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ising and dimer models were among the first models in two - dimensional statistical mechanics for where the partition function and for some of the correlation functions were computed explicitly in terms of pfaffians of certain matrices . for this reason both of these models can be regarded as theories of gaussian discrete two dimensional fermi field . the ising model was solved by onsager and for dimer models the pfaffian solution was found by kasteleyn .    the @xmath0-vertex model , a particular case of which is the ice model is interesting for a number of reasons . physically , it is a model of ferro- and antiferro- electricity . it has many equivalent reformulations , one of them ( which we will use ) describe the 6-vertex configurations on a planar connected simply connected region in terms of stepped surfaces . one of the combinatorial reformulations of the 6-vertex model for specific value of parameters @xcite is related to alternating sign matrices @xcite . the 6-vertex model generalizes dimer models and can be regraded as the theory of gaussian discrete fermions with four fermionic interaction . the partition function of the 6-vertex model with periodic boundary conditions was computed in @xcite using the bethe ansatz method . the computation of correlation functions in the @xmath0-vertex model is highly non - trivial and to the large degree is still a challenge . there are two known approaches to this problem based on the internal symmetry of the model , i.e. on the representation theory of quantum affine algebras . first approach is based on determinantal formulae for certain matrix elements @xcite . for some recent results based on this method see @xcite and @xcite . the second approach is based on form - factor formulae derived in @xcite . for an overview of this approach see @xcite and for the latest results see @xcite .    the 6-vertex model on a planar simply regions can be reformulated as the theory of random stepped surfaces . a configuration of arrows in a 6-vertex model can be interpreted as a configuration of paths which can be viewed as level corves of a _ height function _ defining a stepped surface . gibbs measure on 6-vertex configurations define a gibbs measure on stepped surfaces .    for certain class of such gibbs measures , random surfaces in the thermodynamical limit develop the limit shape phenomenon @xcite also known as the acric circle phenomenon @xcite . it means that on macroscopical scale the random surface becomes deterministic . the fluctuations remain at smaller scale and the structure of fluctuations may change depending on how singular the limit shape is at this point . the limit shape phenomenon is studied in details in dimer models @xcite . the numerical results from @xcite@xcite show how limit shapes develop in a 6-vertex model with domain wall boundary conditions . in this paper we will focus on the limit shape phenomenon for the 6-vertex model on a planar connected simply connected regions with fixed boundary conditions . computing the limit shape involves two steps . step one is the derivation of the formula for the free energy of the 6-vertex model as a function of magnetic fields . unlike the partition function of dimer models , the free energy can not be computed explicitly . however , it can be written in terms of the solution to a linear integral equation . many analytical properties of the partition function are known but scattered in the literature , see for example @xcite@xcite@xcite@xcite@xcite @xcite@xcite@xcite . we collected most of them in the section [ f - energy ] together with some new results . step two is to derive and solve the variational principle which determine the limit shape for given boundary conditions . the free energy of the model as a function of magnetic fields determine the functional in the variational problem . such variational problem was first introduced for dimer models in @xcite . the idea of using this variational problem in the 6-vertex first appeared in @xcite where some interesting partial results were obtained for correlation functions in the bulk of the limit shape . the structure of fluctuations near the limit shape is determined by the asymptotical behavior of correlation functions at smaller scales . we will discuss this problem for the 6-vertex model in the last section . finally let us mention the special case of domain wall boundary conditions . these boundary conditions first appeared in the computation of norms of bethe vectors @xcite . the remarkable fact about them is that the partition function of the 6-vertex model with these boundary conditions can be written as a determinant @xcite . another remarkable fact is that exactly these boundary conditions relate the 6-vertex model with alternating sign matrices @xcite . the large volume asymptotic of the partition function of the 6-vertex mode with these boundary conditions was computed in @xcite@xcite . here is the outline of the paper . in the first two sections we recall some basic facts about the @xmath0-vertex model , about its reformulation it in terms of height functions , and about the thermodynamical limit in the model . the third section contains the description of the free energy per site for as the function of electric fields in thermodynamical limit for periodic boundary conditions . some asymptotical behaviors of the free energy are computed in section 4 . this section is a combination of an overview and original results . in section 5 we study the asymptotical behavior of the limit shapes near the `` freezing point '' . in the last section we discuss fluctuations . we thank c. evans , r. kenyon , a. okounkov , and s. sheffield for interesting discussions . the work of n.r . was supported by the nsf grant dms 0307599 , by the niels bohr initiative at aarhus university , by the humboldt foundation and by the crdf grant rumi1 - 2622 the work of k.p . was supported by the nsf rtg grant and dms 0307599 . first , let us fix the notation . a square @xmath1 grid @xmath2 is a graph with @xmath3- and @xmath4-valent vertices embedded into @xmath5 such that @xmath3-valent vertices are located at points @xmath6 ( see fig . [ dwbc_conf ] ) and @xmath4-valent vertices are located at @xmath7 and at @xmath8 . an edge connecting two @xmath3-valent vertices is called an _ inner _ edge and an edge connecting a @xmath3-valent vertex with a @xmath4-valent vertex is called an _ outer _ edge . states of the @xmath0-vertex model on @xmath2 are configurations of arrows assigned to each edge ( i.e. orientations of @xmath2 ) . they satisfy the ice rule : at any vertex the number of incoming arrows should be equal to the number of outgoing arrows . six possible configurations at a vertex are shown on fig . [ vertices ] . configurations of arrows on boundary edges are called boundary conditions . each configuration of arrows on the lattice can be equivalently described as the configuration of `` thin '' and `` thick '' edges or empty and occupied edges shown on fig . [ vertices ] . there should be an even number of thick edges at each vertex as a consequence of the ice rule . the thick edges form paths . we assume that paths do not intersect ( two paths may meet at an @xmath9-vertex ) . so , equivalently , configurations of the @xmath0-vertex model can be regarded as configurations of paths satisfying the rules from fig . [ vertices ] .    to each configuration of arrows on edges adjacent to a vertex we assign a boltzmann weight , which we denote by the same letters . the physical meaning of a boltzmann weight is @xmath10 , where @xmath11 is the energy of a state and @xmath12 is the temperature ( in the appropriate units ) . thus , all numbers @xmath9 , @xmath13 , @xmath14 , @xmath15 , @xmath16 , and @xmath17 should be positive . choosing the scale such that @xmath18 , it is natural to write boltzmann weights in the exponential form . @xmath19 where @xmath20 , @xmath21 , and @xmath22 are dimensionless interaction energies of arrows at different types of vertices , and @xmath23 and @xmath24 are dimensionless horizontal and vertical components of the magnetic field , respectively . in this interpretation arrows are spins interacting with the magnetic field . we set @xmath25 because for the types of boundary conditions we will consider the difference between the number of @xmath16-vertices and @xmath17 vertices is the same for all states and , therefore , the probability does not depend on the ratio @xmath26 . we also use the standard notation @xmath27 these are the weights of the model when there is no magnetic field .    