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we consider the unitary random matrix model @xmath0 defined on hermitian @xmath1 matrices @xmath2 in a critical regime where the limiting mean density of eigenvalues vanishes at an interior point . it is a basic fact of random matrix theory @xcite that the eigenvalues of the random matrix ensemble ( [ 1-randommatrixmodel ] ) follow a determinantal point process with correlation kernel @xmath3 where @xmath4 denotes the @xmath5th degree orthonormal polynomial with respect to the weight @xmath6 on @xmath7 . we assume in this paper that the confining potential @xmath8 in ( [ 1-randommatrixmodel ] ) is real analytic and that it satisfies the growth condition @xmath9 these assumptions ensure that the mean eigenvalue density @xmath10 has a limit as @xmath11 , @xmath12 , see e.g.  @xcite , which we denote by @xmath13 . it is known that @xmath14 is the density of the measure @xmath15 which minimizes the weighted energy @xmath16 among all probability measure on @xmath7 . the measure @xmath15 is called the equilibrium measure in the external field @xmath17 . the fact that @xmath17 is real analytic ensures that the support @xmath18 consists of a finite union of intervals @xcite . it is a remarkable fact that local scaling limits of the kernel ( [ 1-kern ] ) depend only on the nature of the density @xmath14 . this has been proved rigorously in the bulk of the spectrum for a quartic @xmath17 in @xcite and for general real analytic @xmath17 in @xcite . indeed , if @xmath19 , then @xmath20 exists , and @xmath21 the scaling limits are different at special points of the spectrum . at edge points of the spectrum the density @xmath14 typically vanishes like a square root , and then it is known that for some constant @xmath22 , @xmath23 where @xmath24 and @xmath25 is the airy function . the airy kernel is related to the tracy - widom distribution @xcite . in ( [ edge - scaling ] ) we have assumed that @xmath26 is a right edge point . for a left edge point we change @xmath27 , @xmath28 in the left - hand side of ( [ edge - scaling ] ) . other special points in the spectrum include    * edge points where the density vanishes to a higher order . the possible edge point behaviors ( at a right end point @xmath26 ) are @xmath29 where @xmath22 and @xmath5 is a non - negative integer . * interior points where the density vanishes . then @xmath30 where @xmath22 and @xmath5 is a positive integer .    in these critical cases it is believed that the local scaling limit at @xmath26 of the kernel only depends on the order of vanishing of the density at @xmath26 @xcite . the case where @xmath14 vanishes quadratically at an interior point of @xmath31 , that is , the case @xmath32 in ( [ interior - vanishing ] ) , was considered by bleher and its @xcite for the case of a critical quartic potential @xmath33 , with @xmath34 and @xmath35 . then @xmath36,\ ] ] so that @xmath14 vanishes quadratically at the origin . bleher and its consider the double scaling limit where @xmath37 changes with @xmath38 and tends to @xmath39 as @xmath40 in such a way that @xmath41 remains constant . for the quartic potential this is equivalent to considering ( [ 1-randommatrixmodel ] ) where @xmath11 , @xmath12 , such that @xmath42 exists . bleher and its gave a one - paremeter family @xmath43 of limiting kernels , depending on @xmath44 , so that for some @xmath22 , @xmath45 where @xmath46 is proportional to the value of the limit ( [ nn - to - infty ] ) . the critical kernels are expressed in terms of so - called @xmath47-functions associated with the hastings - mcleod solution of the painlev ii equation @xcite . consider as in @xcite the linear differential equations for a @xmath48-vector ( or @xmath49 matrix ) @xmath50 , @xmath51 where @xmath52 and @xmath53 the compatibility condition for ( [ psi - diffeqns ] ) is that @xmath54 satisfies the painlev ii equation @xmath55 and that @xmath56 . we assume that @xmath57 is the hastings - mcleod solution of painlev ii , which is characterized by the asymptotic condition @xmath58 the critical kernels op @xcite are given by @xmath59 where @xmath60 is the special solution to ( [ psi - diffeqns ] ) which is real for real @xmath61 , satisfies @xmath62 and has asymptotics on the real line @xmath63    if we put @xmath64 then @xmath65 and @xmath66 is a special solution of the differential equations @xmath67 and @xmath68 the equations ( [ 1-diiffvglpsi1])-([1-diiffvglpsi2 ] ) for the @xmath47-functions correspond to the ones used by flaschka and newell @xcite and we will also use those in what follows . the vector @xmath66 is the unique solution of ( [ 1-diiffvglpsi1 ] ) with asymptotics @xmath69 as @xmath70 uniformly in @xmath71 for any @xmath72 .    before discussing our results , we like to point out an integral formula for the kernel @xmath43 . if we take a derivative of ( [ critk1 ] ) with respect to @xmath46 and use ( [ psi - diffeqns ] ) and ( [ psi - diffb ] ) , we get after some calculations @xmath73.\ ] ] using the deift / zhou steepest - descent method for @xmath74 as done in @xcite , one can show that @xmath75 as @xmath76 , so that we get @xmath77    d\sigma.\ ] ] since @xmath78 and @xmath79 are real for real @xmath61 , formula ( [ critk3 ] ) clearly shows that @xmath80 , as it should be . it is the aim of this paper to show that the kernel @xmath43 is a universal limit . whenever the limiting mean eigenvalue density @xmath14 vanishes quadratically at an interior point , the correlation kernel @xmath81 has a double scaling limit given by ( [ 1-phikernel ] ) .    