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linked cluster expansions ( lces ) have a long tradition in statistical physics . originally applied to classical fluids , later to magnetic systems ( @xcite,@xcite,@xcite and references therein ) , they were generalized to applications in particle physics in the eighties @xcite . there they have been used to study the continuum limit of a lattice @xmath0 field theory in 4 dimensions at zero temperature . in @xcite they were further generalized to field theories at finite temperature , simultaneously the highest order in the expansion parameter was increased to 18 . usually the analytic expansions are obtained as graphical expansions . because of the progress in computer facilities and the development of efficient algorithms for generating the graphs , it is nowadays possible to handle of the order of billions of graphs . the whole range from high temperatures down to the critical region becomes available , and thermodynamic quantities like critical indices and critical temperatures are determined with high precision ( the precision is comparable or even better than in corresponding high quality monte carlo results ) @xcite-@xcite . an extension of lces to a finite volume in combination with a high order in the expansion parameter turned out to be a particularly powerful tool for investigating the phase structure of systems with first and second order transitions by means of a finite size scaling analysis @xcite . linked cluster expansions are series expansions of the free energy and connected correlation functions about an ultralocal , decoupled theory in terms of a hopping parameter @xmath1 . the corresponding graphical representation is a sum in terms of connected graphs . the value of @xmath1 parametrizes the strength of interactions between fields at different lattice sites . usually they are chosen as nearest neighbours , but also more general , less local couplings lead to convergent expansions ( @xcite and references therein ) under appropriate conditions on the decay of the interactions . in contrast to the ultralocal terms of a generic interaction we will sometimes refer to hopping terms as non - ultralocal .    in this paper we develop dynamical linked cluster expansions ( dlces ) . these are linked cluster expansions with hopping parameter terms that are endowed with their own dynamics . such systems are realized in spin glasses or partially annealed neural networks with ( fast ) spins and ( slow ) interactions @xcite-@xcite . they also occur in variational estimates for su(n)-gauge - higgs systems as we will show later in this paper . like lces they are expected to converge for a large class of interactions . formally dlces amount to a generalization of an expansion scheme from 2-point to point - link - point - interactions . these are interactions between fields associated with two points and with one pair of points called link . the points and links are not necessarily embedded on a lattice , and the links need not be restricted to nearest neighbours . we develop a new theory in which a generalized notion of connectivity plays a central role . standard notions of equivalence classes of graphs like 1-line irreducible and 1-vertex irreducible graphs have to be generalized , and new notions like 1-multiple - line irreducible graphs must be defined in order to give a systematic classification . we describe algorithms for generating these classes . this is essential , since the number of graphs rapidly increases with the order in the expansion parameter . it typically exceeds the number of lce - graphs by more than an order of magnitude . the paper is organized as follows . in section 2 we specify the models that admit a dlce . we introduce and explain the idea behind the abstract notions of theory . detailed definitions of , related notions and the computation of weights are given in section 3 . section 4 treats the issue of renormalization in the sense of suitable resummations of graphs . algorithms for their generation are described in section 5 . applications to spin glasses and ( partially annealed ) neural networks are indicated in section 6 . there it is of particular interest that dlces allow for the possibility of avoiding the replica trick . in section 7 we present results for the transition line of an su(2 ) higgs model . these are obtained by dlces applied to gap equations that follow from convexity estimates of the free energy density . the results are in good agreement with corresponding high precision monte carlo results @xcite . section 8 contains a summary and outlook . the reader who is more interested in our actual applications of dlces may skip sections 3 - 5 in a first reading . these sections are of interest from a systematic point of view and essential if he is interested in applying dlces . in this section we first specify the class of models for which we develop dynamical linked cluster expansions . next we illustrate some basic notions of multiple - line graph theory , in particular the need for a new notion of connectivity . by @xmath2 we denote a finite or infinite set of points . one of its realizations is a hypercubic lattice in @xmath3 dimensions , infinite or finite in some directions with the topology of a torus . @xmath4 denotes the set of unordered pairs @xmath5 of sites @xmath6 , @xmath7 , also called unoriented links , and @xmath8 a subset of @xmath4 . we consider physical systems with a partition function of the generic form [ 2.zgen ] & & z(h , i , v ) + & & = u , with measures [ 2 m ] = _ x_0 d(x ) , u = _ l _ 1 du(l ) and action [ 2act ] s(,u , v ) = _ x _ s^((x ) ) + _ l _ 1s^1(u(l ) ) - _ x , y _ v(x , y)(x)u(x , y)(y ) , with non - ultralocal couplings & & v(x , y ) = v(y , x ) = 0 , + & & v(x , x ) = 0 . for later convenience the normalization via @xmath9 is chosen such that @xmath10=0 $ ] .    the field @xmath11 is associated with the sites @xmath12 and the field @xmath13 lives on the links @xmath14 , and we write @xmath15 for @xmath16 . for definiteness and for simplicity of the notation here we assume @xmath17 and @xmath18 . in our actual applications @xmath19 is a 4-component scalar field and @xmath20 an su(2)-valued gauge field , or @xmath19 and @xmath20 both are ising spins , or the @xmath19s are the ( fast ) ising spins and the @xmath20s @xmath21 the ( slow ) interactions . the action is split into two ultralocal parts , @xmath22 depending on fields on single sites , and @xmath23 depending on fields on single links @xmath24 . for simplicity we choose @xmath23 as the same function for all links @xmath14 . we may identify @xmath8 with the support of @xmath25 , _ 1 = \ { l=(x , y ) v(x , y ) = 0 } . the support of @xmath26 need not be restricted to nearest neighbours , also the precise form of @xmath22 and @xmath23 does not matter for the generic description of dlces , @xmath22 and @xmath23 can be any polynomials in @xmath19 and @xmath20 , respectively . the only restriction is the existence of the partition function . in one of our applications @xmath22 will be a @xmath27-type theory with o(n ) symmetry , @xmath23 consists of a term linear or quadratic in @xmath20 . note that the interaction term @xmath28 contains a point - link - point - interaction and generalizes the 2-point - interactions @xmath29 of usual hopping parameter expansions . the effective coupling of the @xmath19 fields has its own dynamics governed by @xmath30 , the reason why we have called our new expansion scheme _ dynamical _ lce . dynamical linked cluster expansions are induced from a taylor expansion of @xmath31 about @xmath32 , the limit of a completely decoupled system . we want to express the series for @xmath33 in terms of connected graphs . let us consider the generating equation [ 2.genequation ] w / v(xy ) & = & 1/2 < ( x ) u(x , y ) ( y ) > + & = & 1/2 ( w_h(x ) i(x , y ) h(y ) + w_h(x)h(y)w_i(x , y ) + & + & w_h(x)i(x , y)w_h(y ) + w_i(x , y)h(y)w_h(x ) + & + & w_h(x)w_h(y)w_i(x , y ) ) .    here @xmath34 denotes the normalized expectation value w.r.t . the partition function of eq . ( [ 2.zgen ] ) . subscripts @xmath35 and @xmath36 denote the derivatives of @xmath33 w.r.t . @xmath35 and @xmath37 , respectively . next we would like to represent the right hand side of eq . ( [ 2.genequation ] ) in terms of connected graphs . once we have such a representation for the first derivative of @xmath33 w.r.t . @xmath25 , grapical expansions for the higher derivatives can be traced back to the first one .    for each @xmath33 in eq . ( [ 2.genequation ] ) we draw a shaded bubble , for each derivative w.r.t . @xmath38 a solid line , called a @xmath19-line , with endpoint vertex @xmath39 , and for each derivative w.r.t . @xmath40 a dashed line , called a @xmath20-line , with link label @xmath16 . the main graphical constituents are shown in fig .  1 . two @xmath19-lines with endpoints @xmath39 and @xmath41 are then joined by means of a dashed @xmath20-line with label @xmath42 , if the link @xmath42 has @xmath39 and @xmath41 as its endpoints , i.e. @xmath16 . according to these rules eq . ( [ 2.genequation ] ) , multiplied by @xmath26 and summed over @xmath39 and @xmath41 , is represented by fig .  2 . note that , because of the taylor operation , each solid line from @xmath39 to @xmath41 carries a factor @xmath26 .    since the actual need for a new type of connectivity is not quite obvious from fig .  2 , because eq .  ( [ 2.genequation ] ) does not contain higher than first order derivatives w.r.t . @xmath40 , let us consider a term    ( 15.0,3.0 )    ( 15.0,7.0 )    w_h(x)w_h(y)w_h(r)w_h(s ) w_i(x , y)i(r , s ) occurring in the second derivative of @xmath33 w.r.t . @xmath26 , @xmath43 . according to the above rules this term would be represented as shown in fig . 3a . while the 2 vertices in the last term of fig . 2 are connected in the usual sense via a common ( solid ) line ( the dashed line with an attached bubble could be omitted in this case ) , the graph in fig . 3a would be disconnected in the old sense , since neither @xmath39 nor @xmath41 are line - connected with @xmath44 and @xmath45 , but -as a new feature of dlce graphs- the lines from @xmath39 to @xmath41 and from @xmath44 to @xmath45 are connected via the dashed lines emerging from a common bubble shown in the middle of the graph . as we see from fig . 3a , we need an additional notion of connectivity referring to the possibility of multiple - line connectivity . while the analytic expression is fixed , it is a matter of convenience to further simplify the graphical notation of fig . 3a at @xmath32 . two possibilities are shown in fig . 3b and fig . 3c . to fig . 3b we later refer in the formal definition of the new type of multiple - line connectivity . in the familiar standard notion of connectivity two vertices of a graph are connected via lines . the vertices are line - connected . already there , in a dual language , one could call two lines connected via vertices . the second formulation is just appropriate for our need to define when two lines are connected . the corresponding vertices mediating the connectivity of lines are visualized by tubes , in fig . 3b we have just one of them . the tubes should be distinguished from the former type of vertices represented as full dots which are connected via bare @xmath19-lines . in fig . 3c we show a simplified representation of fig . 3b that we actually use in graphical expansions . ( 15.0,7.0 )    the derivative terms have to be evaluated at @xmath32 . for @xmath32 we have a decomposition of @xmath33 according to w(h , i , v=0)= _ x _ 0 w^(h(x ) ) + _ l _ 1 w^1(i(l ) ) with [ 2.w0 ] z^(h ) = and [ 2.w1 ] z^1(i ) = . in eq.s  ( [ 2.w0],[2.w1 ] ) we have omitted any single site or single link dependence , because we assume that @xmath22 and @xmath23 are the same for all @xmath46 and @xmath47 , respectively . therefore , at @xmath32 , the only non - vanishing derivatives of @xmath33 are [ 2.wd1 ] . w_h(x_1)h(x_2) ... h(x_n ) _ v=0 = _ x_1,x_2, ... x_n and [ 2.wd2 ] . w_i(l_1)i(l_2) ... l_m , but mixed derivatives w.r.t . h and i vanish . as anticipated in fig.s  3b and 3c , for @xmath32 we replace the dashed bubbles and graphically distinguish between bubbles with @xmath19-lines and bubbles with @xmath20-lines . we define . ( 3.0,1.0 )    ( 0.0,0.0 )    ( 4.0,0.0 )    ( 0.0,0.0 ) ( 0.0,0.0)(1.0,0.8)(2.0,0.8 ) ( 0.0,0.0)(1.0,0.5)(2.0,0.5 ) ( 0.0,0.0)(1.0,-0.7)(2.0,-0.7 )    ( 1.2,0.2)(0.2,0)@xmath48 ( 1.2,-0.1)(0.2,0)@xmath48 ( 1.2,-0.4)(0.2,0)@xmath48    } n = v^c_n = ( ) _ h=0 for a connected n - point vertex with @xmath49 bare @xmath19-lines emerging from it and . ( 4.0,1.0 )    ( 0.0,-0.6 )    ( 4.0,0.0 )    ( 1.0,1.5)(2.0,2.1)(3.0,1.5 ) ( 1.0,1.0)(2.0,1.6)(3.0,1.0 ) ( 1.0,0.0)(2.0,0.6)(3.0,0.0 )    ( 1.6,1.1)(0.2,0)@xmath48 ( 1.6,0.8)(0.2,0)@xmath48 ( 1.6,0.5)(0.2,0)@xmath48    ( 2.0,1.85)(0,1)0.2 ( 2.0,1.6)(0,1)0.2 ( 2.0,1.35)(0,1)0.2 ( 2.0,1.1)(0,1)0.2 ( 2.0,0.85)(0,1)0.2 ( 2.0,0.6)(0,1)0.2 ( 2.0,0.35)(0,1)0.2 ( 2.0,0.1)(0,1)0.2    } = m^1 c _ = ( ) _ i=0 for a connected @xmath50-line consisting of @xmath50 bare lines . if @xmath51 , we often omit the dashed line . if the bare lines of a @xmath50-line are internal @xmath19-lines , they get vertices attached to their endpoints , if they are external @xmath20-lines , no vertices will be attached . let @xmath52 denote the lattice volume in @xmath3 dimensions . the taylor expansion of @xmath33 about @xmath32 to second order in @xmath25 then reads [ 2.long ] w(h , i , v ) & = & w(h , i , v=0 ) + & + & _ x , y _ 0 v(x , y ) w_h(x ) w_h(y)w_i(x , y ) + & + & _ x , y , r , s _ 0 v(x , y ) v(r , s ) + & ( & 4 w_h(y)w_h(s)w_h(r)h(x)w_i(x , y)w_i(r , s ) + & + & 2 w_h(x)h(r)w_h(y)h(s)w_i(x , y)w_i(r , s ) + & + & 4 w_h(y)w_h(s)w_h(r)h(x)w_i(x , y)i(r , s ) + & + & 2 w_h(r)h(x)w_h(y)h(s)w_i(r , y)i(x , s ) + & + & w_h(x)w_h(y)w_h(r)w_h(s)w_i(x , y)i(r , s ) ) _ v=0 + & + & o(v^3 ) , where we have used that @xmath53 . for each @xmath33 in the products of @xmath33s we now insert eq.s  ( [ 2.wd1]),([2.wd2 ] ) . if we choose @xmath25 in a standard way as next - neighbour couplings v(x , y ) = 2k _ = 0^d-1 ( _ x+,y+ _ x-,y ) with @xmath54 denoting the unit vector in @xmath55-direction , eq . ( [ 2.long ] ) becomes in a graphical representation at @xmath56 [ 2.graph ] & = & ( 2k ) ( 2d )    ( 4.0,1.0 )    ( 0.0,0.0 )    ( 4.0,1.0 )    ( 0.0,0.2 ) ( 2.0,0.2 ) ( 0.0,0.2)(1,0)2.0     + & + & ( 2k)^2 \ { ( 2d)^2    ( 3.5,1.0 )    ( 0.0,0.0 )    ( 4.0,1.0 )    ( 0.0,-0.2 ) ( 0.0,0.6 ) ( 2.0,0.2 ) ( 0.0,-0.2)(1.0,-0.2)(2.0,0.2 ) ( 0.0,0.6)(1.0,0.6)(2.0,0.2 )    + ( 2d )    ( 4.0,1.0 )    ( 0.0,0.0 )    ( 4.0,1.0 )    ( 0.0,0.2 ) ( 2.0,0.2 ) ( 0.0,0.2)(1.0,-0.4)(2.0,0.2 ) ( 0.0,0.2)(1.0,0.8)(2.0,0.2 )     + & & + ( 2d)^2    ( 3.5,1.0 )    ( 0.0,0.0 )    ( 4.0,1.0 )    ( 0.0,-0.2 ) ( 0.0,0.6 ) ( 2.0,0.2 ) ( 0.0,-0.2)(1.0,-0.2)(2.0,0.2 ) ( 0.0,0.6)(1.0,0.6)(2.0,0.2 )    ( 0.9,0.2)(0,1)0.15 ( 0.9,0.4)(0,1)0.15 ( 0.9,0.6)(0,1)0.15 ( 0.9,0.0)(0,1)0.15 ( 0.9,-0.2)(0,1)0.15 ( 0.9,-0.4)(0,1)0.15    + ( 2d )    ( 4.0,1.0 )    ( 0.0,0.0 )    ( 4.0,1.0 )    ( 0.0,0.2 ) ( 2.0,0.2 ) ( 0.0,0.2)(1.0,-0.4)(2.0,0.2 ) ( 0.0,0.2)(1.0,0.8)(2.0,0.2 )    ( 1.0,0.2)(0,1)0.15 ( 1.0,0.4)(0,1)0.15 ( 1.0,0.6)(0,1)0.15 ( 1.0,0.0)(0,1)0.15 ( 1.0,-0.2)(0,1)0.15 ( 1.0,-0.4)(0,1)0.15     + & & + 2(2d )    ( 3.5,1.0 )    ( 0.0,0.0 )    ( 4.0,1.0 )    ( 0.0,0.0 ) ( 0.0,0.3 ) ( 2.0,0.0 ) ( 2.0,0.3 ) ( 0.0,0.0)(1.0,-0.5)(2.0,0.0 ) ( 0.0,0.3)(1.0,0.8)(2.0,0.3 )    ( 1.1,0.2)(0,1)0.15 ( 1.1,0.4)(0,1)0.15 ( 1.1,0.6)(0,1)0.15 ( 1.1,0.0)(0,1)0.15 ( 1.1,-0.2)(0,1)0.15 ( 1.1,-0.4)(0,1)0.15    } + & + & o(k^3 ) . for clarity , here we have written explicitly the topological symmetry factors and the lattice embedding numbers . ( usually graphs represent their full weights including these factors . ) note that the first two graphs of the second order contribution also occur in a usual lce with frozen @xmath20-dynamics , the next two differ by an additional dashed 2-line and the last one becomes even disconnected without the dashed line .    as usual , graphical expansions for correlation functions , in particular susceptibilities , are generated from @xmath57 by taking derivatives w.r.t . the external fields @xmath38 and @xmath40 . graphically this amounts to attaching external @xmath19-lines and @xmath20-lines with in passing we remark that the conventional lce is included as a special case of the dlce , if the @xmath20-dynamics is `` frozen '' to some value @xmath58 , so that w^1(i ) & = & -s_1(u_0)+ i u_0 , + & = & u_0 , + & = & 0 , i.e. , no n - lines do occur with @xmath59 . in this case it becomes redundant to attach dashed lines to bare lines . as mentioned above , in an lce only the first three contributions would be left in eq . ( [ 2.graph ] ) . the definition of a as it will be given here is adapted to the computation of susceptibilities , where the sum is taken over all possible locations of the fields . the definition easily generalizes to correlation functions .    for details of the standard definiton of graphs in the framework of linked cluster expansions and related notions we refer e.g.  to @xcite . here , for convenience , we briefly recall the very definition of a graph to point out the new properties of as defined below in this section . a ( standard lce ) graph or diagram is a structure = ( _ , _ , e _ , _ ) , where @xmath60 and @xmath61 are disjoint sets of internal lines and vertices of @xmath62 , respectively . @xmath63 is a map e _ : _ & & \{0,1,2 ,  } , + v & & e_(v ) that assigns to every vertex @xmath25 the number of external lines @xmath64 attached to it . finally , @xmath65 is the incidence relation that assigns internal lines to their two endpoints . a or is a structure = ( _ , _ , _ , e_^ ( ) , e_^(u ) , _ , _ ) . @xmath66 , @xmath67 and @xmath68 are three mutually disjoint sets , _ & = & , + _ & = & , + _ & = & . @xmath69 is a map e_^ ( ) : _ & & \{0,1,2 ,  } , + v & & e_^()(v ) that assigns to every vertex @xmath25 the number of bare external @xmath19-lines @xmath70 attached to @xmath25 . every such @xmath19-line represents a field @xmath19 . the number of external @xmath19-lines of @xmath71 is denoted by @xmath72 . similarly , @xmath73 is a map e_^(u ) : _ & & \{0,1,2 ,  } , + m & & e_^(u)(m ) that assigns to every multiple line @xmath74 the number of external @xmath20-lines @xmath75 attached to @xmath74 . every such @xmath20-line represents a field @xmath20 associated with a lattice link . the number of external @xmath20-lines of @xmath71 is given by @xmath76 . furthermore , @xmath77 and @xmath78 are incidence relations that assign bare internal lines to their endpoint vertices and to their multiple lines , respectively . we treat lines as unoriented . the generalization to oriented lines is easily done . more precisely , let @xmath79 be the set of unordered pairs of vertices @xmath80 with @xmath81 , @xmath82 . ( the bar implies unordered pairs , the prime the exclusions of @xmath83 , @xmath84 . ) as in the standard linked cluster expansion , self - lines are excluded . every bare internal line is then mapped onto its pair of endpoints via _ : _ ^. we say that @xmath25 and @xmath85 are the endpoint vertices of @xmath86 if @xmath87 . if there is such an @xmath86 , @xmath25 and @xmath85 are called neighbours . similarly , @xmath78 is a map _ : _ & & _ , + l & & _ ( l ) that maps every bare internal line to a multiple line . a multiple line @xmath88 is composed of bare internal lines @xmath86 which belong to @xmath74 in the sense that @xmath89 . @xmath90 is the total number of bare internal lines belonging to @xmath74 . with @xmath91 , @xmath74 is called a @xmath50-line . we always require that @xmath92 . on the other hand , every bare internal line belongs to one and only one multiple line . for simplicity we often identify a @xmath93-line with the only one bare line that belongs to it . next we introduce some further notions that will be used later . external vertices are vertices having external @xmath19-lines attached , _ , ext = \ { v _ e_^()(v)=0 } , whereas internal vertices do not , @xmath94 . similarly , external multiple lines have external @xmath20-lines attached , _ , ext = \ { m _ e_^(u)(m)=0 } , and the complement in @xmath67 are the internal multiple lines , @xmath95 .    