the weight of a state is the product of weights of vertices in the state . the weight of a state on @xmath28 ( up to a constant factor ) can be written in terms of energies and magnetic fields as @xmath29 where @xmath30 is the total number of @xmath31-vertices , @xmath32 is the total number of @xmath33-vertices , @xmath34 is the total number of @xmath35-vertices , @xmath36 is the total number of horizontal edges occupied by paths , and @xmath37 is the total number of vertical edges occupied by paths . the partition function is the sum of weights of all states of the model @xmath38 where @xmath39 is one of the weights from fig . [ vertices ] . weights define the probabilistic measure on the set of states of the @xmath0-vertex model . the probability of a state is given by the ratio of the weight of the state to the partition function of the model @xmath40 this is the gibbs measure of the @xmath0-vertex model . types of vertices and the corresponding thin and thick edges configurations ]    let us define the characteristic function of an edge @xmath41 as @xmath42    a local correlation function is the expectation value of the product of such characteristic functions : @xmath43      [ [ section ] ]    let us fix arrows ( or , equivalently , think edges ) on outer edges of @xmath2 . states in the 6-vertex with the same configurations of arrows on the boundary are called states with _ fixed _ boundary conditions . the difference between two such states can occur only at inner edges .    the space of states with fixed boundary conditions is empty unless the boundary values satisfy the ice rule : the total number of incoming arrows on the boundary edges should be equal to the total number of outgoing arrows . in the path formulation this means that the number of paths through north and west boundaries should be equal to the number paths through the south and east boundaries . an example of such boundary conditions is the domain wall ( dw ) boundary conditions . for the dw boundary conditions the arrows on the boundary of the lattice are going into the lattice at the top and bottom of the lattice and are going out of the lattice at the right and left of it . a configuration of paths on a @xmath44 lattice with dw boundary conditions is presented on fig . [ dwbc_conf ] . notice that the differences @xmath45 are the same for all configurations with given fixed boundary conditions . here @xmath46 is the total number of vertices of type @xmath47 in the configuration . in particular , the partition function for fixed boundary conditions trivially depends on magnetic fields : @xmath48     square grid for the dw boundary conditions ]    in this paper we will focus on the @xmath0-vertex model with fixed boundary conditions . [ [ section-1 ] ]    another important type of boundary conditions are the _ periodic boundary conditions_. in this case the edges at opposite sides of @xmath2 are identified so that the configuration of arrows on the left and right boundary is the same as well as the configuration of arrows on the top and bottom boundary . the @xmath0-vertex model with periodic boundary conditions is an example of an `` integrable '' ( solvable ) model in statistical mechanics and has been studied extensively , see @xcite,@xcite and references therein . in particular , it means that the row - to - row transfer - matrix of the model can be diagonalized by the bethe ansatz .      by _ outer faces _ we mean unit squares centered at @xmath49 , @xmath50 , with @xmath51 and @xmath52 , @xmath53 , with @xmath54 . each corner outer face has two edges in their boundary , other outer faces have three edges in their boundary . a height function @xmath55 is an integer - valued function on the faces @xmath56 of the grid @xmath28 ( including the outer faces ) , which is    * zero at the southwest corner of the grid , * non - decreasing when going up or right , * if @xmath57 and @xmath58 are neighboring faces , then @xmath59 . the _ boundary value _ of the height function is its restriction to the `` outer faces '' . denote the set of outer faces by @xmath60 . given a function @xmath61 on @xmath60 denote @xmath62 the space of all height functions with the boundary value @xmath61 . if we enumerate faces by the coordinates of their centers the height function can be regarded as @xmath63 matrix with non - negative entries . it is clear that there is a bijection between states of the @xmath0-vertex model with fixed boundary conditions and height functions with fixed boundary values . indeed , given a height function consider its `` level curves '' , i.e. paths on the grid @xmath28 , where the height function changes its value by @xmath4 , see fig . [ hf_example ] . clearly , this defines a state for the @xmath0-vertex model on @xmath28 with boundary conditions determined by the boundary values of the height function .    on the other hand , given a state in the @xmath0-vertex model , consider the corresponding configuration of paths . it is clear that there is a unique height function whose level curves are these paths and which satisfies the condition @xmath64 at the southwest corner . it is clear that this correspondence is a bijection . there is a natural partial order on the set of height functions with given boundary values . one function is bigger then the other if it is entirely above the other . there exist the minimum @xmath65 and the maximum @xmath66 height functions such that @xmath67 for all height functions @xmath55 . thus , we can consider the @xmath0-vertex model as a theory of fluctuating discrete surfaces constrained between @xmath66 and @xmath65 . each surface occurs with probability given by the boltzmann weights of the @xmath0-vertex model .          in the inhomogeneous @xmath0-vertex model the boltzmann weights depend on the edge . thus , we have @xmath68 parameters @xmath69 , @xmath70 , and @xmath71 . let us assume that the inhomogeneity is only in magnetic fields , i.e. weights @xmath31,@xmath33 , and @xmath35 do not change from edge to edge , but the magnetic fields @xmath72 , @xmath73 do . let @xmath74 be the collection of paths corresponding to a state in the @xmath0-vertex model and @xmath75 be the corresponding height function . let us assign weights @xmath76 to the edges of the lattice and @xmath4 to the outer edges , then @xmath77 where @xmath78 for inner faces , @xmath79 for the outer face @xmath80 adjacent to the edge @xmath41 , @xmath81 for outer faces not adjacent to any edge of the lattice ( corner faces ) with @xmath82 for edges at the upper and right sides of the boundary , @xmath83 for edges at the lower and left sides of the boundary . the proof is an elementary exercise . let @xmath84 for horizontal edges and @xmath85 for vertical edges . then the probability of the state with the height function @xmath55 in such a model is @xmath86    if @xmath87 and @xmath88 , the weights @xmath89 are the same for all faces @xmath90 inside the lattice and the probability is given by @xmath91 where @xmath92 are the boltzmann weights with constant magnetic fields and @xmath93 is the volume `` under '' the height function @xmath55 . [ [ section-2 ] ]    let @xmath94 . we place the grid @xmath2 inside of the rectangle @xmath95 so that the vertices of the grid are the points with coordinates @xmath96 where @xmath97 . we recall that a height function is a monotonic integer - valued function on the faces of the grid , which satisfies lipshitz condition ( it changes at most by @xmath4 on any two adjacent faces ) . the height function can be regarded as a function on the centers of the faces of the grid , i.e on @xmath98 , where @xmath99 . the points @xmath100 , @xmath101 , @xmath102 , and @xmath103 , where @xmath104 correspond to the `` outer '' faces of @xmath2 . we introduce the normalized height function as a piecewise linear function on the unit square with the value @xmath105 for @xmath106 and @xmath107 . here @xmath108 is a height function on @xmath2 . normalized height functions are nondecreasing in @xmath109 and @xmath110 directions and they satisfy : @xmath111 if @xmath112 and @xmath113 .    