in our theorem [ theorem-1-maintheorem ] below , we use the equilibrium measure @xmath82 of a compact set @xmath83 . this is the unique probability measure on @xmath84 that minimizes the logarithmic energy @xmath85 among all borel probability measures @xmath86 on @xmath84 . if @xmath84 is a single interval @xmath87 $ ] , then @xmath88 has a density @xmath89 given by @xmath90 if @xmath84 is a finite union of disjoint intervals , say @xmath91 $ ] with @xmath92 . then @xmath88 has density @xmath93 where @xmath94 is a monic polynomial of degree @xmath95 with exactly one zero in each of the gaps @xmath96 , @xmath97 , see e.g.  ( * ? ? ? * lemma 4.4.1 ) . note that ( [ wsdensity ] ) has an extension to an analytic function in @xmath98 , where @xmath99 , which is a fact that we will use in what follows . the following is our main result . [ theorem-1-maintheorem ] let @xmath17 be real analytic on @xmath7 such that @xmath100 . let @xmath14 be the density of the equilibrium measure in the external field , and let @xmath26 be an interior point of @xmath101 which is such that @xmath102 let @xmath11 such that the limit @xmath103 exists with @xmath104 . let @xmath81 be the correlation kernel ( [ 1-kern ] ) for the eigenvalues of the random matrix model ( [ 1-randommatrixmodel ] ) . then there exist constants @xmath22 and @xmath44 such that @xmath105 uniformly for @xmath106 in compact subsets of @xmath7 . explicit formulas for the constants @xmath107 and @xmath46 are @xmath108 and @xmath109 where @xmath110 is the density of the equilibrium measure of @xmath31 .    as noted before , bleher and its @xcite proved ( [ main1 ] ) for the case of a critical quartic @xmath17 . see @xcite for a rigorous expansion of the free energy in this critical case . the random matrix model ( [ 1-randommatrixmodel ] ) may be generalized to include a spectral singularity at the origin @xmath111 if @xmath112 and @xmath113 , then the scaled limit of the correlation kernels is a bessel kernel which involves bessel functions of order @xmath114 , see @xcite . in the multicritical case where @xmath14 vanishes quadratically at @xmath115 , an analogue of theorem 2.1 is valid . in work in progress @xcite , we are considering the double scaling limit of ( [ spec - sing ] ) and we show that the limiting kernels are expressed in terms of the @xmath47-functions associated with a special solution of the general painlev ii equation @xmath116    the main ingredients in the proof of theorem [ theorem-1-maintheorem ] are equilibrium measures and riemann - hilbert problems . we give some comments on both . [ [ equilibrium - measures . ] ] equilibrium measures . + + + + + + + + + + + + + + + + + + + + +    recall that the equilibrium measure in external field @xmath17 minimizes ( [ 1-minenergie ] ) . we need to know how the equilibrium measure @xmath15 in the external field changes as a result of a change in @xmath17 . the particular modification we consider here is @xmath117 so that @xmath118 . let us put @xmath119 then it is known that @xmath120 and @xmath121 are increasing as a function of @xmath37 , see e.g.  @xcite . we also have the buyarov - rakhmanov formula @xcite @xmath122 which expresses the equilibrium measure in the external field as an average of equilibrium measures of sets . a consequence of ( [ 1-buraformule ] ) is that @xmath123 which partly explains why the equilibrium measure @xmath124 plays a role in the formula ( [ main3 ] ) for @xmath46 .     for @xmath125 with @xmath37 equal to @xmath126 , @xmath127 and @xmath128 , respectively.,title="fig : " ]   for @xmath125 with @xmath37 equal to @xmath126 , @xmath127 and @xmath128 , respectively.,title="fig : " ]   for @xmath125 with @xmath37 equal to @xmath126 , @xmath127 and @xmath128 , respectively.,title="fig : " ]    for the case of interest in this paper we have that @xmath14 vanishes at @xmath129 . then for @xmath130 , there is a positive density at @xmath26 , while for @xmath131 , @xmath26 is out of the support of @xmath132 . for @xmath37 slightly less than @xmath127 , there is a gap in @xmath133 , see figure [ figure - tweedeschaling ] . an asymptotic analysis based on the equilibrium measure @xmath132 would require a discussion of the two different situations @xmath130 and @xmath131 , as is done in @xcite . therefore we found it convenient to introduce a modification of the equilibrium problem in external field , which will enable us to treat both cases simultaneously . the modification we make is that we do not require the measure to be non - negative in a neighborhood of the point @xmath26 . for a sufficiently small @xmath134 , we consider the problem to minimize @xmath135 among all signed measures @xmath136 on @xmath7 , where @xmath137 are nonnegative measures , such that @xmath138.\ ] ] we denote the minimizer by @xmath139 and we let @xmath140 .     for @xmath125 with @xmath37 equal to @xmath126 , @xmath127 and @xmath128 , respectively ( compare with figure [ figure - tweedeschaling]).,title="fig : " ]   for @xmath125 with @xmath37 equal to @xmath126 , @xmath127 and @xmath128 , respectively ( compare with figure [ figure - tweedeschaling]).,title="fig : " ]   for @xmath125 with @xmath37 equal to @xmath126 , @xmath127 and @xmath128 , respectively ( compare with figure [ figure - tweedeschaling]).,title="fig : " ]    then @xmath141 for @xmath142 , but for @xmath131 there is a clear distinction between @xmath139 and @xmath132 , see figure 2 . however we still have the analogue of ( [ 1-bura2 ] ) ( as we prove ) @xmath143 what s more , we also have ( [ 1-bura3 ] ) at the level of densities , that is , if @xmath144 denotes the density of @xmath139 , and if @xmath145 is an interior point of @xmath31 ( in particular if @xmath129 ) , then @xmath146 where @xmath110 is the density of the equilibrium measure of @xmath31 , which is what we need for the proof of theorem [ theorem-1-maintheorem ] . note that we do not have ( [ 1-bura4 ] ) for @xmath129 if @xmath144 is the density of @xmath132 . [ [ riemann - hilbert - problem . ] ] riemann - hilbert problem . + + + + + + + + + + + + + + + + + + + + + + + +    the second main tool for the proof of theorem [ theorem-1-maintheorem ] is the