You are an expert at summarizing long articles. Proceed to summarize the following text: 
polymers , whether linear or branched , typically display a two phase structure in solvent with a @xmath0-transition @xcite , whose position depends on temperature and solvent composition , intervening between an extended and a collapsed phase . the case of linear polymers has proved amenable to various analytical treatments such as conformal field theory and coulomb gas methods @xcite , but branched polymers have been rather more stubborn .    a discretized model of branched polymers is provided by lattice animals , in essence graphs of connected sites on some lattice . contact interactions exist between nearest neighbour sites that are not directly linked by a bond of the animal and solvent interactions exist between occupied sites and unoccupied nearest - neighbour sites . all the ingredients of the continuum polymer in solvent problem are thus present . it is known that in @xmath3 dimensions the lattice animal exponents are related to those of the yang - lee edge singularity in @xmath4 dimensions which allows the calculation of the bulk entropic exponent for the so - called strongly embedded lattice animal @xcite . it is also known that the model corresponds exactly to the @xmath1 limit of a certain extended @xmath5-state potts model @xcite , which has led to the conjecture that a percolation critical point separates two different branches of the @xmath0 line @xcite . more controversially , exact enumeration techniques have suggested that _ two _ collapsed phases might exist @xcite whereas transfer matrix investigations have found no sign of this @xcite . this conflict motivated henkel and seno @xcite to conduct a two - pronged investigation of lattice animals using the extended potts model formalism . they attacked the infinite dimensional , mean - field problem by solving the model on a bethe lattice and the low - dimensional problem by using migdal - kadanoff @xcite recursion relations . in both cases they found only one collapsed phase , and observed that it was unlikely that the topology of the phase diagram would change in intermediate dimensions . in this paper we also use the extended potts model formalism , this time on @xmath2 random graphs of arbitrary topology , to show that only one collapsed phase exists for these . a couple of interesting technical points emerge along the way : firstly , ratios of the saddle point equations in the random graph model are isomorphic to the fixed points of the recursion relations that solve the model on the bethe lattice , so the phase structure on @xmath2 graphs in the large @xmath6 limit is identical in all respects to the bethe lattice ; secondly , the symmetry breaking pattern that emerges at the @xmath0-transition is not that which one might have naively expected given earlier results for potts models and percolation on random graphs . the former phenomenon also occurs in other models on random graphs @xcite and is heuristically explained by noting that the loops in the random graph model are predominantly large in the thermodynamic limit @xcite , so the random graphs look locally like the corresponding bethe lattice and one generically obtains mean field behaviour .    the partition function for lattice animals may be written in several equivalent ways , but we restrict ourselves to that in equ.([an1 ] ) to facilitate easy comparison with @xcite . if an animal contains @xmath6 sites , @xmath7 bonds and @xmath8 contact interactions we can write @xmath9 where the site , bond and contact fugacities are given by @xmath10 respectively and the @xmath11 s give the number of different animals with the specified numbers of sites , bonds and contacts . there is an exact correspondence between the lattice animal partition function of equ.([an1 ] ) and the @xmath1 limit of an extended @xmath5 state potts model @xcite with hamiltonian @xmath12 where the @xmath13 sums are over nearest neighbour vertices and the spins @xmath14 take on @xmath5 values . the @xmath15 term is non - standard , in its absence we would have the usual potts hamiltonian ( with an external field ) . the potts model parameters are related to @xmath16 and @xmath17 in equ.