the boundary value of the normalized height function defines a piece - wise constant monotonic function on each side of the region which changes by @xmath114 or do not change between two neighboring boundary sites . denote the space of such normalized height functions with the boundary value @xmath115 by @xmath116 . there is a natural partial ordering on the set of all normalized height functions with given boundary values descending from the partial order on height functions : @xmath117 if @xmath118 for all @xmath119 . we define the operations @xmath120 it is clear that @xmath121 it is also clear that in this partial order there is unique minimal and unique maximal height functions , which we denote by @xmath122 and @xmath123 , respectively .    a _ sequence of stabilizing fixed boundary conditions _ is a sequence of functions @xmath124 which are boundary values of a normalized height function , and which converges as @xmath125 to a function @xmath126 on the boundary of @xmath127 which is non - decreasing along @xmath109 and @xmath110 direction , @xmath128 , and satisfies the condition :    @xmath129    we will call the function @xmath126 the boundary condition for the domain @xmath127 ( or simply the boundary condition ) . any function on @xmath127 which satisfies ( [ lip - cont ] ) , which is non - decreasing in @xmath109 and @xmath110 directions , and coinside with @xmath126 at the boundary is called a _ _ h__eight function on @xmath127 with boundary condition @xmath126 .    among all possible boundary conditions @xmath126 we will distinguish piecewise linear boundary conditions with the slope @xmath130 or @xmath4 along the coordinate axes . we will call these boundary values * critical*. it is clear that any boundary conditions can be approximated critical boundary conditions . if @xmath131 is fixed and is not equal to @xmath4 in the large volume limit , the system will be in a neighborhood of the minimal height function for @xmath132 and in the neighborhood of the maximal height function for @xmath133 . one should expect that the partition function and local correlation functions will have finite limit . when @xmath134 for some @xmath135 , one should expect the existence of the limit shape . we will study this limit in section [ section_var ] . the gibbs measure of the @xmath0-vertex model on an infinite lattice has the _ slope _ @xmath136 if @xmath137 and @xmath138    it is clear from the definition of the height function that the slope should satisfy conditions @xmath139 . the slope is simply the average number of horizontal and vertical edges occupied by paths per length . the important corollary of the result of @xcite is that , when a gradient gibbs measure satisfies certain convexity conditions , there exists a unique translationally invariant measure . this implies that for the @xmath0-vertex model one should expect the uniqueness of such a measure for the generic slope . translationally invariant measures can be obtained by taking the thermodynamic limit of the 6-vertex model with magnetic fields on a torus . then the slope is the legendre conjugate to magnetic fields . @xmath140 for a horizontal @xmath41 and @xmath141 for a vertical @xmath41 . the free energy per site in the thermodynamic limit is @xmath142 where @xmath143 is the partition function with the periodic boundary conditions on the rectangular grid @xmath2 . it is a function of the boltzmann weights and magnetic fields . normally , it is expected that the free energy is not identically zero . physically , this means that the `` excitations '' have the characteristic length which is much smaller than the characteristic length of the system . in some cases the free energy is identically zero , then one expects that a typical excitation will be comparable with the size of the system .    for generic @xmath23 and @xmath24 the @xmath0-vertex model in the thermodynamic limit has the translationally invariant gibbs measure with the slope @xmath136 : @xmath144    the parameter @xmath145 defines many characteristics of the @xmath0-vertex model in the thermodynamic limit . the weights @xmath31 , @xmath33 , and @xmath35 in this region satisfy one of the two inequalities , either @xmath147 or @xmath148 .    if @xmath147 , the boltzmann weights @xmath31 , @xmath33 , and @xmath35 can be parameterized as @xmath149 with @xmath150 . if @xmath151 , the boltzmann weights can be parameterized as @xmath152 with @xmath153 .    for both of these parametrization of weights @xmath154 . the phase diagram of the model for @xmath147 ( and , therefore , @xmath155 ) is shown on fig . [ ferro_diagram_1 ] and for @xmath148 ( and , therefore , @xmath156 ) on fig . [ ferro_diagram_2 ] . when magnetic fields @xmath157 are in one of the regions @xmath158 of the phase diagram , the system in the thermodynamic limit has the translationally invariant gibbs measure supported on the corresponding frozen configuration . there are four frozen configurations @xmath159 , @xmath160 , @xmath161 , and @xmath162 , shown on fig . [ ferroelectric_phase ] . for a finite but large grid the probability of any other state is at most of order @xmath163 for some positive @xmath164 . local correlation functions are given by the value of the corresponding observable on the frozen state @xmath165 where @xmath166 is the one of the ferromagnetic states @xmath167 . the frozen regions in the @xmath157-plane are described by the set of inequalities . the boundaries of these regions can be derived by analyzing the next to the largest eigenvalue of the row - to - row transfer matrix . the description is separated into two cases : @xmath147 and @xmath148 . notice that @xmath168 since @xmath146 . @xmath169 @xmath147 , see fig . [ ferro_diagram_1 ] , @xmath170    -plane for @xmath171 , @xmath172 , and @xmath173,width=302,height=302 ]    @xmath169 @xmath148 , see fig . [ ferro_diagram_2 ] , @xmath174    -plane for @xmath175 , @xmath176 , and @xmath173,width=302,height=302 ]    the free energy is a linear function in @xmath23 and @xmath24 in the four frozen regions : @xmath177    if @xmath157 is a region @xmath178 or @xmath179 the corresponding translationally invariant gibbs measure has the slope @xmath136 given by ( [ slope ] ) . in this phase the system is disordered , which means that local correlation functions decay as a power of the distance @xmath180 between @xmath181 and @xmath182 when @xmath183 .    in the regions @xmath178 and @xmath179 the free energy is given by @xcite : @xmath184 where @xmath185 can be found from the integral equation @xmath186 in which @xmath187 @xmath185 satisfies the following normalization condition : @xmath188 the contour of integration @xmath189 ( in the complex @xmath190-plane ) is symmetric with respect to the conjugation @xmath191 , is dependent on @xmath23 ( see appendix [ a_free_energy_hom ] ) and is defined by the condition that the form @xmath192 has purely imaginary values on the vectors tangent to @xmath189 : @xmath193    the formula ( [ free_energy ] ) for the free energy follows from the bethe ansatz diagonalization of the row - to - row transfer - matrix . its derivation is outlined in appendix [ a_free_energy_hom ] . it relies on a number of conjectures that are supported by numerical and analytical evidence and in physics are taken for granted . however , there is no rigorous proof . there are two points where three phases coexist ( two frozen and one disordered phase ) . these points are called _ tricritical_. the angle @xmath194 between the boundaries of @xmath178 ( or @xmath179 ) at a tricritical point is given by @xmath195 the existence of such points makes the @xmath0-vertex model ( and its degeneration known as the @xmath196-vertex model @xcite ) remarkably different from dimer models @xcite where generic singularities in the phase diagram are cusps . physically , the existence of singular points where two curves meet at the finite angle manifests the presence of interaction in the @xmath0-vertex model . notice that when @xmath197 the phase diagram of the model has a cusp at the point @xmath198 . this is the transitional point between the region @xmath146 and the region @xmath199 which is described below .      