characterization of orthonormal polynomials by means of a riemann - hilbert problem , due to fokas , its , and kitaev @xcite , and the subsequent asymptotic analysis of the riemann - hilbert problem by means of the deift / zhou steepest descent analysis of riemann - hilbert problems , introduced in @xcite , and further developed in @xcite , and other more recent papers . the deift / zhou steepest descent analysis of the riemann - hilbert problem consists of a sequence of explicit transformations , which result in a riemann - hilbert problem that is explicitly solvable in terms of neumann series . ( in fact , in this paper , we only need the first term of this series . ) the critical point @xmath26 needs special attention . of particular interest for us is the paper @xcite by baik , deift , and johansson on the length of the longest increasing subsequence of a random permutation of @xmath147 . these authors show that the fluctuations of this random variable are distributed according to the tracy - widom distribution @xcite in the limit as @xmath40 . one of the technical tools in this important paper is the asymptotic analysis of a riemann - hilbert problem on the unit circle which is related to an equilibrium measure ( also on the unit circle ) whose density vanishes at the point @xmath148 . this situation is comparable to ours . the authors of @xcite , see also subsequent papers @xcite , construct a local parametrix near @xmath148 with the aid of the @xmath47-functions associated with the hastings - mcleod solution of painlev ii . these @xmath47-functions satisfy a model riemann - hilbert problem and the local parametrix is constructed by appropriately mapping the model riemann - hilbert problem onto a neighborhood of @xmath148 so that it satisfies certain desired jump properties approximately . we follow the approach of @xcite but we introduce a modification in the construction of the local parametrix so that it has the desired jump properties _ exactly _ , in contrast to @xcite where the desired jump properties only hold _ the fact that we have the exact jump properties simplifies the arguments considerably and we feel that this is also a main contribution of the present paper . [ [ outline - of - the - rest - of - the - paper . ] ] outline of the rest of the paper . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    in section 3 we collect the necessary facts about equilibrium measures . in particular we study the modified equilibrium problem with external field in some detail . in section 4 we discuss the riemann - hilbert problem satisfied by the @xmath47-functions . here we follow @xcite . then in section 5 we state the riemann - hilbert problem for orthogonal polynomials , discuss the relation with the correlation kernel @xmath81 , and perform the transformations in the steepest descent analysis . finally in section 6 we give the proof of theorem [ theorem-1-maintheorem ] . as explained in the previous section , we consider a modification of the equilibrium problem where we drop the non - negativity condition in a small neighborhood of @xmath26 . we take @xmath134 sufficiently small so that @xmath149 \setminus \{x^*\}.\ ] ] and we use @xmath139 to denote the signed measure that minimizes ( [ mod - energy1 ] ) under the conditions ( [ mod - energy2 ] ) . we define @xmath150 the existence and uniqueness of @xmath139 follows as in @xcite . let @xmath151 be the logarithmic potential of @xmath152 . then standard arguments of potential theory @xcite show that @xmath139 is the unique signed measure satisfying ( [ mod - energy1 ] ) with the property that @xmath153 , \\ \label{variational - nut2 }     2u^{\nu_t}(x ) + \frac{1}{t } v(x ) & \geq   \ell_t ,      \qquad x \in \mathbb r.\end{aligned}\ ] ] for some constant @xmath154 .    as before we use @xmath132 to denote the equilibrium measure in the external field @xmath155 . this is a probability measure that satisfies for some constant @xmath156 , @xmath157 it is known that @xmath120 and @xmath133 are increasing with @xmath158 , see @xcite . for @xmath142 , we have @xmath159 which implies that @xmath141 in view of ( [ delta0 ] ) and the variational conditions ( [ variational - nut1])([variational - mut2 ] ) . for @xmath131 , we have that @xmath26 is outside the support of @xmath132 , and in fact the strict inequality in ( [ variational - mut2 ] ) holds for @xmath160 if @xmath131 .    in general there is the following inequality between @xmath132 and @xmath139 .    for every @xmath161 we have @xmath162    let @xmath163 . from ( [ variational - nut1 ] ) , ( [ variational - nut2 ] ) , ( [ variational - mut1 ] ) , and ( [ variational - mut2 ] ) it follows that @xmath164 the potential @xmath165 is subharmonic on @xmath166 and since @xmath167 , it is subharmonic at infinity as well . by the maximum principle for subharmonic functions @xcite , the maximum of @xmath165 is attained in @xmath168 only . since @xmath169 we then have by ( [ ulambda6 ] ) that equality in ( [ ulambda7 ] ) holds for every @xmath170 , and so @xmath171 . this implies ( [ mut - leq - nut ] ) . it follows from ( [ mut - leq - nut ] ) that @xmath172 is empty . since for @xmath37 slightly less than @xmath127 a gap opens in @xmath133 , which depends continuously on @xmath37 , see @xcite , it follows that for any given @xmath173 , there is @xmath174 such that @xmath175 ,      \qquad \mbox { for } t > t_0.