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convergent expansions such as linked cluster , hopping parameter or high temperature expansions provide an analytic alternative to monte carlo simulations . originally they have been developed in the infinite volume , meanwhile they have been extended to finite volumes as well @xcite . in contrast to generic perturbation theory , hopping parameter expansions ( hpes ) are convergent expansions about completely disordered lattice systems . the expansion parameter @xmath5 is the coefficient of the ( pair ) interaction term . since we calculate free energies and connected correlations in the hopping parameter expansion , we generate linked cluster expansions ( lces ) . lces have been generalized to scalar field theories at _ finite _ temperature by @xcite . compared to the zero temperature results , the finite temperature induces a tiny shift in the critical temperature ( hopping parameter ) . thus one has to face a similar problem as the improvement of a bad signal to noise ratio in monte carlo simulations . here the signal corresponds to the finite temperature effect and the noise to the `` background '' of temperature zero graphs . the price to describe the critical region , in particular critical exponents , with an accurracy of @xmath6 % is to go to the 18th order in the expansion in @xmath5 . this accurracy has been achieved in @xcite . thus the question arises , why we are still interested in a generalization of lces to a finite volume . the reason is twofold . @xmath7 the first one is to identify 1st order transitions from series expansions in the high temperature phase . for example a measurement of a vanishing critical exponent @xmath8 is compatible with a gaussian fixed point , but not conclusive for the onset of a first order transition region at a tricritical point . ( the latter interpretation actually applies to our measurements . ) @xmath7 the second reason is to distinguish second order transitions associated with different universality classes . ideally critical exponents should show a clear gap between plateaus of different universality classes . practically the gap is smeared out because of the truncation of the series expansions at finite order . the truncation has a similar rounding effect on this gap as a finite volume on thermodynamic singularities . in hpes the action is split into a sum of ultralocal parts @xmath9 and a next neighbour part @xmath10 with next neighbour couplings @xmath11 . a taylor expansion in @xmath5 of the partition function @xmath12 about the ultralocal contribution to @xmath12 finally leads to graphical expansions of n - point susceptibilities @xmath13 [ exa.chi ] _ n ( ) = _ 0 a_^(n ) ^. the coefficients @xmath14 are a sum of graphs each of which consists of a product of the inverse symmetry factor , an internal symmetry factor , a lattice embedding factor , and a polynomial of vertex contributions depending on the couplings involved in @xmath9 . it is only the embedding factor that depends on the topology of the particular lattice , that will change in passing from an infinite to a finite volume .      from our series expansions we measure a _ shift in @xmath15 _ from a fit according to [ abc ] ( | _ c(l ) - _ c ( ) | ) - y_t . here @xmath16 is a constant , @xmath17 for 2nd order transitions and @xmath18 for 1st order transitions . the critical couplings @xmath19 and @xmath20 are determined as the radius of convergence of the series expansions in a finite or infinite volume , respectively . since an extrapolation of the scaling behaviour from the height and width of the critical region is not feasible within the series expansions , we have proposed the following _ monotony criterion_.      consider two volumes @xmath21 and @xmath22 such that @xmath23 , @xmath24 , @xmath25 stands for a generic coupling , and @xmath26 for the reduced `` temperature '' . we define    [ fss.rml ] r(v_1,v_2 ) : = 1 - ,    the monotony criterion says that    @xmath27    the different behaviour of @xmath28 comes from the _ singular _ part of @xmath29 . the underlying observation is a different monotony behaviour of response functions @xmath30 for 1st and 2nd order transitions that is seen for @xmath30s with nonanalytic behaviour in the infinite volume limit . the @xmath30s are increasing in volume in a certain neighbourhood of @xmath31 for 2nd order transitions and decreasing for 1st order transitions . for a precise specification of the scaling region , i.e. the bounds that @xmath21 , @xmath22,@xmath32 have to satisfy , we refer to @xcite . the difference