in this case , the boltzmann weights have a convenient parameterization by trigonometric functions . when @xmath200 @xmath201 where @xmath202 , @xmath203 , and @xmath204 .    when @xmath205 @xmath206 where @xmath202 , @xmath207 , and @xmath208 .    the phase diagram of the @xmath0-vertex model with @xmath199 is shown on fig . [ disordered_diagram ] . the phases @xmath158 are frozen and identical to the frozen phases for @xmath146 . the phase @xmath127 is disordered . for magnetic fields @xmath157 the gibbs measure is translationally invariant with the slope @xmath209 . -plane for @xmath175 , @xmath176 , and @xmath210,width=377,height=377 ]    the frozen phases can be described by the following inequalities : @xmath211    the free energy function in the frozen regions is still given by the formulae ( [ linear_energy ] ) . the first derivatives of the free energy are continuous at the boundary of frozen phases , the second derivative is continuous in the tangent direction at the boundary of frozen phases and is singular in the normal direction . it is smooth in the disordered region where it is given by ( [ free_energy ] ) which , as in case @xmath146 involves a solution to the integral equation ( [ int - eqn ] ) . the contour of integration in ( [ int - eqn ] ) is closed for zero magnetic fields and , therefore , the equation ( [ int - eqn ] ) can be solved explicitly by the fourier transformation @xcite . the @xmath0-vertex gibbs measure with zero magnetic fields converges in the thermodynamic limit to the superposition of translationally invariant gibbs measures with the slope @xmath212 . there are two such measures . they correspond to the double degeneracy of the largest eigenvalue of the row - to - row transfer - matrix @xcite . there is a very interesting relationship between the @xmath0-vertex model in zero magnetic fields and the highest weight representation theory of the corresponding quantum affine algebra . the double degeneracy of the gibbs measure with the slope @xmath212 corresponds to the fact that there are two integrable irreducible representations of @xmath213 at level one . correlation functions in this case can be computed using @xmath131-vertex operators @xcite . for latest developments see @xcite . the boltzmann weights for these values of @xmath215 can be conveniently parameterized as @xmath216 where @xmath217 and @xmath218 . the gibbs measure in thermodynamic limit depends on the value of magnetic fields . the phase diagram in this case is shown on fig . [ antiferroelectric_diagram ] for @xmath219 . in the parameterization ( [ hyp - param ] ) this correspond to @xmath220 . when @xmath221 the @xmath3-tentacled `` amoeba '' is tilted in the opposite direction as on fig . [ ferro_diagram_1 ] . when @xmath157 is in one of the @xmath167 regions in the phase diagram the gibbs measure is supported on the corresponding frozen configuration , see fig . [ ferroelectric_phase ] . -plane for @xmath175 , @xmath176 , and @xmath222,width=377,height=377 ]    the @xmath167 regions on the phase diagram are defined by inequalities ( [ boundaries_phases ] ) . the free energy in these regions is linear and is given by ( [ linear_energy ] ) . if @xmath157 is in the region @xmath127 , the gibbs measure is the translationally invariant measure with the slope @xmath136 determined by ( [ slope ] ) . the free energy in this case is determined by solutions to the linear integral equation ( [ int - eqn ] ) and is given by the formula ( [ free_energy ] ) . if @xmath157 is in the region @xmath223 , the gibbs measure is the superposition of two gibbs measures with the slope @xmath212 . in the limit @xmath224 these two measures degenerate to two measures supported on configurations @xmath225 , respectively , shown on fig . [ antiferr ] . for a finite @xmath215 the support of these measures consists of configurations which differ from @xmath226 and @xmath227 in finitely many places on the lattice . we notice that any two configurations lying in the support of each of these gibbs measures can be obtained from @xmath226 or @xmath227 via flipping the path at a vertex `` up '' or `` down '' as it is shown on fig . [ fluct ] finitely many times . it is also clear that it takes infinitely many flips to go from @xmath226 to @xmath227 . the @xmath0-vertex model in the phase @xmath223 is disordered and is also noncritical .    here non - criticality means that the local correlation function @xmath228 decays as @xmath229 with some positive @xmath164 as the distance @xmath180 between @xmath181 and @xmath182 increases to infinity .    the free energy in the @xmath223-region can be explicitly computed by solving the equation ( [ int - eqn ] ) via the fourier transform @xcite . this boundary of the antiferromagnetic region @xmath223 can be derived similarly to the boundaries of the ferromagnetic regions @xmath230 and @xmath231 by analyzing next to the largest eigenvalue of the row - to - row transfer - matrix . the difference is that for the region @xmath223 the largest eigenvalue will correspond to @xmath232 and to compute it we should use the solution to the integral equation ( [ int - eqn ] ) in the case when the contour of integration is closed . this computation was done in @xcite , @xcite . the result is a simple closed curve , which can be described parameterically as @xmath233 where @xmath234 @xmath235 and @xmath236 the parameter @xmath237 is defined by the equation @xmath238 , where @xmath239 the curve is invariant with respect to the reflections @xmath240 and @xmath241 since the function @xmath242 satisfies the identities @xmath243 this function is also @xmath244-periodic : @xmath245 . the boundary of the antiferromagnetic region is a real algebraic curve in @xmath246 and @xmath247 given by @xmath248 where @xmath249 is the positive value of @xmath24 on the curve when @xmath250 . notice that @xmath237 depends on the boltzmann weights @xmath251 only through @xmath252 . the parametric description of the boundary curve implies that @xmath253 the addition formula for the jacobi elliptic function @xmath254 @xcite @xmath255 can be used to express @xmath256 and @xmath257 for @xmath258 and @xmath259 in terms of @xmath260 and @xmath24 . using elementary identities for the elliptic functions , we obtain @xmath261 for @xmath258 and @xmath259 this identity turns into @xmath262 denote @xmath263 , then this identity becomes ( [ antiferroelectic_boundary ] ) . here we study the asymptotics of the free energy of the @xmath0-vertex model as the point @xmath157 inside a disordered region approaches its boundary on the phase diagram of the model . let us consider the interface between the disordered region @xmath127 ( @xmath179 for @xmath146 ) and the @xmath159-region , see fig . [ disordered_diagram ] . it is given by @xmath264 , where @xmath265    let @xmath266 be a regular point on the interface , i.e. the interface can be parameterized by real analytic functions in its neighborhood , then @xmath267 . we denote the normal vector to the interface at @xmath268 by @xmath269 and the tangent vector by @xmath270 . a point @xmath271 in the vicinity of @xmath272 can be represented as @xmath273 , where @xmath274 and @xmath275 are local coordinates in the normal and tangent directions , respectively , and @xmath276 is a scaling factor such that @xmath277 . let us choose the normal vector @xmath269 so that it points in the direction of the disordered region @xmath127 , then the point @xmath278 belongs to @xmath127 if @xmath279 . let @xmath278 be defined as above . the asymptotics of the free energy of the @xmath0-vertex model in the limit @xmath277 is given by @xmath280 where @xmath281 and @xmath282 here the constants @xmath283 and @xmath194 depend on the boltzmann weights of the model and on @xmath284 and are given by @xmath285 and @xmath286 where @xmath287 is defined in ( [ inter_d_f ] ) . moreover , @xmath288 and , therefore , @xmath289 . we refer the reader to @xcite for the details . assume that @xmath155 . the theorem below describes the asymptotic of the free energy function when @xmath290 and @xmath291 these values of @xmath157 describe points inside the right `` tentacle '' on the fig . [ ferro_diagram_1 ] . let us parameterize these values of @xmath24 as @xmath292 where @xmath293 $ ] . when @xmath294 and @xmath295 $ ] the asymptotic of the free energy is given by the following formula : @xmath296    from the integral equation for @xmath185 we can derive the large @xmath23 asymptotics of the density function : @xmath297 . the contour is a small deformation of the segment of the circle of radius @xmath298 centered at the origin with endpoints having arguments @xmath299 .    