\ ] ] this shows that for @xmath131 sufficiently close to @xmath127 , the definition of @xmath139 is independent of the choice of @xmath176 . a very useful fact is that for real analytic @xmath17 , say @xmath17 is analytic in a neighborhood @xmath177 of the real line , the measures @xmath132 have densities @xmath178 which can be expressed in terms of the negative part of an analytic function in @xmath177 . indeed , if @xmath179 then it was shown in @xcite that @xmath180 where @xmath181 denotes the negative part of @xmath182 . a consequence of ( [ tilde - psit ] ) is that @xmath133 is the closure of the set where @xmath182 is negative . the arguments of @xcite can be readily extended to the signed measures @xmath139 , provided that @xmath183 $ ] for some @xmath184 . we define @xmath185 and then the following holds . [ prop2 - 1 ] there exists @xmath186 such that for every @xmath187 , the signed measure @xmath139 has a density @xmath144 and @xmath188 ^ 2 , \qquad \mbox { for } x \in s_t.\ ] ] in addition we have @xmath189 ^ 2      \qquad \mbox { for } x \in \mathbb r \setminus s_t.\ ] ]    this follows as in ( * ? ? ? * proposition 2.51 ) . obviously , for @xmath142 we have @xmath190 . since @xmath191 has a double zero at @xmath26 , we see from ( [ qt1 ] ) that @xmath192 has a zero at @xmath26 of order four . for @xmath37 slightly bigger than @xmath127 , this fourth order zero splits into two double zeros in the complex plane away from the real axis .    for @xmath37 slightly less than @xmath127 , there is a difference in the behavior of the zeros of @xmath193 and @xmath182 . indeed , @xmath182 has two simple real zeros near @xmath26 which are endpoints of the support of @xmath132 , and in addition there is a double real zero in between them . on the other hand , we have that @xmath193 has two double real zeros near @xmath26 , which are endpoints of the support of @xmath194 . the fact that @xmath195 has only double zeros near @xmath26 allows us to take an analytic square root , and in view of ( [ qt1 ] ) we choose it so that for @xmath196 , @xmath197 where the sign of the square root at @xmath129 is taken negative if @xmath131 and positive if @xmath130 . the right - hand side of ( [ qt3 ] ) has an analytic extension to a neighborhood of @xmath26 , which is independent of the value of @xmath187 . thus @xmath144 has an analytic extension to a fixed neighborhood of @xmath26 , which will also be denoted by @xmath144 . recall from the discussion before the statement of theorem 2.1 , that the density @xmath110 of the equilibrium measure of @xmath31 also has an analytic extension to a neighborhood of @xmath26 , which we also denote by @xmath110 . the remaining part of this section is devoted to the proof of the following proposition . [ prop2 - 2 ] we have @xmath198 uniformly for @xmath199 in a neighborhood of @xmath26 . we start with a lemma which contains a weaker form of ( [ diff - tpsi ] ) . we have @xmath200 where @xmath124 is the equilibrium measure of @xmath31 . buyarov and rakhmanov @xcite proved ( for a very general class of @xmath17 ) that @xmath201 and @xmath202 where @xmath203 is the equilibrium measure of the set @xmath204 see ( [ variational - mut1 ] ) with @xmath205 . for real analytic @xmath17 , the sets @xmath31 and @xmath206 differ by an at most finite number of points , so that @xmath207 . thus by ( [ bura1 ] ) and ( [ bura2 ] ) we have @xmath208    now write @xmath209 in view of ( [ bura3 ] ) it suffices to prove that @xmath210 where the norm denotes the total variation of a signed measure . we may assume that @xmath131 . because of ( [ mut - leq - nut ] ) we have @xmath211 for @xmath131 , the support of @xmath194 is contained in an interval @xmath212 $ ] , with @xmath213 see ( * ? ? ? * lemma 8.1(iii ) ) . from ( [ qt3 ] ) and the fact that @xmath193 has two double zeros in @xmath212 $ ] which are the end - points of the support of @xmath194 , we then easily get that @xmath214 combining ( [ deltat1 ] ) and ( [ deltat2 ] ) , we find @xmath215 as @xmath216 , which by ( [ nut - mut2 ] ) implies ( [ nut - mut ] ) . this completes the proof of the lemma . we now give a characterization of @xmath14 , which will be of use in the proof of proposition [ prop2 - 2 ] . [ lemmaclaim ] for @xmath217 we have @xmath218 where @xmath110 is the density of @xmath124 .    by the sokhotski - plemelj formulas , see e.g.  @xcite , we have that @xmath219 satisfies @xmath220 let @xmath221 $ ] and set @xmath222 where the square root is positive for @xmath223 . using ( [ sp1 ] ) and the fact that @xmath224 as @xmath225 , we see that @xmath226 for any polynomial @xmath227 . suppose @xmath227 has degree at most @xmath95 . then ( [ sprongpfr ] ) and the fact that @xmath228 as @xmath229 imply that @xmath230 which we can rewrite for @xmath231 as @xmath232 by contour integration . now we use ( [ eq : fz - lemma ] ) and ( [ sp2 ] ) to obtain @xmath233 and this holds for any polynomial @xmath227 of degree at most @xmath95 . since we know by ( [ wsdensity ] ) that @xmath234 for some monic polynomial @xmath227 of degree @xmath95 , we get ( [ eq : psiv - lemma ] ) . now we can give the proof of proposition [ prop2 - 2 ] . * of proposition [ prop2 - 2 ] . * by ( [ qt ] ) we have for @xmath231 , @xmath235 so that in view of ( [ diff - tnu ] ) we find , @xmath236 thus by ( [ qt3 ] ) for @xmath237 in a neighborhood of @xmath26 , @xmath238 by ( [ eq : psiv - lemma ] ) , the right - hand side of ( [ diff - tpsi2 ] ) is @xmath239 in case @xmath237 is real , and by analytic continuation it continues to be @xmath239 in a punctured neighborhood of @xmath26 . thus ( [ diff - tpsi ] ) holds for @xmath237 in a neighborhood of @xmath26 , and then it easily follows for @xmath240 as well . in this section we recall the rh problem associated with the second painlev equation , see @xcite . we transform this rh problem into a rh problem that will be used in the next section to construct a parametrix around @xmath26 . consider the differential equation of ( [ 1-diiffvglpsi1 ] ) , @xmath241 where @xmath242 is a @xmath243 complex matrix - valued function and @xmath46 , @xmath244 and @xmath245 are considered as parameters . all solutions of ( [ 4-diffvglpsi ] ) are entire functions of @xmath61 .    