for every pair of vertices @xmath81 , @xmath82 , let @xmath96 be the set of lines with endpoint vertices @xmath25 and @xmath85 , and @xmath97 the number of these lines . thus @xmath96 is the set of lines @xmath25 and @xmath85 have in common . with @xmath70 denoting the number of external @xmath19-lines attached to @xmath84 , t__(v ) = _ w _ ^1(v , w ) + e_^()(v ) is the total number of bare lines attached to @xmath25 . some topological notions and global properties of graphs will be of major interest in the following . a central notion is the connectivity of a . recall that we want to consider the dlce expansion of the free energy and of truncated correlation functions as an expansion in connected graphs . as indicated in section 2 , the main generalization compared to the common notion of connectivity of a graph which is required here is that an additional type of connectivity is provided by multiple - lines . to define the connectivity of a @xmath71 , @xmath71 first is mapped to a ( standard ) lce graph @xmath98 to which the standard notion of connectivity applies . there are various equivalent ways to define such a map . we choose the following one . * for every multiple - line @xmath88 define a new vertex @xmath99 . let @xmath100 and define @xmath101 as the union of the vertices of @xmath71 and the new set of vertices originating from the multiple - lines . * for every bare internal line @xmath86 define two new internal lines @xmath102 and incidence relations ( l_1 ) & = & ( v_1 , w(_(l ) ) ) , + ( l_2 ) & = & ( v_2 , w(_(l ) ) ) , where @xmath103 and @xmath104 are the two endpoint vertices of @xmath42 . the set of all lines @xmath102 , for all @xmath86 , is denoted by @xmath105 . * define the external incidence relations : & & \{0,1,2 ,  } , + ( v ) & = & e_^()(v ) , , + ( v ) & = & e_^(u)(m ) , . now , @xmath106 is defined by = ( , , , ) . having defined the standard lce graph @xmath106 for any @xmath71 , we call @xmath71 multiple - line connected or just connected if @xmath106 is connected ( in the usual sense ) . in fig .  4 we have given two examples for a connected ( upper graph ) and a disconnected ( lower graph ) .    ( 15.0,7.0 )    the next important notion is the topological equivalence of two . two [ top.equiv ] _ i = ( _ i , _ i , _ i , e^()_i , e^(u)_i , _ i , _ i ) , i=1,2 are called ( topologically ) equivalent if there are three invertible maps _ 1:_1 & & _ 2 , + _ 2:_1 & & _ 2 , + _ 3:_1 & & _ 2 , such that _ 2 _ 2 & = & _ 1 _ 1 , + _ 2 _ 2 & = & _ 3 _ 1 , and e_2^ ( ) _ 1 & = & e_1^ ( ) , + e_2^(u ) _ 3 & = & e_1^(u ) . here @xmath107 means decomposition of maps , and _ 1 : ^ & & ^ + _ 1(v , w ) & = & ( _ 1(v),_1(w ) ) . a symmetry of a @xmath108 is a triple of maps @xmath109 , @xmath110 and @xmath111 such that _ 2 & = & _ 1 , + _ 2 & = & _ 3 , and e^ ( ) _ 1 & = & e^ ( ) + e^(u ) _ 3 & = & e^(u ) . the number of these maps is called the symmetry number of @xmath71 .    we denote by @xmath112 the set of equivalence classes of connected with @xmath113 bare internal lines , @xmath114 external @xmath19-lines and @xmath115 external @xmath20-lines . furthermore we set _ e_1,e_2 : = _ l0 _ e_1,e_2(l ) . a multiple line graph @xmath71 does not need to have a vertex . if @xmath116 , we have @xmath117 as well . if in addition @xmath71 is connected , @xmath67 consists of only one element , with all external @xmath20-lines attached to it . ( we anticipate that @xmath71 is 1-multiple - line irreducible ( 1mli ) by definition . for the definition of 1mli cf . section 4 below . ) the only graph of @xmath118 is given by = . ( 4.0,1.0 )    ( 0.0,0.0 )    ( 10.0,0.0 )    ( 0.9,0.6)(1.5,0.5)(2.1,0.6 ) ( 0.9,0.4)(1.5,0.3)(2.1,0.4 ) ( 0.9,-0.5)(1.5,-0.4)(2.1,-0.5 ) ( 0.9,-0.3)(1.5,-0.2)(2.1,-0.3 ) ( 1.5,-0.55)(0,1)0.15 ( 1.5,-0.35)(0,1)0.15 ( 1.5,-0.15)(0,1)0.15 ( 1.5,0.05)(0,1)0.15 ( 1.5,0.25)(0,1)0.15 ( 1.5,0.45)(0,1)0.15 ( 1.5,0.65)(0,1)0.15 ( 1.2,0.15)(0.2,0)@xmath48 ( 1.2,-0.05)(0.2,0)@xmath48 ( 1.2,0.05)(0.2,0)@xmath48    } e . it represents the leading term of the susceptibility [ susc.0e ] _ 0,e = _ l_1,  ,l_e _ 1 < u(l_1 ) u(l_e ) > ^c and is given by @xmath119 . the index @xmath120 in ( [ susc.0e ] ) stands for truncated ( connected ) correlation .    by removal of a @xmath50-line @xmath88 we mean that @xmath74 is dropped together with all bare internal lines and all external @xmath20-lines that belong to @xmath74 . this notion is explained in fig . ( it is used in section 4 for @xmath93-lines to define 1-particle irreducible ( 1pi ) and 1-line irreducible ( 1li ) . )    on the other hand , by decomposition of a @xmath50-line @xmath88 we mean that @xmath74 is dropped together with the external @xmath20-lines of @xmath74 , but all bare internal lines that belong to @xmath74 are kept in the graph , being identified now with @xmath93-lines . this notion will be used below to define 1mli and renormalized . it is illustrated in fig . 5b .    similarly , decomposition of a vertex @xmath84 means to remove the vertex @xmath25 and to attach the free end of every line that entered @xmath25 before to a new vertex , a separate one for each line . this notion is used to define 1-vertex - irreducible ( 1vi ) and renormalized vertex moments for . for an example see fig . ( 15.0,7.0 )      in the last section we have defined and the notions of connectivity and equivalence of such graphs . the definition is chosen in such a way that the series expansions of the free energy and of truncated correlation functions are obtained as a sum over equivalence classes of connected . the number @xmath113 of bare internal lines of a @xmath71 counts the order in the expansion parameter @xmath26 to which @xmath71 contributes . if @xmath26 is of the form [ susc.vform ] v(x , y ) = 2k _ z(x ) _ y , z , with @xmath121 any finite @xmath39-dependent set of lattice sites , the contribution of @xmath71 is a multiple of @xmath122 . often used special cases are the nearest neighbour interactions [ susc.nn ] v(x , y ) = 2k _ = 0^d-1 ( _ x , y+ + _ x , y- ) and the uniform interaction [ susc.mf ] v(x , y ) = 2k ( 1 - _ x , y ) , which is used in models of spin glasses and partially annealed neural networks . susceptibilities of the @xmath19 and @xmath20 fields will be represented as [ susc.susc ] _ e_1,e_2 & = & _ x_1,  ,x_e_1 _ 0 _ l_1,  ,l_e_2 _ 1 < ( x_1 ) ( x_e_1 ) u(l_1 ) u(l_e_2 ) > ^c + & & _ x_1,  ,x_e_1 _ 0 _ l_1,  ,l_e_2 _ 1 . _ h = i=0 + & = & _ l0 ( 2k)^l _ _ e_1,e_2(l ) w ( ) with lattice volume @xmath52 and dimension @xmath3 . similar representations hold for higher moments @xmath55 . the weight @xmath123 of a @xmath124 is given as the product of the following factors    * for every vertex @xmath84 a factor v^c_n = ( ) _ h=0 , where @xmath125 is the total number of bare lines attached to @xmath25 . * for every multiple line @xmath88 a factor [ susc.m1c ] m^1 c _ = ( ) _ i=0 , where @xmath91 , that is @xmath74 is a @xmath50-line , * a factor @xmath126 , where @xmath127 is the topological symmetry number of @xmath71 , * a factor counting the permutation symmetry of external @xmath19-lines , , * a factor counting the permutation symmetry of external @xmath20-lines , , * the lattice embedding number of @xmath71 , which is the number of ways @xmath71 can be embedded on a lattice of given geometry , e.  g.  on a hypercubic lattice . to this end , the vertices of @xmath71 ( if any ) are placed onto lattice sites . one arbitrary vertex is placed at a fixed lattice site , in order to account for the volume factor @xmath128 in ( [ susc.susc ] ) . a priori there is no exclusion principle . this means that any number of vertices can be placed at the same lattice site . ( this is sometimes called free embedding . ) two restrictions apply to the embeddings . the first constraint results from the fact that a bare internal line represents a hopping propagator @xmath26 , with lattice sites @xmath39 and @xmath41 at which the two endpoint vertices of the line are placed at . a reasonable computation of the embedding number takes into account the particular form of @xmath26 from the very beginning . the second constraint is that bare lines of the same multiple - line have to be mapped on the same pair of sites . + for example , if @xmath26 is the nearest neighbour interaction ( [ susc.nn ] ) , two vertices which have at least one line in common are to be placed at nearest neighbour lattice sites . on the other hand , a propagator @xmath26 of the form ( [ susc.mf ] ) implies a rather weak constraint in that @xmath39 and @xmath41 must be different , but otherwise can be freely placed over the lattice . we remark that in case of a non - trivial internal symmetry ( such as considered in section 7 ) the expressions of eq.s  ( [ susc.susc])-([susc.m1c ] ) must be modified appropriately . in particular , the weight ( [ susc.m1c ] ) of a multiple - line does no longer take such a simple form . ( [ su2.labgraph ] ) below is an example for the case of a hopping term originating in an su(2 ) higgs model . truncated correlation functions , susceptibilities and other moments are obtained as sums over that are connected . their number rapidly grows with increasing order , that is with increasing number of bare internal lines . the procedure of `` renormalization '' means that the connected moments are represented in terms of reduced ones . the reduced moments are obtained by summation over classes which are more restricted than just by their property of being connected . of course the number of graphs of such classes is smaller . only the most restricted classes must be constructed . the subsequent steps towards the moment computation are most conveniently done by operating analytically with the reduced moments . in particular , it is no longer necessary to generate all connected and the corresponding intermediate classes . a connected @xmath71 is called 1-particle irreducible ( 1pi ) if it satisfies the following condition . remove an arbitrary @xmath93-line of @xmath71 . there is at most one connected component left that has external lines attached . ( this notion is the same as the one used in the context of feynman graphs . ) on the other hand , if in addition the remaining graph is still connected , then @xmath71 is called 1-line irreducible ( 1li ) . in many cases it is sufficient to use only the second notion . it is for instance sufficient that all vertices are constrained to have only an even number of lines attached , or more generally , if graphs and subgraphs with one external line are forbidden . for notational simplicity we assume in the following that this is the case and henceforth refer only to the notion 1li . the generalization to the case in which 1li and 1pi graphs must be distinguished goes along the same lines as for lces , which was discussed in @xcite .    by @xmath129 we denote the subset of @xmath124 that are 1li . 1li - susceptibilities are defined as series in the hopping parameter similarly as in ( [ susc.susc ] ) by restricting the summation to 1li graphs , _ e_1,e_2 ^ 1li = _ l0 ( 2k)^l _ _ e_1,e_2 ^ 1li(l ) w ( ) . susceptibilities are easily obtained in a closed form in terms of 1li - susceptibilities @xmath130 . it can be shown that the @xmath130s can be obtained by an appropriate legendre transform . for instance _ 2,0 & = & , + _ 2,1 & = & , where @xmath131 is the fourier transform of the hopping propagator @xmath26 , v(x , y ) = _ -^ e^-ik(x - y ) ( k ) .    in lces the second important resummation comes from so called vertex renormalizations . this means partial resummation of graphs with specific properties such as having one external vertex only . these sums then are considered as `` renormalized vertices '' replacing the vertices of graphs with complementary properties . the procedure naturally leads to the notion of 1-vertex irreducibility ( 1vi ) and renormalized moments .    in dlce we follow this procedure . the very definition of 1vi has to be modified slightly for because of the enhanced connectivity properties due to multiple - lines . in addition , as a natural generalization , we supplement vertex renormalization by multiple - line renormalization . a @xmath71 is called 1-vertex irreducible ( 1vi ) if it satisfies the following condition . decompose an arbitrary vertex @xmath84 . every connected component of the remaining graph has then at least one external line attached . it can be a @xmath19-line or a @xmath20-line . we write _ e_1,e_2 ^ 1vi(l ) = \ { _ e_1,e_2 ^ 1li(l ) } for the set of equivalence classes of graphs that are both 1li and 1vi , with @xmath114 external @xmath19-lines , @xmath115 external @xmath20-lines and @xmath113 bare internal lines . the renormalized vertex moment graphs are 1li graphs that have precisely one external vertex and no external multiple line , q_k(l ) = \ { _ k,0 ^ 1li(l ) } .    a @xmath71 is called 1-multiple - line irreducible ( 1mli ) if it satisfies the following criterion . decompose an arbitrary multiple - line @xmath88 . every remaining connected component has then at least one external line attached . it can be a @xmath19-line or a @xmath20-line . we write _ e_1,e_2 ^ 1mli(l ) = \ { _ e_1,e_2 ^ 1li(l ) } . the renormalized multiple - line moment graphs are graphs that are 1li and have precisely one external multiple - line , but no external vertex , r_k(l ) = \ { _ 0,k^1li(l ) } . the equivalence classes of graphs that are both 1vi and 1mli are denoted by s_e_1,e_2(l ) = _ e_1,e_2 ^ 1vi(l ) _ e_1,e_2 ^ 1mli(l ) .    with the renormalized moment graphs as defined above , the 1li - susceptibilities are now obtained in the form _ e_1,e_2 ^ 1li = _ l0 ( 2k)^l _ _ e_1,e_2(l ) ( ) . the weights @xmath132 are given as a product of factors as described in the last subsection , with the following two exceptions . * the vertex coupling constants @xmath133 are replaced by the renormalized vertex moments v^c_n v^c_n = _ l0 ( 2k)^l _ q_n(l ) w ( ) . * the multiple line coupling constants @xmath134 are replaced by the renormalized multiple line moments m^1 c _ m^c_= _ l0 ( 2k)^l _ r_(l ) w ( ) .    in the series representations above , the @xmath123 are computed according to the rules of subsection 3.2 . in this section we describe a mechanism to generate the classes @xmath135 , @xmath136 and @xmath137 with @xmath138 internal lines . similar as in lces the idea is to define appropriate classes of vacuum graphs ( no external lines ) . the generation of the graphs of @xmath139 , @xmath140 and @xmath141 is then done by attaching external lines in various ways . by defining appropriate recursion relations , the order by order construction can then be restricted to the vacuum classes . fortunately we can profit from graphs that have been generated already in standard lces . this way we completely avoid a cumbersome recursive construction , but obtain the vacuum by operating on the lce vacuum graphs of the same order .    the graphs of @xmath139 , @xmath140 and @xmath141 will be obtained through a sequence of simpler graph classes . at each step , any method of construction should satisfy the following two conditions . * it should be surjective or complete . at least one graph of every equivalence class should be generated . * for every equivalence class precisely one representant should be kept . all generated graphs of a particular class of graphs should be mutually inequivalent . this requires a so called non - equivalence test for every newly generated graph . the graph should be kept if and only if it is not equivalent to a graph already generated . in general it must be compared with all that have been generated before . if this is necessary , the non - equivalence test is rather time consuming . the algorithm we describe below circumvents an extensive comparison of . it is defined in such a way that two graphs can be equivalent only if they have their origin in the same graph of the prior graph class . essentially we proceed in the following steps . the starting point are the vacuum diagrams of @xmath142 as computed in @xcite . @xmath142 is the set of ( standard ) lce equivalence classes of connected , 1li vacuum graphs with @xmath113 lines . operating on @xmath142 , we generate all mutually inequivalent with @xmath113 lines that are connected , 1li and have no external lines ( neither @xmath19-lines nor @xmath20-lines ) , _ 2(l ) _ 0,0 ^ 1li(l ) . this is done in two steps . in a first step lines with the same endpoint vertices are arranged in all possible ways into multiple lines . we obtain a subset of @xmath143 which we denote by @xmath144 . a @xmath145 belongs to @xmath144 , @xmath146 , if and only if all bare lines of @xmath71 that belong to the same multiple line have the same endpoint vertices . the second step is to relax the additional constraint on @xmath144 . this is achieved by resolving the vertices . this means that every vertex is split into several ones , and that the bare lines that were attached to the original vertex before now are distributed over the new vertices in all possible ways . as a general constraint on the resolutions , the resulting should be 1li in order to stay in @xmath143 . we impose an additional constraint on the resolutions , allowing only for so - called admissible vertex resolutions . they are defined in such a way that two of @xmath147 can be equivalent only if they originate from the same graph of @xmath144 . this considerably simplifies the non - equivalence test . once we have obtained @xmath147 , the final classes @xmath135 , @xmath136 and @xmath137 are obtained by attaching external @xmath19-lines and @xmath20-lines in the appropriate way to all graphs of @xmath147 . the steps in the generation of the dlce graph classes are shown schematically in fig . ( 15.0,2.5 )    ( 0.0,1.0)(2.0,0)@xmath142 ( 2.3,1.0)(1,0)0.9 ( 4.0,1.0)(2.0,0)@xmath148 ( 6.3,1.0)(1,0)0.9 ( 8.0,1.0)(2.0,0)@xmath143 ( 10.3,1.0)(1,1)0.9 ( 10.3,1.0)(1,0)0.9 ( 10.3,1.0)(1,-1)0.9 ( 12.0,2.0)(2.0,0)@xmath135 ( 12.0,1.0)(2.0,0)@xmath136 ( 12.0,0.0)(2.0,0)@xmath137    in the following we describe the construction in detail . below we use the following notation . for any finite set @xmath149 let @xmath150 denote the set of partitions of @xmath149 into mutually disjoint nonempty subsets of @xmath149 . for every @xmath151 we have = _ p p.      starting with the lce vacuum graphs of @xmath142 , we first of all arrange internal lines into multiple lines . consider an arbitrary graph @xmath153 . simultaneously for every pair of neighbouring vertices @xmath81 choose a partition @xmath154 , where @xmath155 is the set of common lines between the vertices @xmath25 and @xmath85 . every @xmath156 belongs to a unique @xmath157 . for every @xmath158 we introduce a multiple line @xmath159 such that all @xmath160 are precisely the bare internal lines that belong to @xmath159 . this procedure defines the incidence relation @xmath161 for all @xmath86 and , along with that , a that belongs to @xmath144 .    