for large @xmath23 the free energy function is given by @xmath300    as @xmath294 the density is given by ( [ tent - dens ] ) and , taking into account the asymptotical description of the contour of integration we obtain @xmath301 where @xmath302 this integral is easy to compute : @xmath303 the minimum occur at @xmath304 or , at @xmath305    the formula ( [ tent - ass ] ) follows after the substitution of this into the expression for @xmath306 . the @xmath196-vertex model can be obtained as the limit of the @xmath0-vertex model when @xmath307 . magnetic fields in this limit behave as follows :    * @xmath147 . in the parameterization ( [ dg1par-1 ] ) after changing variables @xmath308 , and @xmath309 take the limit @xmath310 keeping @xmath135 fixed . the weights will converge ( up to a common factor ) to : @xmath311 * @xmath151 . in the parameterization ( [ dg1par-2 ] ) after changing variables @xmath308 , and @xmath312 take the limit @xmath310 keeping @xmath313 fixed . the weights will converge ( up to a common factor ) to : @xmath314    the two limits are related by inverting horizontal arrows . from now on we will focus on the 5-vertex model obtained by the limit from the 6-vertex one when @xmath147 . the phase diagram of the 5-vertex model is easier then the one for the 6-vertex model but still sufficiently interesting . perhaps the most interesting feature is that the existence of the tricritical point in the phase diagram . we will use the parameter @xmath315 notice that @xmath316 . the frozen regions on the phase diagram of the @xmath196-vertex model , denoted on fig . [ amoeba_5v ] as @xmath159 , @xmath160 , and @xmath161 , can be described by the following inequalities : @xmath317    -vertex model with @xmath318 ( @xmath319).,width=302,height=302 ]      the disordered region @xmath127 near the tricritical point forms a corner @xmath320 the angle @xmath194 between the boundaries of the disordered region at this point is given by @xmath321    one can argue that the finiteness of the angle @xmath194 manifests the presence of interaction in the model . in comparison , translationary invariant dimer models can only have cusps as such singularities . as it follows from results @xcite the limit from the 6-vertex model to the 5-vertex model commutes with the thermodynamical limit and for the free energy of the @xmath196-vertex model we can use the formula @xmath322 where @xmath323 is the free energy of the @xmath0-vertex model . let @xmath324 and @xmath325 the asymptotics of the free energy along this ray inside the corner near the tricritical point is given by @xmath326 where @xmath327 and @xmath328    the proof is computational . the details can be found in @xcite . the scaling along any ray inside the corner near the tricritical point in the 6-vertex model differ from this only by in coefficients . the exponent @xmath329 is the same . the details will be given in a separate publication . if @xmath331 , the region @xmath223 consists of one point located at the origin . when @xmath332 we have @xmath333 and @xmath334 . moreover , @xmath335 , @xmath336 , and @xmath337 . since @xmath338 , when @xmath339 , we have @xmath340 since @xmath341 is an odd function we obtain the following asymptotic of @xmath242 @xcite : @xmath342 in this limit the antiferromagnetic region degenerates into the origin @xmath250 and @xmath343 exponentially fast . we note that the point @xmath198 is special for @xmath199 . the following identity holds in the region @xmath127 @xcite : @xmath344    here @xmath345 . the constant @xmath346 does not vanish in the @xmath127-region including its boundary . it is determined by the solution to the integral equation for the density @xmath185(see appendix c ) .    directly from the definition of the free energy we have @xmath347 where @xmath348 and @xmath349 are the number of arrows pointing to the left and the number of arrows pointing to the right , respectively . therefore , the matrix @xmath350 of second derivatives with respect to @xmath23 and @xmath24 is positive definite .    as it follows from the asymptotical behavior of the free energy near the boundary of the @xmath127-phase , despite the fact that the hessian is nonzero and finite at the boundary of the interface , the second derivative of the free energy in the transversal direction at a generic point of the interface develops a singularity . the legendre transform of the free energy @xmath351 as a function of @xmath352 is defined for @xmath353 .    the variables @xmath109 and @xmath110 are known as polarizations and are related to the slope of the gibbs measure as @xmath354 and @xmath355 . we will write the legendre transform of the free energy as a function of @xmath136 @xmath356 @xmath357 is defined on @xmath358 .    for the periodic boundary conditions the surface tension function has the following symmetries : @xmath359 the last two equalities follow from the fact that if all arrows are reversed , @xmath360 is the same , but the signs of @xmath109 and @xmath110 are changed . it follows that @xmath361 and @xmath362 . the function @xmath306 is linear in the domains that correspond to conic and corner singularities of @xmath360 . outside of these domains ( in the disordered domain @xmath127 ) we have @xmath363 here we gradient of a function as a mapping @xmath364 .    when the 6-vertex model is formulated in terms of the height function , the legendre transform of the free energy can be regarded as a surface tension . the surface in this terminology is the graph of of the height function . [ [ section-3 ] ]    now let us describe some analytical properties of the function @xmath357 is obtained as the legendre transform of the free energy . the legendre transform maps the regions where the free energy is linear with the slope @xmath365 to the corners of the unit square @xmath366 . for example , the region @xmath159 is mapped to the corner @xmath367 and @xmath368 and the region @xmath161 is mapped to the corner @xmath367 and @xmath369 . the legendre transform maps the tentacles of the disordered region to the regions adjacent to the boundary of the unit square . for example , the tentacle between @xmath159 and @xmath161 frozen regions is mapped into a neighborhood of @xmath367 boundary of @xmath370 , i.e. @xmath371 and @xmath372 .    