for @xmath246 , let @xmath247 be the sector @xmath248 there exists a unique solution @xmath249 of equation ( [ 4-diffvglpsi ] ) so that @xmath250 as @xmath251 in the sector @xmath247 . here we use @xmath252 to denote the third pauli matrix . there exist complex values @xmath253 , @xmath246 ( called stokes multipliers ) so that @xmath254 with @xmath255 and @xmath256 furthermore , we have @xmath257 now define the rays @xmath258 oriented away from the origin , and the matrix - valued function @xmath259 then @xmath242 satisfies the following rh problem    * @xmath242 is analytic on @xmath260 . * @xmath261 on @xmath262 , * @xmath263 as @xmath251 , * @xmath242 is bounded near @xmath115 . [ fig : jumpspsialgemeen ]    ( 100,100)(0,0 ) ( 77,57)@xmath264 ( 51,80)@xmath265 ( 14,72)@xmath266 ( 7,41)@xmath267 ( 33,18)@xmath268 ( 68,26)@xmath269 ( 50,50)(0,1)23 ( 50,50)(0,-1)23 ( 50,73)(0,1)27 ( 50,27)(0,-1)27 ( 50,50)(2,1)23 ( 50,50)(2,-1)23 ( 73,61.5)(2,1)27 ( 73,38.5)(2,-1)27 ( 50,50)(-2,1)23 ( 50,50)(-2,-1)23 ( 27,61.5)(-2,1)27 ( 27,38.5)(-2,-1)27    the stokes multipliers @xmath253 depend on @xmath46 , @xmath244 and @xmath245 . an isomonodromy deformation is a variation of these parameters such that the stokes multipliers remain constant . flaschka and newell @xcite showed that the isomonodromy deformations are given by the painlev ii equation @xmath270 and @xmath271 . any solution of the painlev ii equation is a meromorphic function with an infinite number of poles . we write @xmath272 for the @xmath242 function ( [ def - psi ] ) with parameters @xmath46 , @xmath54 , @xmath273 , where @xmath244 is the hastings - mcleod solution and @xmath274 . let @xmath275 be the set of poles of @xmath244 . then @xmath272 is defined and analytic for @xmath276 and for @xmath277 . it is known that there are no poles on the real line @xcite . the stokes multipliers corresponding to the hastings - mcleod solution are @xmath278 thus @xmath272 is analytic across the imaginary axis . we reverse the orientation of @xmath279 and @xmath280 , and we define @xmath281 then @xmath272 solves the following rh problem , see also figure [ fig : jumpspsi ] .    * @xmath242 is analytic on @xmath282 , * @xmath283 on @xmath284 , + @xmath285 * @xmath286 on @xmath287 , * @xmath288 as @xmath251 , * @xmath242 is bounded near @xmath115 . the rh problem has a solution if and only if @xmath277 . the properties ( c ) and ( d ) are valid uniformly for @xmath46 in compact subsets of @xmath289 . ( 100,50)(0,0 ) ( 17,2)@xmath290 ( 17,46)@xmath291 ( 64,2)@xmath290 ( 67,46)@xmath291 ( 0,0)(2,1)33.1 ( 33.33,16.665)(2,1)36 ( 69.5,34.72)(2,1)31 ( 0,50)(2,-1)33.1 ( 33.33,33.33)(2,-1)36 ( 69.5,15.275)(2,-1)31      we modify the rh problem for @xmath242 so that it resembles the rh problem that we will need locally near @xmath26 . we introduce an additional parameter @xmath292 and define @xmath293 for any given @xmath277 and @xmath294 , we then have that @xmath2 is defined for @xmath295 , see figure [ fig : jumpsm ] , and satisfies the following rh problem    * @xmath296 is analytic for @xmath297 and @xmath277 . * @xmath298 for @xmath299 , * @xmath300 for @xmath301 , * @xmath302 for @xmath303 . * @xmath304 as @xmath305 in the upper half plane , + @xmath306 * @xmath307 as @xmath70 in the lower half plane , + @xmath308 * @xmath309 is bounded for @xmath61 near @xmath115 .    the properties ( c1 ) , ( c2 ) , and ( d ) hold uniformly for @xmath46 in compact subsets of @xmath310 and for @xmath292 . ( 140,50)(-20,0 ) ( 30,6)@xmath311 ( 31.6,42)@xmath312 ( -19,23.6)@xmath313 ( 101,23.6)@xmath313 ( 0,0)(2,1)33.1 ( 30,8)(-2,1)6 ( 70.5,8)(2,1)6 ( 29.8,43)(-2,-1)7(70.7,43)(2,-1)7 ( 0,0)(2,1)33.1 ( 33.33,16.665)(2,1)36 ( 69.5,34.72)(2,1)31 ( 0,50)(2,-1)33.1 ( 33.33,33.33)(2,-1)36 ( 69.5,15.275)(2,-1)31 ( 0,25)(1,0)33.1 ( 33.33,25)(1,0)36 ( 69.5,25)(1,0)31 the proof of theorem [ theorem-1-maintheorem ] is based on the steepest descent analysis of the riemann - hilbert problem for orthogonal polynomials .    since the main point of the present discussion is the treatment of the critical point @xmath26 , we will restrict ourselves to the one - interval case . we also assume that there are no other singular points besides @xmath26 . thus @xmath314 $ ] and @xmath14 vanishes like a square root at @xmath315 and @xmath316 , and quadratically at @xmath26 but at no other points of @xmath31 . in addition , we assume that the inequality in the variational condition ( [ variational - mut2 ] ) is strict for @xmath317 $ ] . we then have that @xmath318\ ] ] for certain @xmath319 for every @xmath37 close to @xmath320 . note that @xmath321 is an increasing and @xmath322 a decreasing function of @xmath37 . we will comment below on the modifications that have to be made in the multi - interval case , see remark [ remark-5-vereenvoudiging ] .      for each @xmath38 and @xmath323 , we consider a riemann - hilbert problem , introduced by fokas , its and kitaev @xcite . we will look for a @xmath243 matrix - valued function @xmath324 ( we drop the subscripts for simplicity ) that satisfies the following conditions :    * @xmath325 is analytic in @xmath326 , * @xmath327 for @xmath328 , + @xmath329 * @xmath330 as @xmath229 . here @xmath331 ( resp . @xmath332 ) denotes the limit as we approach @xmath333 from the upper ( resp . lower ) half - plane . the rh problem possesses a unique solution which is given by @xmath334    -2\pi i \kappa_{n-1,n } p_{n-1,n}(z )    & { \displaystyle}-\kappa_{n-1,n } \int_{\mathbb r}\frac{p_{n-1,n}(s ) e^{- n v(s)}}{s - z}ds \end{pmatrix } \label{3-fik}\ ] ] where @xmath335 denotes the orthonormal polynomial as before , see also @xcite . the correlation kernel ( [ 1-kern ] ) can be expressed directly in terms of the solution of the rh problem . indeed , by the christoffel - darboux formula for orthogonal polynomials @xmath336 which involves the orthogonal polynomials of degrees @xmath38 and @xmath95 only . using ( [ 3-fik ] ) and the fact that @xmath337 for every @xmath338 , we then get @xmath339 applying the deift - zhou steepest - descent method to the rh problem will allow us to find the asymptotics of @xmath81 as given in theorem [ theorem-1-maintheorem ] . we note that an expression like ( [ 3-knny ] ) was first given in the context of random matrices with external source @xcite where the eigenvalue correlation kernel is given in terms of the solution of a @xmath340 matrix valued rh problem , and the expression similar to ( [ 3-knny ] ) was found to be very convenient for asymptotic analysis . also in the present @xmath243-case we find it helpful to work with ( [ 3-knny ] ) .      