in fig . 7 we give examples for this construction . in the first one , an lce graph leads to 5 mutually inequivalent dlce gaphs , in the second one to 3 dlce graphs of @xmath148 .    proceeding this way for all partitions of lines for all pairs of neighboured vertices of @xmath71 , and for all @xmath153 , we obtain the class of @xmath144 . the non - equivalence check for every newly generated @xmath62 can be drastically restricted . it is sufficient to compare @xmath62 only with those that have been constructed before from the same graph @xmath153 as @xmath62 results from . other graphs of @xmath144 are inequivalent to @xmath62 . ( 15.0,7.0 )      the second step is to relax the constraint on @xmath163 that bare lines which belong to the same multiple line must have the same endpoint vertices . this is achieved by vertex resolutions of all graphs of @xmath144 . to efficiently reduce the non - equivalence check of new graphs we allow only for so - called admissible vertex resolutions . let @xmath164 , @xmath165 any vertex of @xmath62 and @xmath166 any partition of the set of lines @xmath167 entering @xmath25 . we remove the vertex @xmath25 and draw for every @xmath158 a new vertex @xmath168 so that all lines @xmath160 enter the vertex @xmath168 rather than @xmath25 before its removal . this procedure is called a vertex resolution of @xmath25 . the resulting graph @xmath71 must be 1li in order to belong to @xmath143 . the vertex resolution of @xmath25 is called admissible if in addition the ( standard lce ) graph @xmath169 is connected with @xmath169 defined in the following way . for every @xmath170 there is precisely one @xmath171 with @xmath172 , and there is exactly one @xmath158 with @xmath160 . draw a vertex @xmath173 for every such multiple line , a vertex @xmath168 for every @xmath158 , and define a line @xmath174 with incidence relation ( ) = ( w(_(l ) ) , v(p ) ) . this defines a graph @xmath175 with _ & = & \ { v(p ) p } \ { w(_(l ) ) l_v } + _ & = & \ { ( l ) l_v } . a vertex resolution is called admissible if the graph @xmath169 is connected . roughly speaking this implies that no vertex drops off after vertex resolution .    fig . 8 shows two admissable and one non - admissible vertex resolutions ( the auxiliary graphs @xmath169 are not displayed ) . ( 15.0,7.0 )    the class @xmath143 is now generated as follows . consider an arbitrary graph @xmath176 . simultaneously for all vertices @xmath165 apply an admissible vertex resolution . we obtain a @xmath145 . doing this for all possible resolutions of all vertices and for all @xmath176 , we get the complete graph class @xmath143 . again , due to the restriction to admissible vertex resolutions , the non - equivalence check of a newly generated graph @xmath145 can be restricted to graphs that have been generated from the same graph of @xmath144 as @xmath71 was .    in fig . 9 we show an example for the application of admissible vertex resolutions . here one graph of @xmath148 leads to eight graphs of @xmath143 , indicating the proliferation of dlce graphs . note that the starting graph was only one of five of fig . ( 15.0,5.5 )      the final step is to construct the classes of @xmath180 , @xmath181 and @xmath182 out of the vacuum of @xmath183 . this step is realized by attaching external @xmath19-lines and external @xmath20-lines in different ways to every @xmath145 . @xmath179 is obtained by attaching @xmath184 external @xmath20-lines to just one multiple line @xmath88 , for all multiple lines @xmath74 and for all @xmath71 . @xmath178 is obtained by attaching @xmath184 external @xmath19-lines to just one vertex @xmath84 , for all vertices @xmath25 and for all @xmath71 . @xmath135 is obtained by attaching @xmath114 external @xmath19-lines to the vertices of @xmath71 and @xmath115 external @xmath20-lines to the multiple lines of @xmath71 , for all @xmath145 . this is done under the constraint that the resulting have to be 1vi and 1mli . again , the non - equivalence test of can be confined to pairs of graphs that have their origin in the same graph of @xmath143 .    in fig . 10 we illustrate the attachment of external lines to a vacuum dlce graph of @xmath143 . ( 15.0,5.5 ) in this section we indicate applications of dlces to spin glasses and partially annealed neural networks with `` slow '' interactions and `` fast '' spins @xcite-@xcite , @xcite .      for simplicity we consider the sherrington - kirkpatrick ( sk ) model which is a spin glass model with infinite connectivity @xcite . following the notation of section 2 , the fast spins are represented by the @xmath185-field @xmath186 , @xmath187 , taking values in @xmath188 . @xmath189 denotes the total number of spins . the slow interactions are mediated by the @xmath20-field @xmath190 , @xmath191 . the sk model is described by the partition function [ neur.z ] z_^^ = _ -^_i < j = 1^n du_(i , j ) , with @xmath9 some normalization that will be specified below . the effective hamiltonian @xmath192 of @xmath20 is given by [ neur.h ] ( u ) = - + n _ i < j=1^n u_(i , j)^2 , where @xmath55 is a positive coupling constant . @xmath193 is the partition function of a spin system [ neur.zspin ] z_(u ) = _ \ { _ i = 1 } ) at frozen spin - spin couplings @xmath190 and temperature @xmath194 , controlling the fast spin fluctuations . in ( [ neur.z ] ) it has been assumed that the equilibrium distribution of the slow variables is a boltzmann distribution governed by a second temperature @xmath195 . this assumption is justified , if the time evolution of the @xmath20s is determined by a dissipative langevin equation ( for the precise conditions see e.g.  @xcite ) . such a langevin equation can be actually derived for the @xmath20s from an ansatz which is motivated by neural networks @xcite-@xcite . one of the major quantities of interest is @xmath196 . let us first rewrite @xmath197 in the form [ neur.zre ] z_n^ & = & _ -^_i < j = 1^n ( du_(i , j ) ) ) z_(u)^n + & & [ [ z_(u)^n ] ] , where we have introduced @xmath198 and @xmath199 . the normalization has been chosen such that @xmath200 = 1 $ ] . the limiting case @xmath201 with @xmath202 kept fixed corresponds to a quenching of the @xmath20s in a spin glass , in this limit the @xmath20 degrees of freedom decouple from the spin dynamics , while @xmath203 corresponds to a pure annealing in the disordered system ( see e.g.  @xcite ) . partial annealing then refers to a value of @xmath204 with @xmath205 . in particular the limit @xmath206 with fixed @xmath202 and @xmath207 amounts to the replica trick in spin glasses @xcite . assuming that averaging @xmath208 $ ] and limes @xmath206 commute , we have [ neur.rep ] [ [ ( u ) ] ] = _ n0 . the left hand side of ( [ neur.rep ] ) is calculated by means of the right hand side . so far @xmath204 has been a positive real number . for integer @xmath209 , @xmath210 is the partition function of an n - times replicated system . let us first consider this case . we rewrite [ neur.zspin1 ] z_(u)^n = _ \{_i^(a ) } , with @xmath211 labelling the replicated spin variables , so that [ neur.zspin2 ] z_n^ & = & _ -^_i < j = 1^n du_(i , j ) _ \{_i^(a ) = 1 } , + s(u,^(a ) ) & = & -_a=1^n _ j=1^n u_(i , j ) _ i^(a ) _ j^(a ) + j n _ i < j u_(i , j)^2 . linear terms in @xmath185 and @xmath20 may be included according to [ neur.zspin3 ] s_lin = - h _ a=1^n _ i=1^n _ i^(a ) + c _ i < j=1^n u_(i , j ) with constant external fields @xmath212 and @xmath120 . apparently @xmath213 has the form of models to which dlce applies , with a hopping term [ neur.hop ] s_hop(u,^(a ) ) = - _ a=1^n _ i < j=1^n u_(i , j ) _ i^(a ) _ j^(a ) , a single link action [ neur.link ] s^1(u_(i , j ) ) = c u_(i , j ) + j n u_(i , j)^2 , and a single site action [ neur.site ] s^(_i^(a ) ) = - h _ a=1^n _ i^(a ) . application of dlce to @xmath214 and derived quantities yields their series expansions in @xmath207 . so far , merely for simplicity , we have considered the sk model which is a spin glass model with infinite and uniform connectivity , in which all spins are coupled with equal variance @xmath215 = 1/(j n)$ ] . for integer @xmath216 , this system becomes exactly solvable in the thermodynamic limit corresponding to the limit @xmath217 , so that the leading order of a large-@xmath189 expansion becomes exact , cf . e.g.  @xcite . in dlce this amounts to a resummation of appropriate tree graphs .    in cases of finite or non - uniform connectivity it is no longer possible to obtain solutions in a closed form . examples are multilayered neural networks with couplings between adjacent layers , and spin glasses with short range interactions . models of this kind are suited for an application of dlces and will be studied in future work .    finally we come to the limiting case of @xmath201 . usually it is obtained by extrapolating the results obtained for @xmath209 towards @xmath201 . for instance , @xmath218 $ ] is obtained from @xmath219 $ ] for @xmath209 . this is a rather subtle point because the extrapolation is not unique without further assumptions . in the next subsection we show how it is possible to completely avoid both the extrapolation and the replica trick and to directly compute the quantity of interest by means of dynamical linked cluster expansions . the task is to appropriately identify the classes of contributing graphs . first we adapt the notation to section 2 to include more general cases . @xmath2 denotes the support of the spins , that is the set of lattice sites , with @xmath220 denoting their total number . @xmath8 @xmath221 are the pairs of sites whose spins interact . in accordance with ( [ neur.zre ] ) , we write for the normalized link - average of a function @xmath222 [ sg.average ] [ [ f(u ) ] ] = u f(u ) with u & = & _ l_1 d(u(l ) ) , + d(u ) & = & _ 1 du , _ -^d(u ) = 1 . it is convenient to introduce the single link expectation values < g(u ) > _ 1 d(u ) g(u ) and the generating function @xmath223 by < > _ 1 . the way in which the replica trick can be avoided is examplified for the free energy density @xmath224 of the spin system averaged over the link couplings . the partition function of the spin system for a given distribution of the link interactions @xmath225 is given by = _ sp , where @xmath226 and s_sp(,u ) & = & - _ x , y_0 v(x , y ) ( x ) ( y ) u(x , y ) , + & = & _ x_0 d(x ) . without loss of generality we identify the support of the interaction @xmath227 with the set @xmath228 of lattice sites where @xmath229 is supported , _ 1 = \ { l=(x , y ) v(x , y)= 0}. for simplicity we assume @xmath26 to be of the form ( [ susc.vform ] ) , so that @xmath1 is a measure of the strength of the interactions @xmath26 . the free energy density of the spin system allows for a series expansion in the standard lce sense , with the link field @xmath13 playing the role of a `` background field '' , [ sg.wsp_series ] w_sp(u ) = _ l0 ( 2k)^l _ ^sp_0(l)w^sp(,u ) . here @xmath230 ( with @xmath231 ) denotes the set of equivalence classes of connected lce graphs with @xmath232 external lines and @xmath113 internal lines . the spin - weights @xmath233 are of the form [ sg.wsp_weight ] w^sp(,u ) = r^sp ( ) _ _ _ 1^ _ l_1 u(l)^m(l ) . the sum is taken over all non - vanishing lattice embeddings of the graph @xmath71 . it runs over all maps of internal lines of the graph @xmath71 to pairs of lattice sites of @xmath8 that are consistent with the graph topology in the sense discussed in section 3 . for every @xmath14 , @xmath234 denotes the number of lines of @xmath71 that are mapped onto the link @xmath42 by the embedding . all other factors that contribute to the weight are collected in the prefactor @xmath235 , including the inverse topological symmetry number of @xmath71 . next we want to express @xmath236 $ ] as a series in @xmath1 by means of dlce . toward this end we set @xmath237 and insert the series ( [ sg.wsp_series ] ) with ( [ sg.wsp_weight ] ) into ( [ sg.average ] ) . at this stage we are not concerned with question of ( uniform or dominated ) convergence and obtain [ sg.wspint ] [ [ w_sp(u ) ] ] & = & _ l0 ( 2k)^l _ _ 0^sp(l ) u w^sp(,u ) + & = & _ l0 ( 2k)^l _ _ 0^sp(l ) r^sp ( ) _ _ _ 1^ _ l_1 < u(l)^m(l)>_1 . the next step is to express the full single link expectation values in terms of the connected ones . they are related by [ sg.conn ] < u^m > _ 1 = _ ( ) _ p < u^|p| > _ 1^c , where @xmath238 denotes the set of all partitions of @xmath239 into non - empty , mutually disjoint subsets of @xmath240 . @xmath241 is the number of elements of the set @xmath242 . the relation ( [ sg.conn ] ) is equivalent to the partition of all lines of @xmath71 that are mapped to the same lattice link into multiple lines , with every multiple line contributing a factor [ sg.wdiff ] < u^|p| > _ 1^c = . _ i=0 = m^1 c_|p| . using ( [ sg.conn ] ) , ( [ sg.wdiff ] ) we rewrite ( [ sg.wspint ] ) as [ sg.wsptodlce ] [ [ w_sp(u ) ] ] = _ l0 ( 2k)^l _ _ 0^sp(l ) r^sp ( ) _ ( _ ) ( _ p m^1 c_|p| ) ( _ _ 1^ _ l_1 1 ) . the last summation in ( [ sg.wsptodlce ] ) is over all maps @xmath243 of the lines of @xmath71 to the lattice links of @xmath8 subject to the constraint that all lines that belong to the same multiple - line corresponding to some @xmath158 are mapped onto the same lattice link . finally we rewrite ( [ sg.wsptodlce ] ) as a sum over . to this end , we first observe that for every @xmath244 , every partition @xmath245 of the lines of @xmath71 into multiple - lines generates a @xmath246 in the obvious way . let us denote by @xmath247 the subset of of @xmath248 that stay connected after decomposition of all multiple lines . ( these are the which stay connected in the usual graph theoretical sense , when the dashed lines are omitted . ) for every @xmath249 there is a unique @xmath250 and at least one @xmath251 such that @xmath252 . let @xmath253 be the ( uniquely determined ) number of partitions @xmath251 with @xmath252 , and @xmath254 such an arbitrary partition . ( [ sg.wsptodlce ] ) then becomes [ sg.wdlce ] [ [ w_sp(u ) ] ] = _ l0 ( 2k)^l _ _ 0,0(l ) n_r^sp ( ( ) ) ( _ p ( ) m^1 c_|p| ) ( _ ( ) _ 1^ 1 ) . the last bracket of ( [ sg.wdlce ] ) is the lattice embedding factor of the @xmath255 . the second bracket from the right does not depend on the choice of @xmath254 and is the product of the multiple - line coupling constants as defined in section 3 . finally , @xmath256 is precisely the remaining part of the weight of @xmath255 that was described in detail in section 3 , endowed with the correct inverse topological symmetry number of the @xmath255 ( because of the factor @xmath253 ) .    in summary , we obtain the series expansion of the link - averaged free energy density in terms of dlce graphs , [ sg.wseries ] [ [ w_sp(u ) ] ] = _ l0 ( 2k)^l _ _ 0,0(l ) w ( ) . the weight @xmath257 of a @xmath255 is defined and computed according to the rules given in section 3 . ( [ sg.wseries ] ) is the series representation of the link - averaged free energy density of the spin system , i.e.  the free energy density of the @xmath201 replica system , in terms of dlce graphs . it looks much like the series representation of the @xmath93-replica system , which is given by [ sg.w1series ] w_1-repl = _ l0 ( 2k)^l _ _ 0,0(l ) w ( ) according to the discussion of section 2 . we recall that @xmath248 is the set of dlce vacuum graphs with @xmath113 bare lines that are connected in the generalized dlce sense . comparing ( [ sg.wseries ] ) and ( [ sg.w1series ] ) , the transition from @xmath203 to @xmath201 replicas is achieved by keeping only the subset @xmath258 of that are connected in the original ( lce ) sense . we expect that the series ( [ sg.wseries ] ) are convergent for a large class of interactions @xmath30 and @xmath26 if the coupling constant @xmath1 is sufficiently small . for special interactions most of the yield vanishing contributions so that we can further restrict the sum to a subset of @xmath259 . an example is given by the mean field type of interaction of the sk model where all pairs of sites are coupled with the same strength . that is , in the notation of this subsection , v(x , y ) & = & k ( 1-_x , y ) , + _ 1 & & _ 1 + s^1(u ) & = & v u^2 . in the thermodynamic limit @xmath260 only those survive that consist exclusively of @xmath261-lines , and that do not contain any loop of @xmath261-lines . the resulting tree structure provides the possibility of summing the series ( [ sg.wseries ] ) and of analytically continuing the result to the spin glass phase of large coupling @xmath1 . work in this direction is in progress . in this section we discuss applications of dlces to study the higgs transition within a variational estimate for the gauged su(2 ) higgs model . the estimate serves as an effective description of the electroweak standard model . we consider a 4-dimensional hypercubic finite temperature lattice @xmath2 of size @xmath262 , with @xmath263 the inverse temperature in lattice units and @xmath264 the spatial volume . the lattice links are the set of nearest neighbour lattice sites , given by _ 1 = \ { ( x ; ) x_0 , = 0,  ,3 } . the gauge field @xmath265 is an su(2 ) valued field living on the lattice links @xmath8 . it is convenient to parametrize such an su(2 ) matrix by [ su2.uparam ] u & = & _ 0(u ) _ 2 + i ( u ) + & & u_0 + i with @xmath266 . the higgs field @xmath185 lives on the lattice sites @xmath2 . its values are real multiples of su(2 ) , [ su2.phiparam ] ( x ) & = & _ 0((x ) ) _ 2 + i ( ( x ) ) + & & _ 0(x ) + i ( x ) , with @xmath267 . we say that @xmath185 is @xmath268 valued .    