applying the legendre transform to asymptotics of the free energy in the tentacle between @xmath159 and @xmath161 frozen regions we get @xmath373 and @xmath374 here @xmath375 and @xmath372 . from ( [ sur_ten_boundary ] ) we see that @xmath376 , i.e. @xmath360 is linear on the boundary @xmath367 of @xmath370 . therefore , its asymptotics near the boundary @xmath367 is given by @xmath377 as @xmath375 and @xmath372 . we note that this expansion is valid when @xmath378 . similarly , considering other tentacles of the region @xmath127 , we conclude that the surface tension function is linear on the boundary of @xmath370 . [ [ section-4 ] ]    next let us find the asymptotics of @xmath360 at the corners of @xmath370 in the case when all points of the interfaces between frozen and disordered regions are regular , i.e. when @xmath379 . we use the asymptotics of the free energy near the interface between @xmath159 and @xmath127 regions ( [ asympt_free_energy_interface ] ) . first let us fix the point @xmath284 on the interface and the scaling factor @xmath276 in ( [ asympt_free_energy_interface ] ) . then from the legendre transform we get @xmath380 and @xmath381 it follows that @xmath382 in the vicinity of the boundary @xmath383 and , hence , @xmath384 as @xmath385 . thus , under the legendre transform , the slope of the line which approaches the corner @xmath386 depends on the boundary point on the interface between the frozen and disordered regions . it follows that the first terms of the asymptotics of @xmath360 at the corner @xmath386 are given by @xmath387 where @xmath388 and @xmath389 can be found from ( [ slope_corner ] ) and @xmath267 .    when @xmath390 the function @xmath360 is strictly convex and smooth for all @xmath391 . it develops conical singularities near the boundary .    when @xmath392 , in addition to the singularities on the boundary , @xmath360 has a conical singularity at the point @xmath212 . it corresponds to the `` central flat part '' of the free energy @xmath80 , see fig . [ antiferroelectric_diagram ] .    when @xmath393 the function @xmath360 has corner singularities along the boundary as in the other cases . in addition to this , it has a corner singularity along the diagonal @xmath394 if @xmath155 and @xmath395 if @xmath156 . we refer the reader to @xcite for further details on singularities of @xmath360 in the case when @xmath146 . [ [ section-5 ] ]    let @xmath360 be the surface tension function of the @xmath0-vertex model with the periodic boundary conditions defined in ( [ surface_tension ] ) . we consider the functional @xmath396=\int_d\sigma(\nabla \varphi)d^2x+\lambda\int_d \varphi d^2x,\ ] ]    let @xmath397 be a minimizer of this functional on the space @xmath398 of functions nondecreasing in @xmath109 and @xmath110 directions and satisfying the condition @xmath399 and the boundary condition @xmath400 notice that height functions are lipschitz with @xmath401 . the functional @xmath402 $ ] has unique minimizer . indeed , since @xmath360 is convex , the minimizer is unique when it exists . the existence of the minimizer follows from compactness of the space @xmath398 in the sup norm . the arguments are completely parallel to those in @xcite . [ [ section-6 ] ]    if the vector @xmath403 is not a singular point of @xmath360 , the minimizer @xmath55 satisfies the euler - lagrange equation in a neighborhood of @xmath352 @xmath404 we can also rewrite this equation in the form @xmath405 where @xmath406 is an unknown function such that @xmath407 . it is determined by the boundary conditions for @xmath55 .    applying ( [ f - sig - comp ] ) , we see that @xmath408    from the definition of the slope , see ( [ slope ] ) we have @xmath409 and @xmath410 . thus , if the minimizer @xmath55 is differentiable at @xmath352 , it satisfies the constrains @xmath411 and @xmath412 .      the following statement is a minor variation of the theorem 4.3 from @xcite . let @xmath125 , @xmath135 be finite and @xmath134 then the sequence of random normalized height functions @xmath413 converges in probability to the minimizer of ( [ var - pr ] ) . the rate of convergence is exponential of @xmath414 . the minimizer of the variational problem ( [ var - pr ] ) is called the _ limit shape _ of the height function . this theorem is the manifestation of the general philosophy of the large deviations principle . the probability of a having state with the height function @xmath55 has the has the following asymptotic @xmath125 : @xmath415 here @xmath360 is the surface tension function for the periodic boundary conditions . clearly this probability has maximum at the limit shape . states with the height function , which macroscopically differ from the limit shape , should be expected to be exponentially improbable . the theorem states that this is exactly what is taking place . the space of normalized height functions on @xmath2 has the partial order described in section 2 . denote the minimal and maximal normalized height functiions with respect to this partial order @xmath417 and @xmath418 , respectively .    for two functions @xmath419 , @xmath420 on @xmath127 define the distance @xmath421 similarly define the the distance between two functions on the boundary of @xmath127 . let @xmath124 be a sequence of functions on the boundary of @xmath127 converging to @xmath422 , and such that @xmath124 is a boundary of a normalized height function of @xmath2 . denote @xmath423 and @xmath424 the minimum and maximum height functions from @xmath425 . the following is clear :    let @xmath426 and @xmath427 be minimal and maximal height functions with the boundary condition @xmath422 . then , @xmath428 and @xmath429 with respect to the distance ( [ distance ] ) .    the functions @xmath426 and @xmath427 minimize the functionals @xmath430 in the sapec @xmath431 . let us assume that the boundary conditions are critical , that is @xmath422 is piece - wise linear , non - decreasing in @xmath109 and @xmath110 direction with the slope @xmath130 or @xmath4 . in this case the minimum and maximum height functions are piecewise linear functions , such that each linear part has the slope either @xmath130 or @xmath432 along coordinate axes . we will say a point @xmath352 is _ regular _ if at this point the function @xmath426 is differentiable . for critical boundary conditions regular points form regions with piece - wise linear boundary where the function @xmath426 has a constant slope . we will call them _ linear domains_.    points where the gradient of @xmath426 is discontinuous will be called _ singular _ points . for critical boundary conditions singular points are the points where several linear domains meet . the _ _ v__alency of a singular point is the number of linear domains which meet at this point . for generic critical boundary conditions the valency of each critical point is at most three . the list of all possible phases at a tricritical point of the @xmath0-vertex model with generic critical boundary conditions is given on fig . [ triple_pts ] .    -vertex model.,width=302,height=377 ]      here we study the asymptotic of minimizer of @xmath433 $ ] as @xmath434 . it is more convenient to divide @xmath435 by @xmath436 , so we are looking for the asymptotics of the minimizer @xmath437 of @xmath438=\frac{1}{|\lambda|}\int_d \sigma(\nabla\varphi)d^2x \pm \int_d \varphi d^2x.\ ] ]    let us focus on the limit @xmath439 . the limit @xmath440 can be treated similarly . when @xmath441 , the minimizer @xmath437 approaches the minimal height function @xmath426 described above . let us look for the asymptotical formula for the minimizer in a small neighborhood of a point @xmath442 of the form @xmath443 where @xmath444 as @xmath441 . let @xmath445 and @xmath446 . because @xmath426 is linear , its second derivatives vanish and we can rewrite the euler - lagrange equation ( [ e - l_eq ] ) as @xmath447    notice that for critical boundary conditions the function @xmath448 is piece - wise constant .    now assume that @xmath442 is a singular point , i.e. a point where two or more linear domains of @xmath426 meet . recall that for generic critical boundary conditions only two or three linear domains can meet at a point . taking the limit @xmath449 in ( [ el - h ] ) we obtain . let @xmath437 be the minimizer of @xmath450 and @xmath442 be a singular point , then for each @xmath451 there exits @xmath452 this function is the solution to @xmath453 with the boundary conditions @xmath454 as @xmath455 along a ray in the @xmath456-th linear domain adjacent to @xmath442 . here @xmath457 in the @xmath456-th linear domain and since the boundary conditions are critical , @xmath458 . the function @xmath23 is once differentiable . there is a smooth curve separating @xmath5 into regions where @xmath23 is linear and regions where @xmath23 is smooth with positive definite matrix of second derivatives .    