in the first transformation we normalize the rh problem at infinity . a standard approach would be to use the equilibrium measure @xmath341 in external field @xmath155 where @xmath342 . if we would do this for the case where @xmath343 , we would have an equilibrium measure with a gap in the support around @xmath26 , and an annoying consequence is that the equality in the variational conditions is not valid near @xmath26 . for this reason we have introduced the signed measures @xmath139 in section 2 and we will use these measures now to normalize the rh problem . we let @xmath342 and assume @xmath37 is sufficiently close to @xmath127 so that ( [ eq : st - interval ] ) holds . as before we use @xmath344 to denote the density of @xmath139 . then we define the @xmath345-function @xmath346 where we take the branch cut of the logarithm along the negative real axis . then the following properties of @xmath347 are easy to check . * @xmath348 is analytic in @xmath349 $ ] , * @xmath350 as @xmath351 . * @xmath352 for @xmath353 , * @xmath354 for @xmath355 $ ] , * @xmath356 for @xmath357 $ ] .    in terms of the analytic function @xmath193 , see ( [ qt ] ) and proposition [ prop2 - 1 ] , we have @xmath358,\ ] ] and @xmath359 @xmath360 where @xmath361 is analytic for @xmath362 $ ] and that square root is taken which is positive for large real @xmath46 . we define @xmath363 and @xmath364 which are defined and analytic in the neighborhood @xmath177 of the real line where @xmath17 is analytic with cuts along @xmath365 and @xmath366 , respectively . note that @xmath193 has simple zeros in @xmath321 and @xmath322 and only double zeros in @xmath367 $ ] , so that @xmath368 is indeed analytic there . it is possible that @xmath193 has double real zeros , and that @xmath368 has sign changes in @xmath369 or @xmath370 . however , by ( [ gt - prop2 ] ) and ( [ gt - prop3 ] ) we have that @xmath371 for @xmath372 and @xmath373 for @xmath374 , since we are in a situation with strict inequality in ( [ variational - nut2 ] ) for @xmath372 and @xmath374 . we will now perform the first transformation of the rh problem : we take @xmath342 and define @xmath375 using the jump condition ( [ rhpyb ] ) of @xmath325 and ( [ 3-ty ] ) , we easily check that @xmath376 for @xmath377 , where @xmath378 because of the properties ( [ gt - prop1])([gt - prop3 ] ) and the definitions ( [ def - phi])([def - varphi ] ) , we see that the jump matrix @xmath379 has the following forms on the respective intervals @xmath380 , @xmath370 , and @xmath381 , @xmath382 thus @xmath383 is the unique solution of the rh problem    * @xmath383 is analytic in @xmath384 , * @xmath385for @xmath386 , * @xmath387 .      the jump matrix @xmath379 on the interval @xmath388 has the factorization @xmath389        now we open lenses around the intervals @xmath390 and @xmath391 as shown in figure [ figure - lens ] . let @xmath392 be the upper lips of the lenses and @xmath393 the lower lips , with orientation as in figure [ figure - lens ] . we open the lenses in such a way that they are fully contained in @xmath177 , the region of analyticity of @xmath17 and @xmath193 . in addition , we can take @xmath392 and @xmath393 in such a way that @xmath394 on @xmath392 and @xmath393 , with the exception of a neighborhood of @xmath26 if @xmath131 . this effect is due to the fact that @xmath139 has a negative density near @xmath26 if @xmath131 . however , if @xmath131 increases to one , the exceptional neighborhood shrinks to a point . it follows that for any given @xmath395 , there is a constant @xmath396 such that for @xmath37 sufficiently close to @xmath127 we have @xmath397 for all @xmath398 with @xmath399 . we define @xmath400 the rh problem for @xmath383 and the factorization ( [ 3-factorisatie1 ] ) imply that @xmath84 solves the following rh problem :    * @xmath84 is analytic in @xmath401 * @xmath402 for @xmath403 , + @xmath404 * @xmath405 as @xmath229 , * @xmath84 remains bounded near @xmath321 , @xmath322 , and @xmath26 ,    where @xmath406 is given by @xmath407 \begin{pmatrix }    1 & e^{-2n \tilde{\varphi}_t } \\    0 & 1 \end{pmatrix } & \textrm { on } ( -\infty , a_t ) , \\[3ex ] \begin{pmatrix }    0 & 1 \\    -1 & 0 \\ \end{pmatrix } & \textrm { on } ( a_t , b_t),\\[3ex ] \begin{pmatrix }    1 & 0 \\    e^{2n \varphi_t(z ) } & 1 \\ \end{pmatrix } & \textrm { on } c_1 \cup c_2 , \end{array } \end{cases}\label{3-sprongvs}\ ] ]    the jump matrices on @xmath408 and on @xmath409 tend to the identity matrix as @xmath40 and @xmath410 . ignoring these jumps , we find the parametrix for the outside region . uniform convergence breaks down in neighborhoods of @xmath321 , @xmath322 , and @xmath26 , so that we will also need to construct local parametrices near those points . the outside parametrix @xmath411 solves the following rh problem    * @xmath412 is analytic in @xmath413 $ ] , * @xmath414 on @xmath380 , * @xmath415 as @xmath229 . as in @xcite it has the solution @xmath416,\ ] ] where @xmath417.\ ] ] note that @xmath418 depends on @xmath37 .      at edge points @xmath321 and @xmath322 the density @xmath144 vanishes like a square root . this allows the construction of local parametrices near @xmath321 and @xmath322 with the use of airy functions . we will not give the details , see @xcite .      