([an1 ] ) by @xmath18 where @xmath19 is the coordination number of the lattice or graph on which the animals live , and the partition functions are related by @xmath20 where the potts partition function is @xmath21 the idea behind the correspondence is essentially identical to the use of the standard potts model hamiltonian in the @xmath1 limit to describe percolation @xcite , which is based fortuin and kastelyn @xcite cluster transcription of potts and related models . having discussed the lattice animal model in general we now move on to formulate it on random graphs of arbitrary topology . a particularly economical way of writing down statistical mechanical models on such random graphs was introduced in @xcite where it was observed that taking the @xmath22 limit in the hermitian matrix models familiar from studies of 2d gravity generated random graphs of arbitrary topology , as opposed to the planar graphs picked out by the @xmath23 limit that is used in 2d gravity . the random graphs appear as feynman diagrams in the perturbative expansion of the partition function . the matrix indices vanish in the @xmath22 limit , leaving `` thin '' graphs rather than the fat , or ribbon , graphs of the matrix model proper . we shall denote the non - planar , arbitrary topology random graphs as thin graphs and the animals which live on them as thin animals throughout the rest of the paper for brevity . in general the partition function of a statistical mechanical model on an ensemble of thin random graphs with @xmath24 vertices is given by @xmath25 where the contour integral over the vertex coupling @xmath26 picks out the graphs with @xmath24 vertices , @xmath27 is an appropriate action , @xmath28 is the inverse of the quadratic form in this action , and the @xmath29 are the various matter variables in the action . as the number of thin random graphs increases factorially with @xmath30 the factor @xmath31 gives the number of undecorated ( i.e. without matter ) graphs in the class of interest and disentangles this growth from any non - analyticity due to phase transitions . for the @xmath2 ( 3-regular ) random graphs we discuss here @xmath32 it is perhaps worth emphasising that the @xmath29 appearing in equ.([part ] ) are _ scalar _ variables , so the evaluation of thin graph partition functions is inherently easier than dealing with the matrix integrals of the planar limit . in the large @xmath30 , thermodynamic , limit saddle point methods may be used to evaluate equ.([part ] ) . the saddle point equation for @xmath26 may be trivialised by scaling it out of the action as an overall factor , leaving any critical behaviour to be revealed by the behaviour of the saddle point equations for the matter fields @xmath29 . phase transitions are manifested by an exchange of dominant saddle point , either continuously or discontinuously . we can now write down the action which gives the boltzmann factors appropriate to the hamiltonian of equ.([ham ] ) for a @xmath5-state potts model on thin @xmath2 random graphs @xmath33 where @xmath34 and we have pre - emptively scaled out the cubic vertex coupling @xmath26 as it plays no role in the determination of the phase structure . the inverse of the quadratic form in equ.([action ] ) has the diagonal terms @xmath35 with all off - diagonal terms 1 , which is precisely what is required . we can also see that when @xmath36 we have @xmath37 , @xmath38 , and we recover ( up to a trivial rescaling ) the action for the standard potts model used in @xcite . the saddle point equations for the action in equ.([action ] ) may be solved for increasing @xmath5 until patience , or algebraic computing power , run out . `` generic '' behaviour sets in at @xmath39 , as opposed to @xmath40 for the standard action with @xmath36 . one finds a disordered phase where @xmath41 is distinct and the @xmath42 @xmath29s are equal to @xmath43 , which we denote as having @xmath44 symmetry . there are two patterns of symmetry breaking from this phase to ordered states : in the first we have @xmath41 , one distinct @xmath45 and the other @xmath46 equal to @xmath43 , which we denote as @xmath47 ; in the second we find @xmath41 , @xmath48 and @xmath49 which we denote by @xmath50 . we now follow the path used in the analysis of @xcite by writing down effective actions that are sufficient to capture these symmetry breaking patterns . in effect , most of the @xmath5 potts variables are redundant as @xmath51 or @xmath52 of them are equal in the ordered phases and @xmath42 in the disordered phase . imposing the appropriate symmetries we find the effective actions @xmath53 and @xmath54 solving the saddle point equations @xmath55 for either effective action to get @xmath56 involves only a quadratic equation in the disordered phase , but one is left with a messy quartic to solve in the ordered phase . a direct attack is not very illuminating , but we can use the same legerdemain exercised in @xcite for bethe lattice to simplify the problem greatly . the useful observation made there was that the lattice animal partition function could be written as @xmath57 where @xmath58 was the canonical ensemble ( fixed @xmath6 ) free energy . the sum in equ.