for non - generic boundary conditions in the thermodynamical limit more then three linear domains can met at one point . in this case one should expect that the conjecture still holds with more then three linear domains meeting at a singular point . [ [ section-7 ] ]    a height function defines the surface @xmath459 in @xmath460 . regions where @xmath55 is linear are planes . here we will describe the solution to the equation ( [ h - eqn ] ) with the boundary conditions ( [ as - cond ] ) in the case when there are only two asymptotic planes meeting at@xmath442 . let @xmath461 and @xmath462 be the directions of the steepest assent of these planes . the numbers @xmath463 and @xmath464 are either @xmath130 or @xmath465 since we assume critical boundary conditions and therefore @xmath426 is piece - wise linear with slope @xmath466 or @xmath467 . the function @xmath468 , defined in ( [ asympt_fun ] ) , has the asymptotic conditions ( [ as - cond])determined by these planes . the function @xmath468 is also invariant with respect to translations in @xmath469-direction . thus , we are looking for a function @xmath283 such that @xmath470 which satisfies the differential equation ( [ h - eqn ] ) with the asymptotic conditions ( [ as - cond ] ) . let us introduce @xmath471 then the differential equation for @xmath23 becomes the first order ode for the function @xmath472 @xmath473 integrating it , we obtain the equation defining the function @xmath474 implicitly @xmath475 with some constant @xmath189 . [ [ section-8 ] ]    for the domain wall boundary conditions minimal and maximal height functions are shown on fig . [ dw_min_max ] . in this case the boundary between two linear domains is a line in the direction @xmath476 for the minimal height function and is the line in the direction @xmath467 for the maximal height function . thus , for these boundary conditions the function @xmath23 describing the asymptotic of the minimizer is constant in the direction @xmath476 when @xmath441 and it is constant in the @xmath467-direction if @xmath477 . let @xmath478 be the solution to ( [ eqn - for - kappa ] ) which is invariant with respect to translations in the @xmath476 direction . the symmetries of the legendre transform @xmath360 of the free energy imply that @xmath361 . using the equation ( [ implic_eq_g ] ) and this symmetry of @xmath360 , we obtain @xmath479 taking into account ( [ f - sig - comp ] ) we obtain @xmath480 and , hence , @xmath481    in the case of two asymptotic planes we have an additional symmetry of @xmath468 with respect to the intersection line of these planes , i.e. @xmath482 . the free energy @xmath80 also has the symmetry @xmath483 . therefore @xmath484 and we proved the following statement .    for dw boundary conditions the minimizer @xmath437 when @xmath485 has the following asymptotic when @xmath486 , @xmath487 : @xmath488    if @xmath486 and @xmath489 and @xmath490 the asymptotic of the minimizer @xmath437 is given by @xmath491    [ [ section-9 ] ]    when @xmath492 ( which is equivalent to @xmath493 ) the height function @xmath437 develops extra linear domains known as facets . these are the regions of the antiferroelectric phase discussed in section [ af - region ] . in dimer models this phenomenon is studied in @xcite . the facets also appear in the function @xmath23 describing the asymptotic of the minimizer as @xmath494 . for the function @xmath23 describing the asymptotic near a double singular point @xmath442 where two linear domains meet along the diagonal @xmath495 the facet is a strip @xmath496 in coordinates @xmath497 . its width @xmath498 is given by the formula @xmath499    in the limit @xmath333 or @xmath332 the asymptotic of @xmath242 gives the following asymptotical value of @xmath498 : @xmath500    the free energy @xmath80 is linear in the antiferromagnetic region . it is growing with exponent @xmath501 in the normal direction to the boundary outside of the antiferromagnetic region . this agrees with the pokrovsky - talapov law which states that @xmath80 should be growing with exponent @xmath501 in the normal direction to the boundary of the facet @xcite . in particular , @xmath55 grows with the exponent @xmath501 in the @xmath467-direction near the boundary of the @xmath35-droplet far enough from the boundary of the square . for @xmath502 we have @xmath503 where @xmath115 is the value of the height function at the boundary of the facet and @xmath283 is a constant which can be computed explicitly . as we have seen in the previous section at the macroscopical distances in the thermodynamical limit the height function is deterministic and is the minimizer for the variational problem ( [ var - pr ] ) . but the height function at smaller distances remain random . their fluctuations are described by the asymptotical behavior of correlation functions in the thermodynamical limit . these asymtotics have been studied extensively in dimer models which describe the @xmath504 case of the 6-vertex model . the thermodynamical limit of correlations functions in the bulk describes translationary invariant gibbs measure with given polarization . these asymptotics of correlation functions have been studied a lot using various methods which are essentially based on representation theory of affine quantum groups . the exact computation of the asymptotic of correlation functions for local observable separated by large distances on the lattice remain one of the main problems . on the other hand this is one of the most important physically relevant information about correlation functions . some information about this asymptotic of correlation functions can be obtained using the arguments of finite - size scaling and the assumption of conformal invariance of the leading terms of the asymptotic . let us consider the 6-vertex configurations of paths which may end at some edges . the weight of such configurations is given the same product of weights as before . define local observables @xmath505 and @xmath506 as @xmath507 @xmath508 the value of a product of such observable when each of the factors correspond to a different edge is the product of values of observables . the following formulae were obtained in @xcite for @xmath250 and when all edges are vertical on the same row using the finite - size scaling and the assumption of conformal invariance : @xmath509 @xmath510 here @xmath511 is the distance between @xmath512 and @xmath513 , @xmath514 . the terms denoted by @xmath515 are with higher powers of @xmath516 and @xmath517 where @xmath518 is the distance between @xmath512 and @xmath513 . these higher order terms may also be oscillating . because @xmath250 the integration contour @xmath189 in the integral equation fro @xmath185 is a segment of the real line in the additive parameterization . the constant @xmath519 is the fermi - momentum and it is also can be expressed in term of @xmath185 . it is also equal to the vertical electrical polarization . the finite size computations were extended to the case @xmath520 in @xcite . they argued that the complete spectrum of effective @xmath521 conformal field theory is given by @xmath522 where @xmath406 is defined by the hessian of the free energy and is related to @xmath164 as @xmath523 .    combining these results we conjecture that for generic electric fields the asymptotical behavior of correlation functions is still given by formulae ( [ as-1 ] ) and ( [ as-2 ] ) with @xmath523 one can show easily that when @xmath504 we have @xmath524 and the asymptotics ( [ as-1 ] ) ( [ as-2 ] agrees with the results from @xcite on dimer models .      