 ( see  ) , we prove that each of the two operators is trace - class .    _ ( a ) operator @xmath450 : _ it is easy to check that the operator can be re - expressed as ( see ( [ eq96 ] ) ) @xmath843 where @xmath844 by choosing the integration paths close enough to @xmath353 , resp . @xmath426 , a simple estimate shows that there exists @xmath845 so that @xmath846 from this it follows immediately that @xmath847 and @xmath848 are hilbert - schmidt operators on @xmath311 . so is the conjugated kernel of @xmath450 trace - class . _ ( b ) operator @xmath449 : _ as @xmath446 for @xmath849 , assume that @xmath279 .    in this case , we have @xmath850 from ( [ eq24 ] ) , by plugging in @xmath851 and @xmath852 and by changing the contour from @xmath402 to @xmath853 , we obtain @xmath854 where @xmath855 hence @xmath856 the fourier transform of @xmath354 . note that @xmath857 where @xmath858 as @xmath859 so then is @xmath860 . moreover , as @xmath861 is well - defined pointwise except at @xmath862 ( since it is conditionally convergent ) , @xmath863 equals   almost everywhere . furthermore , we can compute   explicitly and obtain @xmath864}.\ ] ] similarly , @xmath865}.\ ] ] we have @xmath866 hence @xmath867 here all the steps make sense since @xmath868 . note @xmath869 for @xmath279 , and @xmath870 . by using  , we see that @xmath871 where @xmath872 and @xmath873 are the operators with kernel @xmath874},\\ \mathcal{h}(x , y)&= \frac{(x - y)^{t_i - t_j-1}}{(t_i - t_j-1 ) ! } e^{-\frac{(x - y)}2 } { \mathbbm{1}}_{[x - y>0 ] } , \end{aligned}\ ] ] and @xmath875 is the multiplication operator @xmath876 then @xmath877 changing the variable @xmath137 by @xmath878 , the above equals @xmath879 which is finite . hence @xmath880 is a hilbert - schmidt operator . similarly , @xmath881 is finite , and hence @xmath882 is a hilbert - schmidt operator . therefore , @xmath883 is a trace - class operator . now we consider @xmath288 . from  , @xmath884 but by changing the contour in  , @xmath885 hence there is a constant @xmath383 such that @xmath886 for some constant @xmath887 for @xmath888 . therefore , @xmath889 if @xmath890 . similarly , @xmath891 if @xmath892 . being a product of two hilbert - schmidt operator , @xmath893 is trace - class if   holds . this completes the proof . consider the directed percolation model defined in ( [ eqdp ] ) , but look away from the characteristic line . we set @xmath894 denote @xmath895 the passage time from @xmath896 to @xmath28 . then @xmath897 in section  6 of  @xcite ( see also @xcite ) a last passage percolation with one source only is considered . nevertheless , the arguments given therein can readily be used to prove the following : if @xmath898 @xmath899 where @xmath900 and @xmath901 if @xmath902 then there exists a constant @xmath903 ( see @xcite ) such that @xmath904 where @xmath905 and @xmath906 is the gue tracy - widom distribution . by symmetry , if @xmath902 , setting @xmath907 and @xmath908 , it holds @xmath909 and for @xmath910 @xmath911    consider now the case @xmath912 . then since @xmath913 we have @xmath914 with the notation @xmath915 and ( [ eqgauss0 ] ) we obtain @xmath916 we take the limit @xmath34 on both sides and by ( [ eqgauss1 ] ) and ( [ eqgauss3 ] ) we get @xmath917                                      p.l . ferrari . . in j.  baik , t.  kriecherbauer , l - c . li , k.  mclaughlin , and c.  tomei , editors , _ integrable systems and random matrices : in honor of percy deift _ , contemporary math . , pages 321332 . soc . , 2008 . k.  johansson . random matrices and determinantal processes . in a. bovier , f.  dunlop , a.  van enter , f.  den hollander , and j.  dalibard , editors , _ mathematical statistical physics , session lxxxiii : lecture notes of the les houches summer school 2005 _ , pages 156 . elsevier science , 2006 .
the totally asymmetric simple exclusion process ( tasep ) on @xmath0 with the bernoulli-@xmath1 measure as initial conditions , @xmath2 , is stationary . 
 it is known that along the characteristic line , the current fluctuates as of order @xmath3 . 
 the limiting distribution has also been obtained explicitly . in this paper 
 we determine the limiting multi - point distribution of the current fluctuations moving away from the characteristics by the order @xmath4 . 
 the main tool is the analysis of a related directed last percolation model . 
 we also discuss the process limit in tandem queues in equilibrium .