([gc ] ) can be seen to diverge at a critical value of @xmath59 , which we denote as @xmath60 , which in turn gives the critical canonical ensemble free energy @xmath61 of the infinite lattice animal . therefore , if we can calculate @xmath62 as a function of @xmath63 we obtain the canonical free energy @xmath64 of the infinite lattice animal as a function of these variables .    to see how this helps , consider the solutions of the saddle point equations obtained from equs.([subdom],[dom ] ) when @xmath65 , which are given by @xmath66 for both in the disordered phase , with symmetry @xmath67 , where @xmath68 and where we have not solved the quadratics explicitly for reasons that will become apparent below . the ordered phase solutions are obtained from , for equ.([subdom ] ) with breaking pattern @xmath69 @xmath70 \tilde \phi^3 + \left [ 4 ( \tau - 1)^2 - 6 y ( 1 - \tau ) + y^2 ( 1 + x y ) \right ] \tilde \phi^2 \\ \nonumber & - &   \left [ 4 ( 1 - \tau)^2    + x y^2 ( 2 \tau - y ) + 2 y ( \tau - 1 ) \right ] \tilde \phi + \left [ ( \tau - 1 ) ^2 + \tau x y^2 \right ]   \\ \nonumber & & \\ \nonumber \phi & = &   1 - \tilde \phi \\ \nonumber & & \\ \nonumber \psi & = & - { \tau^2 \over y } \left [ y ( \tilde \phi^2 - \tilde \phi ) - 2 ( \tau   - 1 ) \tilde \phi + ( \tau - 1 ) \right ] \end{aligned}\ ] ] and , for equ.([dom ] ) with breaking pattern @xmath50 @xmath71 \tilde \phi^3 + \left [ 16 ( \tau - 1)^2 + 12 y ( 1 - \tau ) + y^2 ( 1 + x y ) \right ] \tilde \phi^2 \\ \nonumber & - &   \left [ 16 ( 1 - \tau)^2    - x y^2 ( 4 \tau - y ) - 4 y ( \tau - 1 ) \right ] \tilde \phi + \left [ 4 ( \tau - 1 ) ^2   - 2   \tau x y^2 \right ]   \\ \nonumber & & \\ \nonumber   \phi & = &   1 - \tilde \phi \\ \nonumber & & \\ \nonumber   \psi & = & - { \tau^2 \over y } \left [ y ( \tilde \phi^2 - \tilde \phi ) + 4 ( \tau   - 1 ) \tilde \phi - 2 ( \tau - 1 ) \right].\end{aligned}\ ] ] to avoid the onerous task of solving for @xmath72 in all of these we now , as advertised , treat equ.([solhot ] ) and the first of equs.([solsub],[soldom ] ) as equations for @xmath73 , in which variable they are only linear . if we make the substitution @xmath74 to simplify the resulting expressions we find from equs.([solhot],[solsub],[soldom ] ) @xmath75 where we have labelled the solutions by their symmetry breaking pattern . we see here the first hint of a close parallel with the bethe lattice solution of @xcite since our two ordered solutions are related by @xmath76 which mirrors a similar symmetry in @xcite .    the physical solutions @xmath77 are obtained by maximising these expressions with respect to @xmath78 , which plays the role of an order parameter . the appropriate solution for a given @xmath79 is then the largest of these . bearing in mind that we must have the fugacity @xmath59 positive to make sense physically and that the natural ranges of @xmath80 and @xmath17 are @xmath81 , @xmath82 , we see that the physical ranges for @xmath78 are @xmath83 for @xmath84 , @xmath85 for @xmath86 and @xmath87 for @xmath88 .    at this point it might appear that we have succeeded in showing precisely the reverse of what was announced in the introduction in that we have found _ two _ candidate ordered phases , which might be identified with the two contentious collapsed polymer phases . however , an extensive numerical investigation throughout the @xmath79 range reveals that the one of the solutions , @xmath89 , is always larger than the other , @xmath90 , for a given @xmath79 so it is the former which is always dominant .    