reflects the @xmath33-function and power law singularities for 1st or 2nd order transitions , respectively , predicted at @xmath34 in the thermodynamic limit.thus the actual behaviour of @xmath29 we have found in applications to scalar @xmath1 models ( cf . section 3 ) is neither a pecularity of the model nor an artifact of the series expansions . @xmath7 it should be noticed that the criterion is not restricted to series expansions , but also applicable to monte carlo calculations ( since both involved volumes may be finite ) if the regular contribution to @xmath30 is really negligible . @xmath7 it is neither restricted to order parameter susceptibilities as the notation suggests , but similarly applies to other response functions like the specific heat if they diverge in the thermodynamic limit .      the effective potential is defined via the effective action @xmath36 evaluated for a constant background field @xmath37 ( @xmath38 ) . it can be expressed in terms of the 1pi - susceptibilities according to @xmath39 @xmath40 the @xmath41s are obtained from the @xmath42s if the graphical expansion of the @xmath42s is restricted to 1pi graphs . the ( non)convex shape of the effective potential in the symmetric phase indicates the order of the transition . its evaluation in a _ finite _ volume leads to an alternative determination of @xmath19 via the coexistence of different minima of the free energy or a vanishing of the coefficient of the quadratic term . we apply the finite size scaling analysis with hpes to a scalar theory with @xmath45 and @xmath46-terms and @xmath1 symmetry in 3 dimensions , @xmath47 is chosen as @xmath48 and @xmath49 . after a suitable reparametrization the ultralocal part of the action reads [ exa.action ] ( , , ) = ^2 + ( ^2 - 1)^2 + ( ^2 - 1)^3 the hopping parameter term is given as               the value for @xmath50 for the first order transitions confirms results of @xcite for @xmath2 and extends results of @xcite for @xmath3 . it disagrees with the naive expectation of a scaling with @xmath0 rather than @xmath51 . the figure shows results for the monotony      criterion applied to the special case of @xmath52 , @xmath53 , @xmath54 , @xmath3 , @xmath55 . the additional index @xmath37 refers to the order of truncation . a similar figure is obtained for @xmath56 as function of @xmath25 for fixed order of the truncation and varying volumes . thus the intersection of the curves with the @xmath25-axis , which localizes the tricritical point , weakly depends on the order of the truncation and the choice of @xmath21 . the ultimate determination of the tricritical couplings @xmath57 and @xmath58 needs an extrapolation in both parameters . respecting the mutual relation between the truncation effect and the finite volume dependence of the intersection , the final estimate for @xmath57 leads to @xmath59 . for further details on the involved extrapolations we refer to @xcite . this result improves the infinite volume estimate of @xmath60 by two orders of magnitude . it should be noticed that the volume independence of @xmath29 along the tricritical line supports the validity of a mean field analysis of tricritical exponents even in three dimensions as one might have expected from the ginzburg criterion . the localization of @xmath57 from the change in the shape of the effective potential leads to an upper bound on @xmath57 , we find @xmath61 , since the involved systematic errors are less well under control . an extension to _ effective _ scalar models for an underlying salam - weinberg theory with hpes in a finite volume will particularly well work in the range of large higgs masses . it complements the range of higgs masses which has been available in recent monte carlo simulations and allows a determination of the _ critical _ higgs mass above which the electroweak transition turns into a smooth crossover phenomenon @xcite .
linked cluster expansions are generalized from an infinite to a finite volume on a @xmath0-dimensional hypercubic lattice . 
 they are performed to 20th order in the expansion parameter to investigate the phase structure of scalar @xmath1 models for the cases of @xmath2 and @xmath3 in 3 dimensions . in particular 
 we propose a new criterion to distinguish first from second order transitions via the volume dependence of response functions for couplings close to but not at the critical value . 
 the criterion is applicable to monte carlo simulations as well . 
 here it is used to localize the tricritical line in a @xmath4 theory . 
 we indicate further applications to the electroweak transition .