as a special case of eq . ( [ 2.zgen ] ) we consider the partition function [ su2.zdlce ] z_ve1 = u , in which u = _ _ = 0 ^ 3 d_h(u(x ; ) ) , with @xmath269 the normalized haar measure on su(2 ) , and [ su2.phimeasure ] = _ x_0 d((x ) ) , where d ( ) & = & d^4 , + s^ ( ) & = & + ( - 1 ) ^2 + & = & ^2 + ( ^2 - 1 ) ^2 . finally , [ su2.svei ] s_ve1(u , ) & = & - _ x_0 \ { 4 _ link _ 0(u(x;0 ) ) + _ = 1 ^ 3 4 _ cube _ 0(u(x ; ) ) + & & + _ = 1 ^ 3 ( 2 ) + & & + _ 0((x ) ) } , depending on variational parameters @xmath270 , @xmath271 and @xmath272 @xmath21 , and on the hopping parameter @xmath273 . the first two terms are non - gauge invariant substitutes for the wilson plaquette term of the gauge part of the su(2 ) higgs model , cf . ( [ su2.1.wilson ] ) below . we distinguish timelike from spacelike directions . timelike links are denoted as @xmath274 and spacelike as @xmath275 , @xmath276 . the third term is a hopping term coupling @xmath185 and @xmath20 fields only in spatial directions . the last term depends on the third variational parameter @xmath272 associated with the higgs field @xmath185 , while the remaining ultralocal action for @xmath185 has been absorbed in the measure .    in the next subsection we first indicate how we arrive at the action ( [ su2.svei ] ) within a variational estimate for the free energy of the su(2 ) higgs model . we derive equations for the optimal choice of variational parameters and an equation -based on stability arguments- to determine the critical hopping parameter @xmath277 at which the higgs phase transition sets in . these equations will depend on expectation values of n - point correlations that are most suitably evaluated with a dlce in three dimensions . the generic graphical expansion exposed in sections 3 - 5 now has to be adapted to account for the internal symmetry . this is somewhat involved so that we restrict our discussion to some remarks in section 7.2 , but postpone any details to a forthcoming paper @xcite . in section 7.3 we present some results for the critical line in comparision with other variational estimates and monte carlo simulations .      by means of dlces we want to obtain an analytic estimate for the critical line in the su(2 ) higgs model that serves as an effective description of the higgs transition in the electroweak standard model . because of the rather different methodical ansatz in this approach , for a common choice of parameters this estimate will provide an independent check of numerical results based on monte carlo simulations . in addition , larger scalar field couplings are available in our approach and larger values for @xmath278 , the number of time slices . ( in our calculations @xmath279 or @xmath280 , while it was chosen as 2 or 3 in @xcite . ) larger @xmath278s are required for a finite size scaling control .    to fix the notation we introduce the action of the su(2 ) higgs model as [ su2.1.higgs ] z & = & u , + s(u , ) & = & s_w(u ) + s_hop , + s_hop & = & - _ x_0 \ { _ = 0 ^ 3 ( 2 ) } . the higgs self - interaction has been absorbed in the measure @xmath281 of eq . ( [ su2.phimeasure ] ) , @xmath282 is the hopping parameter term in 4 dimensions . @xmath283 is the su(2 ) gauge invariant wilson action [ su2.1.wilson ] s_w(u ) = _ x_0 _ _ ve ) . eq . ( [ su2.1.var ] ) follows from the convexity of the exponential function and holds independently of the specific choice of @xmath284 and @xmath285 , if the measure is positive definite and normalized . @xmath52 is the 4-dimensional volume , @xmath286 denotes the true physical free energy density defined via @xmath284 , @xmath287 the trial free energy density defined via the second equality in ( [ su2.1.var ] ) . @xmath288 refers to the original action in @xmath284 , in our case it is the su(2 ) higgs action , and @xmath289 to an ansatz for the action in @xmath285 depending on a generic set of variational parameters @xmath290 . these parameters should be optimized so that the difference of the trial and the physical free energy density @xmath291 becomes minimal for @xmath292 at which @xmath287 takes its minimum . the most naive ansatz for @xmath289 is a mean field ansatz in the spirit of a molecular field approximation , leading to a factorization of @xmath293 according to [ su2.1.zve0 ] z_ve0 ( , ) = z_link^4 l_0 v_3 ( ) z_site^l_0 v_3 ( ) . the partition function depends on two variational parameters @xmath290 and @xmath272 and factorizes in a product over single link ( @xmath294 ) and single site ( @xmath295 ) partition functions , @xmath264 denotes the three - dimensional volume . in this ansatz there is no space for implementing an asymmetry between temporal and spatial directions . thus the results will be temperature independent by construction . one can think of a variety of improvements of the molecular field ansatz , arguments against or in favour of the various versions will be given in @xcite . here we only consider the ansatz which is most plausible from a physical point of view and leads to the best agreement with monte carlo results for comparable sets of parameters . the ansatz treats the spacelike degrees of freedom of the hopping term in 3 dimensions beyond the mean field level ( that is with dlce ) , but the timelike degrees of freedom within a mean field approach , so that the partition function factorizes over the spacelike hyperplanes . the reason is that the spatial hopping term is supposed to contain the nonperturbative properties of the full model that drive the higgs phase transition . while a mean field approach for all variables will be too rough to produce high quality results for @xmath277 , it appears more reasonable for timelike variables , although a finite temperature effect on the higgs transition gets lost this way . ( we should remark that from results in a scalar @xmath0 theory in four dimensions the finite temperature effect on @xmath277 is expected to be anyway quite small @xcite . ) now it is easily checked that the choice of @xmath296 in eq . ( [ su2.svei ] ) leads to a factorization of @xmath297 according to [ su2.1.zvei ] z_ve1(_link,_cube , ) = z_cube(_cube,)^l_0 z_link(_link)^l_0 v_3 . the partition function @xmath298 of the spacelike degrees of freedom reads [ su2.1.zcube1 ] z_cube = _ x_0^(3 ) ( d((x ) ) e^-s^((x ) ) _ = 1 ^ 3 d_h(u(x ; ) ) ) , with @xmath299 the 3-dimensional lattice and [ su2.1.zcube2 ] s_cube(u , ) & = & - _ x_0^(3 ) \ { _ = 1 ^ 3 4 _ cube _ 0(u(x ; ) ) + _ 0((x ) ) + & & + _ = 1 ^ 3 ( 2 ) } , while @xmath294 is an ultralocal one - link integral [ su2.1.z0 ] z_link = d_h(u ) with [ su2.1.s0 ] s_link(u ) = 4 _ link _ 0(u ) . expectation values @xmath300 , @xmath301 , @xmath302 of observables @xmath303 refer to @xmath298 , @xmath294 and @xmath297 , respectively . minimization of the trial free energy density @xmath304 in terms of these expecation values leads to three equations . the first one @xmath305 is solved by @xmath306 in the symmetric phase . the remaining two equations are given by [ su2.1.mfeq1 ] & & + | ( < _ 0(u ) > _ cube)^3 + ( < _ 0(u ) > _ link)^2 < _ 0(u ) > _ cube - _ cube + & & < _ 0(u ) > _ cube = 0 and [ su2.1.mfeq2 ] ( < _ 0(u ) > _ cube)^2 < _ 0(u ) > _ link - _ link < _ 0(u ) > _ link = 0 . here we have used the shorthand notation [ su2.1.plaq ] < _ 0(u ) > _ cube & & < _ 0(u(x;1 ) ) > _ cube , + _ cube^1 , 2 & & < tr u(x;1 ) u(x+;2 ) u(x+;1)^-1 u(x;2)^-1 > _ cube + & - & ( < _ 0(u(x;1))>_cube)^4 , with @xmath307 . in ( [ su2.1.plaq ] ) we have used the lattice symmetries . eq.s  ( [ su2.1.mfeq1 ] ) , ( [ su2.1.mfeq2 ] ) should be solved for @xmath270 , @xmath271 , and @xmath272 as series in @xmath273 . the stability condition for the symmetric minimum at @xmath306 is given as [ su2.1.stab ] & & 3 _ = 0 + & & + 12 | ( < _ 0(u ) > _ cube)^3 + < _ 0(u ) > _ cube ( < _ 0(u ) > _ link)^2 - _ cube + & & < _ 0(u ) > _ cube _ = 0 + & & + ( 2 ) 2 < _ 0(u ) > _ link < _ 0 ( ) > _ cube _ = 0 - 1 < _ 0 ( ) > _ cube _ = 0 + & & + & & < 0 , with < _ 0 ( ) > _ cube < _ 0((x ) ) > _ cube , @xmath307 . for an equality sign , ( [ su2.1.stab ] ) determines @xmath277 order by order in the expansion . note the two type of expectation values @xmath308 , @xmath309 entering eq.s  ( [ su2.1.mfeq1 ] ) , ( [ su2.1.mfeq2 ] ) , ( [ su2.1.stab ] ) . while the single link expectation values @xmath308 can be evaluated exactly , the @xmath309s must be approximated . the derivatives of @xmath310 w.r.t . @xmath271 or twice w.r.t . @xmath272 induce up to 5-point functions in @xmath20 and 6-point functions in four @xmath20s and two @xmath185s . because of the bad signal / noise ratio monte carlo calculations of such connected correlations would not be feasible . therefore this is the place , where we use a dlce to evaluate expectation values of the type @xmath311 ( all internal and configuration space indices suppressed ) order by order in @xmath273 . clearly , also in this scheme the price is high . for example , the number of connected dlce graphs contributing to order @xmath312 to a 4-point function of 2 @xmath20s and 2 @xmath185s is about 100 . not all correlations in ( [ su2.1.mfeq1]),([su2.1.mfeq2]),([su2.1.stab ] ) appear in the form of susceptibilities . in a product of 4 @xmath20s , for instance , the configuration space indices are fixed to the boundary of a plaquette . such features must be noticed for calculating the graphical weights . these remarks may indicate the complexity of the actual evaluation of eq.s  ( [ su2.1.mfeq1 ] ) , ( [ su2.1.mfeq2 ] ) , ( [ su2.1.stab ] ) . here we omit any further details on the list of contributing graphs and postpone them to @xcite . before we quote some results , we comment on further subtleties , when dlces are adapted to internal symmetries of the hopping term . in this section we focus on the ( gauge ) group @xmath313 , since we are interested in applications to the electroweak phase transition , but it should be obvious , how one could proceed along the same lines for a group @xmath314 with @xmath315 . in the development of the theory in sections 3 - 5 we had assumed that @xmath316 , @xmath317 to simplify the notation . in eqs . ( [ su2.uparam ] ) and ( [ su2.phiparam ] ) we had @xmath318 , @xmath319 . let us write the hopping part of the action in the symmetrized form    [ su2.hop1 ] s_hop = - _ x , y_0 v(x , y ) ( x)^u(x , y ) ( y ) , with hopping parameter v(x , y ) = 2 _ = 0 ^ 3 ( _ y , x+ + _ y , x- ) and    c@=c u(x , x+ ) & u(x ; ) + u(x+,x ) & u(x;)^-1 + u(x , y ) & 0    , = 0,  ,3 , using the parametrizations ( [ su2.uparam ] ) and ( [ su2.phiparam ] ) we have [ su2.hop2 ] & & ( x)^u(x , y ) ( y ) = ( x ) ( y ) u_0(x , y ) + & & + _ 0(y ) ( x ) ( x , y ) - _ 0(x ) ( y ) ( x , y ) + & & + ( x ) ( y ) ( x , y ) .    