for the disordered solution we clearly have a maximum at @xmath91 @xmath92 so the free energy is @xmath93 and we also find @xmath94 we are now in a position to map out the phase structure of thin animals . we have found two phases , whose free energies are given by @xmath95 which we can identify with the extended phase and @xmath96 which we identify with the ( single ) collapsed phase . the transition line between these two phases may be pinned down by noting that @xmath97 iff @xmath98 . we then observe that @xmath99 so @xmath100 has a maximum at @xmath91 if @xmath101 , a minimum if @xmath102 and a turning point if @xmath103 . a glance at figs.1,2 where we plot @xmath104 and @xmath105 for @xmath106 and @xmath107 respectively clarifies what is going on . in fig.1 we see that @xmath104 and @xmath105 share a common maximum at @xmath91 when @xmath106 ( which is the case for all @xmath108 ) , whereas for @xmath107 the maximum of @xmath105 lies at some @xmath109 . this means that we see a continuous transition as @xmath17 is varied across @xmath98 when @xmath110 between extended animals at small @xmath17 and collapsed animals at large @xmath17 . for @xmath102 the position of the critical line in the @xmath79 plane must be determined numerically and the transition is first order . the physical solution jumps discontinuously from the maximum of @xmath88 at @xmath91 to the maximum of @xmath86 at @xmath109 . the appropriate equations for pinpointing the first order line are thus @xmath111 which give the conditions for the maximum in @xmath112 at @xmath109 to be equal to the maximum in @xmath113 at @xmath91 .    a triple point at @xmath114 separates the first and second order transition lines . all of these features are indicated in the schematic phase diagram of fig.3 . the presence of a jump across the first order line is confirmed explicitly by the sample values in figs.4,5 where we see that as we cross the line by varying @xmath17 at fixed @xmath107 we move from @xmath88 dominant in fig.4 to the situation in fig.5 where @xmath86 is just equal to the maximum of @xmath88 ( @xmath115 at the origin ) for a _ non - zero _ @xmath78 value ( @xmath116 ) at @xmath117 . at this point @xmath78 jumps discontinuously and we move into the collapsed phase as @xmath17 increases . when moving across the second order portion of the line however , the maximum of @xmath86 is at the origin and we see no jump in @xmath78 , giving a second order transition . this can be seen by referring back to fig.1 and comparing it with fig.6 . in fig.6 we are _ below _ the second order line at @xmath118 and the two curves look much like fig.4 . as we move up to the line at @xmath119 in fig.1 we can see that the maximum of @xmath86 now lies at the origin and increasing @xmath17 further takes us into the collapsed phase via a second order transition . in closing this section it should be remarked that although the broad features of the random graph phase diagram appear to be in accordance with two - dimensional models , there is one notable difference . the triple point separating the first and second order portions of the @xmath0 line is the percolation point in @xmath120 . that is _ not _ the case here  the percolation point in the random graph model is at @xmath121 and @xmath98 , which in turn implies @xmath122 . model . ] this presumably can be ascribed to the mean - field nature of the model , a comment which also applies to the bethe lattice results of @xcite , whose similarities with the work here we now move on to discuss . anyone familiar with the results of @xcite for bethe lattice animals may at this point be experiencing a strong sense of dj vu . the thin animal phase diagram has proved to be identical to that on the bethe lattice , down to non - universal quantities such as the position of the @xmath0 line and triple point . as we have already noted , even before extracting the physical solutions @xmath60 parallels are apparent since the two ordered solutions on both random graphs and the bethe lattice possess a particular @xmath123 symmetry . our criteria for extracting the @xmath0 line are also identical to those used on the bethe lattice , although the equations for @xmath59 on the bethe lattice are quadratic rather than linear , as above . all this clearly begs an explanation . our saddle point equations for @xmath86 at @xmath65 , which we have so far not written explicitly , are given by @xmath124 at first sight these bear little relation to the fixed points of the recursion relations used in @xcite to obtain the dominant solution to the model on the bethe lattice . the method of derivation is certainly completely different . on the bethe lattice one considers the behaviour of the central spin @xmath125 @xcite , writing @xmath126 where @xmath127 is the spin on the @xmath128th sub - branch and @xmath129 and the primed sums are over the sub - branch with first spin @xmath130 . the solution then follows by defining @xmath131 where @xmath6 labels the `` shell '' of the bethe lattice , and noting that it satisfies the recursion relation @xmath132 defining @xmath133 and taking the limit @xmath134 allows one to write the fixed point of the recursion relations in equ.