in a neighborhood @xmath419 of @xmath26 , we want to construct a parametrix @xmath420 with the following properties    * @xmath420 is analytic in @xmath421 , * @xmath422 on @xmath423 , * @xmath424 as @xmath425 , @xmath410 , uniformly for @xmath426 , * @xmath427 remains bounded for @xmath199 near @xmath26 . we seek a parametrix @xmath420 near @xmath26 in the form @xmath428 where @xmath429 is analytic in @xmath430 , @xmath431 is a conformal map from @xmath430 to a neighborhood of @xmath115 , @xmath432 is analytic in @xmath430 , and @xmath433 is a real constant . recall that @xmath2 is given by ( [ def - m ] ) .    in view of the jump properties of @xmath2 , we seek @xmath431 , @xmath432 , and @xmath433 so that for @xmath434 , @xmath435 since @xmath436 , the right - hand side of ( [ mappings1 ] ) does indeed define an analytic function in @xmath430 . the conformal map @xmath431 will not depend on @xmath37 . recall that @xmath437 has a zero of order four at @xmath26 , and that it is negative on @xmath438 . thus @xmath439 is analytic in @xmath430 and has a zero of order three at @xmath26 . we may assume there are no other zeros in @xmath430 . then we can take a third root and define @xmath440^{1/3 } ,      \qquad z \in \delta.\ ] ] then @xmath431 is analytic in @xmath430 with @xmath441 where @xmath107 is given by ( [ main2 ] ) . taking smaller @xmath430 if necessary , we then have that @xmath431 is indeed a conformal map on @xmath430 . note that @xmath442 and that @xmath431 is real and positive on @xmath443 and real and negative on @xmath444 . we still have some freedom in opening the lenses . we take @xmath392 and @xmath393 in @xmath430 so that @xmath431 maps them to the rays where @xmath2 has its jumps . that is , @xmath445 is mapped into @xmath446 and @xmath447 , and @xmath448 is mapped into @xmath449 and @xmath450 . having @xmath431 with @xmath442 , we take @xmath240 in ( [ mappings1 ] ) and we see that we should take @xmath451 then @xmath433 is real and it is also given by @xmath452 and so @xmath453    having @xmath431 and @xmath433 we finally take @xmath454 so that ( [ mappings1 ] ) holds , that is , @xmath455 where for the last line we used ( [ def - fz ] ) and ( [ def - theta2 ] ) . since the right - hand side of ( [ def - st ] ) is analytic in @xmath430 and vanishes for @xmath240 we can divide by @xmath456 ( which has a simple zero at @xmath457 ) and obtain an analytic function @xmath432 in @xmath430 . then @xmath454 is real for real @xmath199 , and @xmath458 . so the above construction yields analytic functions @xmath431 , @xmath432 and a constant @xmath433 so that ( [ mappings1 ] ) holds . then we can define the parametrix @xmath420 by ( [ 5-constructiep ] ) , provided that @xmath459 for @xmath434 stays away from the poles of the hastings - mcleod solution of painlev ii , since for @xmath460 we have that @xmath296 is not defined . for given values of @xmath38 and @xmath37 , we can take @xmath430 small enough so that this is indeed the case . however , we want to let @xmath40 , @xmath410 so that @xmath461 , and work with a neighborhood @xmath430 that is independent of @xmath38 and @xmath37 , although it may depend on @xmath462 .    note that by ( [ def - st ] ) and ( [ qt3 ] ) @xmath463 because of ( [ diff - tpsi ] ) we have @xmath464 as @xmath410 , uniformly for @xmath465 in a neighborhood of @xmath26 . using this in ( [ def - st2 ] ) we get that @xmath466 where @xmath467 is uniformly in @xmath199 as @xmath410 , and @xmath468 is uniformly in @xmath37 as @xmath469 . by ( [ f - near - xstar ] ) and ( [ def - st3 ] ) we then also have @xmath470 where again @xmath467 is uniformly in @xmath199 as @xmath410 , and @xmath468 is uniformly in @xmath37 as @xmath469 . then if @xmath471 we get @xmath472 where we used the definition ( [ main3 ] ) of the constant @xmath46 . since there are no poles on the real line , we can find a neighborhood @xmath430 of @xmath26 such that @xmath473 for all @xmath434 if @xmath38 is large enough and @xmath461 . note that @xmath430 depends on @xmath462 , but not on @xmath38 and @xmath37 . then @xmath474 is well - defined for @xmath199 in a fixed neighborhood @xmath430 of @xmath26 . finally , we define @xmath429 in such a way that the matching condition at @xmath475 is satisfied . we do this by defining @xmath476 so that @xmath477 is analytic near @xmath26 by the jump property ( [ rhpsinftyb ] ) of @xmath478 . because of the asymptotic behavior of @xmath296 in ( [ rhpmc1 ] ) and ( [ rhpmc2 ] ) ( which is valid uniformly for @xmath46 away from the poles and for @xmath292 ) we have the matching condition @xmath479 uniformly for @xmath480 . this completes the construction of the parametrix @xmath420 in the neighborhood of @xmath26 . we set @xmath481 since @xmath84 and @xmath420 have the same jumps inside each of the disks , and @xmath84 and @xmath478 have the same jumps on @xmath482 $ ] , @xmath483 has jumps on a contour @xmath484 as shown in figure [ figure - contourr ] .     after the third and final transformation . ] @xmath483 solves the following rh problem :    * @xmath483 is analytic in @xmath485 * @xmath486 as @xmath487 , * @xmath488 as @xmath229 , * @xmath483 is bounded ,    for certain jump matrices @xmath489 , which as @xmath40 , @xmath410 so that @xmath461 , satisfy @xmath490    as in @xcite it now follows that @xmath491 uniformly for @xmath492 . now we are ready for the proof of theorem [ theorem-1-maintheorem ] . we start with the expression ( [ 3-knny ] ) for the kernel @xmath81 . after the transformation ( [ 3-ty ] ) , we find that for @xmath493 , @xmath494 using formula ( [ 3-st ] ) for @xmath84 in the upper parts of the lenses , we get for @xmath495 , @xmath496 assume that @xmath145 and @xmath465 are inside the disk @xmath430 around @xmath26 . then by ( [ 5-rsp ] ) , ( [ 5-constructiep ] ) , ( [ def - m ] ) , and ( [ mappings1 ] ) we get @xmath497 and so @xmath498 