here we will list some open question about the 6-vertex model and other related models . * find the classification of generic singularities of limit shapes . for @xmath146 one should expect the presence of corners as generic singularities of limit shapes . * find the scale at which fluctuations near singularities of limits shapes are described by some random process and describe such processes . for example in dimer models such fluctuations near the boundary of the limit shape are described by airy process , and near a generic cusp are described by the pearcey process . + one can argue that the same processes should describe correlation functions near similar singularities of the limit shape in the 6-vertex model but this is still a conjecture . + the scaling near the corner singularity seems particularly interesting problem since such singularities do not appear in dimer models . * understand the role of integrability of the 6-vertex model in the formation of the limit shape . here by integrability we mean that the model can be solved by the bethe ansatz , and that the weights satisfy the yang - 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[ spec_par ] . thus , a pair of spectral parameters @xmath533 is associated to the vertex @xmath534 of the grid . boltzmann weights assigned to this point are : @xmath535 @xmath536 @xmath537 where functions @xmath538 describe the parameterization of boltzmann weights of the homogeneous model . -by-@xmath3 square grid with corresponding spectral parameters . ] the partition function of the model with periodical boundary conditions can be written as @xmath539 where @xmath12 is the @xmath540 row - to - row transfer matrix of the 6-vertex model @xcite . the raw - to - raw transfer - matrix is the trace of the `` quantum monodromy matrix '' : @xmath541 here @xmath542 is the product @xmath543 which acts in @xmath544 . the matrix @xmath545 acts as the matrix @xmath546 in the basis @xmath547 of the product of the 0-th and j - th factors in the tensor product . it acts trivially in other factors . let us denote the number of arrows pointing down in a row of the grid by @xmath548 . denote by @xmath549 the corresponding subspace in the space of all possible states on the raw of length @xmath260 of vertical edges . the action of the transfer - matrix preserve these subspaces . the bethe ansatz method gives the following result for the eigenvalues of @xmath12 : @xmath550 where @xmath551 and @xmath552 satisfy the bethe equations @xmath553    it follows that the partition function of the model is given by @xmath554 where @xmath555 are the eigenvalues of the row - to - row transfer matrix @xmath12 and the sum is taken over all eigenvalues with their multiplicities . thus , in the homogeneous case when all @xmath556 are the same the asymptotic of the partition function as @xmath557 is @xmath558 where @xmath518 is the multiplicity of the largest eigenvalue @xmath559 of the row - to - row transfer - matrix on @xmath260 sites .    therefore , if we can compute the asymptotic of the largest eigenvalue of the transfer - matrix as @xmath125 we can find the asymptotic of the partition function in the thermodynamical limit .    strictly speaking , this logic gives the asymptotic of the partition function in the limit @xmath560 . under mild assumptions one can argue that the leading term of the asymptotic of free energy is uniform . however , in some cases a resonant phenomenon may occur which will make the leading term of the asymptotic of @xmath143 depend on the ratio @xmath561 @xcite . in the homogeneous @xmath0-vertex model the boltzmann weights are the same for all vertices on the grid @xmath28 . when @xmath562 the functions @xmath563 are : @xmath564 other values of @xmath215 can b obtained by analytical continuation . the formula for eigenvalues of the row - to - row transfer matrix given in the previous section becomes @xmath565 where @xmath566 are solutions of the bethe ansatz equations @xmath567^n= e^{2nh}\mathop{\prod_{k=1}^n}_{k\neq j } \frac{\sin(u_k - u_j+\eta)}{\sin(u_k - u_j-\eta ) } , \quad j=1, .. ,n.\ ] ]    introducing new variables @xmath568 we get @xmath569 where @xmath570 satisfy the bethe equations @xmath571    we want to find the asymptotic of the largest eigenvalue of the transfer - matrix when @xmath125 . we shall do it in two steps . first we will find the asymptotic of the largest eigenvalue among @xmath572 in the limit @xmath573 after this we will find for which @xmath574 this eigenvalue is the largest . [ a_bethe_eqs ] let us denote @xmath575 then , taking the logarithm of both sides of each bethe equation ( [ bethe_eq_z ] ) and dividing by @xmath260 , we get @xmath576 where @xmath577 are integers . all theoretical and numerical results on the @xmath0-vertex model suggest that the following is true . [ main_conj ] in the limit @xmath578 with @xmath579 is fixed the roots of the bethe equations corresponding to the maximal eigenvalue of the transfer matrix in the subspace @xmath580 concentrate on a smooth curve @xmath189 in the complex @xmath190-plane . the curve @xmath189 is symmetric with respect to the complex conjugation @xmath191 . the maximal eigenvalue of the transfer - matrix in the subspace @xmath549 corresponds to @xmath581 . these conjectures can be proven in the free - fermion case @xmath504 . the conjecture ( [ main_conj ] ) implies that as @xmath125 one should expect finite limit density of @xmath582 on the contour @xmath189 . if @xmath583 the limit @xmath584 is the value of the limit density function of solutions to the bethe equations on @xmath189 . let us denote the endpoints of the contour @xmath189 by @xmath585 and @xmath586 , @xmath587 and @xmath588 . taking the limit in the equation ( [ b - eqn ] ) we obtain the integral equation for @xmath185 : @xmath589 here @xmath590 . differentiating this equation with respect to @xmath190 , we obtain another , more convenient form of this equation : @xmath591 where @xmath592 the kernel @xmath593 is singular on two curves @xmath594 and it can be written as @xmath595    notice that functions @xmath596 , i.e. @xmath597 and @xmath598 are inverse to each other .    if @xmath504 , coutours @xmath226 and @xmath227 coincide and the kernel is zero . if @xmath599 , there are two cases :    * conours @xmath189,@xmath226 , @xmath227 do not intersect each other ; * contours @xmath189,@xmath226 , @xmath227 , _ all _ intersect at two points @xmath600 which are the solutions of @xmath601 and are conjugate to each other ( these two points may coincide in the degenerate case ) . this is possible only if @xmath602 . the integral equation on @xmath185 can be rewritten as @xmath603 notice that the contour @xmath189 can be deformed as long as it does not intersect the curves @xmath226 and @xmath227 .    the contour @xmath189 is defined by the condition that the form @xmath192 has purely imaginary values on the vectors tangent to @xmath189 @xmath604 and by the normalization condition on @xmath185 @xmath605    the free energy per site is defined as @xmath606    the formula for eigenvaues @xmath572 has two terms . each of them grow exponentially for large @xmath260 . generically , one of the terms will dominate the other . taking into account conjectures about roots of bethe equations in the thermodynamicallimit we obtain the follwoing integral representation for the free energy @xcite : @xmath607 or @xmath608 where @xmath185 is determined bythe integral equation ( [ eq - for - rhoz ] ) with conditions ( [ c - cond ] ) and ( [ rho - norm ] ) . the free - fermion curve in the space of parameters of the 6-vertex model is @xmath504 . since in this case @xmath609 the integral equation of @xmath610 immediately implies @xmath611 . the contour @xmath189 in this case is an arc @xmath612 where @xmath613 $ ] .      minimizing this expression in @xmath164 we get the equation @xmath615 this defines the critical value @xmath616 when the absolute value of the r.h.s is not greater then @xmath4 . otherwise @xmath617 or @xmath602 . taking into account the parameterization of boltzmann weights by electric fields @xmath23 and @xmath24 it is easy to see that ( [ s - int ] ) together with the equation for @xmath624 coincide with the formula for the free energy derived from the bethe ansatz .
we study the 6-vertex model with fixed boundary conditions . in the thermodynamical limit 
 there is a formation of the limit shape . 
 we collect most of the known results about the analytical properties of the free energy of the model as the function of electric fields and study the asymptotical behavior near singularities . 
 we also study the asymptotic of limit shapes and the structure of correlation functions in the bulk .