([curse ] ) as @xmath135 where @xmath136 , @xmath137 . given the identical phase diagram we have found on random graphs to the bethe lattice results of @xcite it is clear that the two sets of equations , equs.([ours ] , [ his ] ) , must somehow be equivalent . this can be made explicit by a simple transformation . equs.([ours ] ) may be symbolically written as @xmath138 , where @xmath139 is the column vector @xmath140 and @xmath141 is @xmath142 . we are at liberty to rewrite these as @xmath143 as @xmath144 is invertible in general @xmath145 if we now divide @xmath146 and @xmath147 @xmath148 and make the identifications @xmath149 we obtain exactly the two recursion equation fixed points of equ.([his ] ) . although we have two distinct physical systems , closed random graphs of arbitrary topology which _ do _ contain loops , and the loop - less bethe lattice the preceding calculation shows that the thermodynamic behaviour of lattice animals on both is identical . in a precise sense the loops do not matter , as the saddle point equations that govern the random graph thermodynamic limit are isomorphic to the recursion relations that arise in solving the loop - less bethe lattice . the identity between the bethe lattice recursion relations and ratios of the random graph saddle point equations also occurs for other systems ( standard ising , potts spins etc . ) @xcite , where the consequences are the same , namely the random graph solutions are identical to those on the bethe lattice . the astute reader might remark that the solution of the lattice animal problem on random graphs presented in this paper does not really provide any further evidence against the existence of a second collapsed phase , as we have also shown that the solutions to the model have the same content as the bethe lattice solution in @xcite , though this is an interesting result in itself . one could counter the objection by stating that we have found a different physical system , and solved it by very different means , in order to arrive at these conclusions . the thin graph solution also clarifies one lacuna in the bethe lattice approach . as we have noted , solving the full set of saddle point equations @xmath150 ( i.e. _ before _ going to the effective actions ) shows that only one disordered and two ordered phases are present for arbitrary @xmath5 . on the bethe lattice enough order parameters are introduced by hand to distinguish the two putative ordered phases , which is then shown to be self - consistent . the possibility of a more exotic phase diagram is not , however , explicitly excluded . the random graph saddle point equations show that this really is enough , as no other patterns of symmetry breaking are observed for any @xmath5 . the observed symmetry breaking pattern itself on random graphs is unexpected , namely @xmath151 . for the standard @xmath5-state potts model one has @xmath152 , so the pattern observed for the sub - dominant solution @xmath153 might have been a more obvious first guess . we have also remarked that the critical percolation point , @xmath154 on thin graphs , is not equal to the tricritical point @xmath155 that separates the two portions of @xmath0 line , in contrast to the @xmath120 model , although the general topology of the phase diagram is similar . although the work presented in this paper is fairly self - contained one obvious generalisation suggests itself . the action of equ.([action ] ) will also serve to represent lattice animals on _ planar _ random graphs if one takes the @xmath156 to be @xmath157 hermitian matrices and performs the limit @xmath23 . the @xmath121 case ( percolation ) has already been solved by kazakov @xcite , which suggest that the animal problem might also be tractable . the interest in doing this is that it could shed light on a another vexed question for animals  the ( lack of ) conformal invariance in the 2d model .    