similarly @xmath499    now we fix @xmath500 and @xmath501 and take @xmath502 so that for @xmath38 large enough , @xmath145 and @xmath465 are inside the disk around @xmath26 , so that ( [ 6-kerns ] ) , ( [ sformula1 ] ) , and ( [ sformula2 ] ) hold . then it follows from ( [ f - near - xstar ] ) and ( [ xy - in - uv ] ) that @xmath503 from ( [ def - st5 ] ) and ( [ xy - in - uv ] ) we get @xmath504 as @xmath40 , @xmath410 such that @xmath461 . furthermore , from ( [ r - asymp ] ) , ( [ xy - in - uv ] ) , and the fact that @xmath483 is analytic near @xmath26 , we get @xmath505 from ( [ def - sinfty ] ) , ( [ defet ] ) , and ( [ xy - in - uv ] ) we easily get @xmath506 as @xmath40 . the constants implied by the @xmath507-symbols in ( [ r - limits ] ) and ( [ et - limits ] ) are independent of @xmath500 and @xmath501 , when @xmath500 and @xmath501 are restricted to a compact subset of @xmath7 . combining ( [ r - limits ] ) and ( [ et - limits ] ) we get that @xmath508 since @xmath433 is real .    then multiplying ( [ sformula2 ] ) and ( [ sformula1 ] ) and letting @xmath40 , @xmath410 such that @xmath471 , we get by using ( [ 6-kerns ] ) , ( [ xy - in - uv ] ) , ( [ f - limits ] ) , ( [ st - limits ] ) , and ( [ randet - limits ] ) that @xmath509 uniformly for @xmath500 and @xmath501 in compact subsets of @xmath7 .    by ( [ 1-asymptotiekphi ] ) , ( [ psij ] ) and ( [ def - psi ] ) we have that @xmath510 in view of the jump ( [ rhppsib1 ] ) satisfied by @xmath272 on @xmath284 we have that @xmath511 since @xmath512 , we also get ( after simple calculation ) @xmath513 note that @xmath514 includes the full real line , so that we can take @xmath515 in ( [ phi - in - psi1 ] ) and @xmath516 in ( [ phi - in - psi2 ] ) which we use in ( [ sformula3 ] ) to obtain ( [ main1 ] ) . this completes the proof of theorem [ theorem-1-maintheorem ] . [ remark-5-vereenvoudiging ] the above proof of theorem [ theorem-1-maintheorem ] was given under the assumption that @xmath31 consists of one interval , and that there are no other singular points except for @xmath26 . here singular point refers to the classification of @xcite according to which there are three types of non - regular behavior for a real - analytic external field @xmath17 , see also @xcite . the singular points of type i are points in @xmath517 where equality in the variational inequality ( [ variational - mut2 ] ) with @xmath205 holds . singular points of type ii are interior points of @xmath31 where the density @xmath14 vanishes , and singular points of type iii are edge points of @xmath31 where @xmath14 vanishes to higher order than a square root . here we indicate briefly the modifications that have to be made if these assumptions are not satisfied . if @xmath31 consists of more than one interval , then the main complication is that the construction of the outside parametrix @xmath418 is more complicated , since it uses @xmath518-functions as in ( * ? ? ? * lemma 4.3 ) . it should be noted that @xmath418 will also depend on @xmath38 . as a result it will follow that @xmath429 as defined in ( [ defet ] ) also depends on @xmath38 . however this will not effect the asymptotic behavior ( [ et - limits ] ) as @xmath40 , so that the above proof goes through .    in case there are other singular points , we have to construct special local parametrices around each of them . for our purposes , we only need to know the existence of those parametrices , which is established in ( * ? ? ? * section 5 ) . we thank maarten vanlessen for useful remarks . the authors are supported by fwo research projects g.0176.02 and g.0455.04 . the second author is also supported by k.u.leuven research grant ot/04/24 , by intas research network necca 03 - 51 - 6637 , by nato collaborative linkage grant pst.clg.979738 , by grant bfm2001 - 3878-c02 - 02 of the ministry of science and technology of spain and by the european science foundation program methods of integrable systems , geometry , applied mathematics ( misgam ) and the european network in geometry , mathematical physics and applications ( enigma ) .    50 a.i . aptekarev , p.m. bleher , and a.b.j . kuijlaars , large @xmath38 limit of gaussian random matrices with external source , part ii , preprint arxiv : math - ph/0408041 . g. akemann , p.h . damgaard , u. magnea , and s.m . nishigaki , multicritical microscopic spectral correlators of hermitian and complex matrices , nuclear phys . b 519 ( 1998 ) , 682714 . j. baik , p. deift , and k. johansson , on the distribution of the length of the longest increasing subsequence of random permutations , j. amer . soc . 12 ( 1999 ) , 11191178 . j. baik , p. deift , and k. johansson , on the distribution of the length of the second row of a young diagram under plancherel measure , geom . 10 ( 2000 ) , 702731 . j. baik , p. deift , and e. rains , a fredholm determinant identity and the convergence of moments for random young tableaux , comm . math . 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we study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support . 
 we establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit . 
 the limiting kernels are constructed out of functions associated with the second painlev equation . 
 this extends a result of bleher and its for the special case of a critical quartic potential . 
 the two main tools we use are equilibrium measures and riemann - hilbert problems . in our treatment of equilibrium measures 
 we allow a negative density near the critical point , which enables us to treat all cases simultaneously . 
 the asymptotic analysis of the riemann - hilbert problem is done with the deift / zhou steepest descent analysis . for the construction of a local parametrix at the critical point 
 we introduce a modification of the approach of baik , deift , and johansson so that we are able to satisfy the required jump properties exactly .