as a technical point , sidestepping the solution of difficult saddle point equations by considering the canonical free energy and solving for the fugacity associated with the external field is an idea which might also be fruitfully applied to the solution of the standard potts ( or even ising ) models in an external field on random graphs , where similarly complicated equations arise . cardy , `` conformal invariance and statistical mechanics '' in _ fields , strings and critical phenomena _ , les houches xlix , e. brzin and j. zinn - justin eds , north holland amsterdam ( 1990 ) ; + p. christe and m. henkel , _ introduction to conformal invariance and its applications to critical phenomena _ , springer verlag heidelberg ( 1993 ) ; + b. nienhuis , `` coulomb gas formulation of two dimensional phase transitions '' in _ phase transitions and critical phenomena _ vol 11 , c. domb and j.l . lebowitz eds , academic press ( 1987 ) ; + b. duplantier and h. saleur , phys . rev . lett . * 59 * , 539 ( 1987 ) ; + c. vanderzande , a.l . stella and f. seno , phys . lett . * 67 * , 2757 ( 1991 ) g. parisi and n. sourlas , phys . lett . * 46 * , 871 ( 1981 ) m. giri , m.j . stephen and g.s . grest , phys . b16 , 4971 ( 1977 ) ; + f.y . wu , j. stat . phys . * 18 * , 115 ( 1978 ) ; rev . phys . * 54 * , 235 ( 1982 ) ; + a. coniglio , j. phys . * a16 * , l187 ( 1983 ) s. flesia , d.s . gaunt , c.e . soteros and s.g . whittington , j . phys . * a25 * , l1169 ( 1992 ) ; + c. vanderzande , phys . rev . lett . * 70 * , 3595 ( 1993 ) f. seno and c. vanderzande , j. phys . * a27 * , 5813 ( 1994 ) ; corr . * a27 * , 7937 ( 1994 ) n. madras , c.e . soteros , s.g . whittington , j.l . martin , m.f . sykes , s. flesia and d.s . gaunt , j. phys . * a23 * , 5327 ( 1990 ) ; + i.s . chang and y. shapir , phys . rev . * b38 * , 6736 ( 1988 ) b. derrida and h.j . herrmann , j. physique * 44 * , 1365 ( 1983 ) ; + b. derrida and l. de seze , j. physique * 43 * , 475 ( 1982 ) m. henkel and f. seno , phys . * e53 * , 3662 ( 1996 ) t.w . burkhardt and j.m.j . van leeuwen , `` progress and problems in real - space renormalization '' in _ real space renormalization _ burkhardt and j.m.j . van leeuwen editors , springer - verlag ( 1982 ) , p. 1 ; + t.w . burkhardt , `` bond - moving and variational methods in real - space renormalization '' , _ ibid . d. johnston and p. plech ' a , j. phys . * a30 * ( 1997 ) 7349 . c. baillie , d. johnston and j - p . kownacki , nucl . phys . * b432 * ( 1994 ) 551 ; + c. baillie , w. janke , d. johnston and p. plech ' a , nucl . phys . * b450 * ( 1995 ) 730 ; + c. baillie and d. johnston , nucl . b47 * ( proc . suppl . ) ( 1996 ) 649 ; + c. baillie , n. dorey , w. janke and d. johnston , phys . lett * b369 * ( 1996 ) 123 . d. johnston and p. plech ' a , j. phys . * a31 * ( 1998 ) 475 . b. bollobs , `` random graphs '' , academic press , 1985 . f. peruggi , j. phys . * a16 * ( 1983 ) l713 . + f. peruggi , f. di liberto and g. monroy , j. phys . * a16 * ( 1983 ) 811 ; + f. peruggi , physica * 141a * ( 1987 ) 140 ; + f. peruggi , f. di liberto and g. monroy , physica * 141a * ( 1987 ) 151 ; + j. essam , j - c lin and p. taylor , phys . rev . * e52 * ( 1995 ) 44 ; + f. peruggi , f. di liberto and g. monroy , z. phys . * b66 * , ( 1987 ) 379 . c. fortuin and p. kasteleyn , physica * 57 * ( 1972 ) 536 ; + p. kasteleyn and c. fortuin , j. phys . jpn ( suppl . ) * 26 * ( 1969 ) 11 . c. bachas , c. de calan and p. petropoulos , j. phys . * a27 * ( 1994 ) 6121 . v. kazakov , mod . * a4 * ( 1989 ) 1691 .
lattice animals provide a discretized model for the @xmath0 transition displayed by branched polymers in solvent . 
 exact graph enumeration studies have given some indications that the phase diagram of such lattice animals may contain two collapsed phases as well as an extended phase . 
 this has not been confirmed by studies using other means . 
 we use the exact correspondence between the @xmath1 limit of an extended potts model and lattice animals to investigate the phase diagram of lattice animals on @xmath2 random graphs of arbitrary topology ( `` thin '' random graphs ) . 
 we find that only a two phase structure exists 
 there is no sign of a second collapsed phase . 
 the random graph model is solved in the thermodynamic limit by saddle point methods . 
 we observe that the ratio of these saddle point equations give precisely the fixed points of the recursion relations that appear in the solution of the model on the bethe lattice by henkel and seno . 
 this explains the equality of non - universal quantities such as the critical lines for the bethe lattice and random graph ensembles 
 .    22.5 cm 0.05 cm 0.2 cm 0.2 cm