note that in the parametrization in terms of @xmath320s and @xmath321s the @xmath22 part of the action is o(4 ) invariant , but @xmath296 is only o(3 ) invariant for @xmath322 or @xmath323 . thus we have to distinguish between singlet @xmath324 and triplet @xmath325 indices in internal space . recall the generating equations ( [ 2.genequation ] ) of the graphical expansion of a dlce in which @xmath326 had been expressed in terms of connected @xmath204-point functions , @xmath327 . using the @xmath328 symmetry it follows from eq . ( [ su2.hop2 ] ) that each derivative @xmath329 now has to be replaced by the differential operator [ su2.lab ] l_ab(x , y ) = _ ab + _ = 1 ^ 3 ( _ a , _ b,0 - _ b , _ a,0 + _ ab ) . latin indices run from @xmath330 to @xmath331 , greek indices from @xmath93 to @xmath331 . furthermore we have introduced the totally antisymmetric @xmath332-symbol which is zero if any of the three indices is zero and @xmath333 . this implies for instance for the one nonvanishing term at @xmath32 in eq . ( [ 2.genequation ] ) a substitution according to [ su2.ww ] w_h(x)w_h(y)w_i(x , y ) w_h^a(x)w_h^b(y ) l_ab(x , y)w . according to the graphical rules an @xmath204-line was generated by @xmath204 iterated derivatives w.r.t . @xmath334 , evaluated at @xmath32 . these derivatives now have to be replaced by a multiple application of @xmath113 yielding an n - line    [ su2.labgraph ]    ( 5.0,1.0 )    ( 0.0,-0.6 )    ( 4.0,0.0 )    ( 1.0,1.5)(2.0,2.1)(3.0,1.5 ) ( 1.0,1.0)(2.0,1.6)(3.0,1.0 ) ( 1.0,0.0)(2.0,0.6)(3.0,0.0 )    ( -0.2,0.7)(0.2,0)@xmath39 ( 4.2,0.7)(0.2,0)@xmath41    ( 0.6,1.5)(0.2,0)@xmath335 ( 3.3,1.5)(0.2,0)@xmath336    ( 0.6,1.0)(0.2,0)@xmath337 ( 3.3,1.0)(0.2,0)@xmath338    ( 0.6,0.0)(0.2,0)@xmath339 ( 3.3,0.0)(0.2,0)@xmath340    ( 1.6,1.1)(0.2,0)@xmath48 ( 1.6,0.8)(0.2,0)@xmath48 ( 1.6,0.5)(0.2,0)@xmath48    ( 2.0,1.85)(0,1)0.2 ( 2.0,1.6)(0,1)0.2 ( 2.0,1.35)(0,1)0.2 ( 2.0,1.1)(0,1)0.2 ( 2.0,0.85)(0,1)0.2 ( 2.0,0.6)(0,1)0.2 ( 2.0,0.35)(0,1)0.2 ( 2.0,0.1)(0,1)0.2    = _ = 0 ^ 3 ( _ y , x+ + _ y , x- ) ( 2)^n . ( _ i=1^n l_a_ib_i ) w^1(i ) _ i=0 where the @xmath93-link generating function @xmath223 is defined by = .    for @xmath341 for all @xmath342 , @xmath204 applications of @xmath343 to @xmath344 reduce to @xmath345 , but otherwise we get a complicated mixing of singlet and triplet terms . to evaluate products of @xmath113s systematically , it is again quite convenient to manipulate graphical rather than analytic expressions . we represent one application of @xmath343 on @xmath344 as shown in fig . ( [ lab ] ) . multiple applications of @xmath343 require an iteration of fig . ( [ lab ] ) . ( 15.0,4.0 )    twiddle lines are bare @xmath19-lines with a singlet index @xmath346 , crossed lines are bare @xmath19-lines with a triplet index @xmath347 . the arrow in the last term stands for the @xmath348 symbol in @xmath343 , its orientation reflects the order of indices @xmath349 .    this way one achieves a further graphical `` decomposition '' of an n - line into terms merely consisting of bare lines , from which the internal symmetry factors can be easily read off . for example a 2-line is further decomposed into 11 terms , a 3-line into 29 and a 4-line into 142 . this may indicate that the number of decomposed contributions rapidly increases with the number @xmath204 of bare lines collected to an n - line . in our actual calculations up to order @xmath312 the maximal number of internal lines was 4 and the highest multiple line was a 6-line with 4 internal and 2 external u - lines . some of the results will be reported in the next section .      in table 1 we compare results for @xmath277 obtained for 2 values of @xmath350 and @xmath351 and various lattice volumes within three methods . the first line refers to a naive molecular field approximation with a partition function @xmath293 factorizing according to eq . ( [ su2.1.zve0 ] ) . because of the complete factorization in space and time directions the ansatz keeps only a trivial volume dependence and is temperature independent by construction . the second line refers to monte carlo simulations of the su(2 ) higgs model in four dimensions @xcite for two lattice extensions @xmath352 and @xmath331 . the third line shows results of the dlces with the variational ansatz ( ve1 ) presented in section 7.2 . again this ansatz is independent of @xmath278 by construction , so that we get an identical result for @xmath353 . the values for @xmath277 have been obtained by a linear regression in @xmath354 , @xmath355 denoting the order in @xmath273 of the dlce . for the regression we used the coefficients of @xmath356 , @xmath357 , @xmath358 and @xmath312 . the value of @xmath277 at order @xmath312 was given by @xmath359 for @xmath360 and @xmath361 . although the fourth order seems to be low for a series expansion , the agreement with the monte carlo results is not too surprising , if we remind the complexity of the expansion , which is manifest in the number of graphs contributing at fourth order to the various expectation values . furthermore , for otherwise fixed parameters , we see a decrease of @xmath277 between the monte carlo and the dlce results for increasing @xmath278 . this is in qualitative agreement with the expectation . .[kc.1 ] results for @xmath277 for 2 values of @xmath362 and @xmath351 , various lattice extensions , and 3 methods . the monte carlo results are taken from [ 15 ] , the mean field and dlce results , extrapolated to infinite order , from this paper . ( the error in the dlce results refers to the extrapolation of @xmath277 . ) [ cols=">,^,^ , < , < " , ]     the partition function @xmath363 factorizes acording to [ su2.zveii ] z_ve2(_(0,string),_(s , link ) , ) = z_(0,string)(_(0,string),)^v_3 z_(s , link)(_(s , link))^3 l_0 v_3 , i.e. , a product of a partition function @xmath364 that couples fields with hopping parameter terms along a string in time direction , and @xmath365 , a partition function for a single spatial link . while expectation values w.r.t . @xmath365 can be evaluated exactly ( within the numerical accuracy ) , expectation values w.r.t . @xmath364 are preferably evaluated with dlce . this way they do depend on @xmath278 , in principle they are sensitive to detect a temperature effect on @xmath277 . so far the couplings along a string are only kept in time direction , while the spacelike directions are incorporated on a mean field level in @xmath365 . the third variational ansatz ve3 therefore is chosen as a symmetrized version of ve2 in that the string now can extend in any of the 4 directions . again we treat space - and timelike degrees of freedom differently , because we are interested in systems at finite temperature with temperature dependent coupling @xmath277 . the number of variational parameters is larger , each of it depending on two indices telling us whether the parameter belongs to a single link action or to a string , and whether the string extends in temporal or spacelike directions .    for both versions ve2 and ve3 the actual @xmath278 dependence ( @xmath366 ) lies outside the systematic error , if we expand the series to order @xmath312 . for @xmath279 the fourth order is the minimal order in @xmath273 at which a finite temperature effect is visible in principle . ( for the graphs it should be possible to wind at least once around the torus in @xmath330-direction . ) at o@xmath367 the part of graphs for which the lattice embedding is sensitive to @xmath278 is about @xmath368 of the total number of graphs . such a small `` signal@xmath369noise '' ratio produces an effect in the 7th or 8th digit , outside the error of the @xmath370 because of the truncation of the dlce series . by increasing the order of the expansion , finite - temperature effects will become more pronouced . for higher orders an algorithmic implementation of the proliferating graphs becomes unavoidable . in this paper we have introduced a new expansion scheme for 3-point interactions or , more precisely , for point - link - point interactions . this scheme generalizes linked cluster expansions for 2-point interactions by including hopping parameter terms endowed with their own dynamics . in chapters 3 - 5 we have developed a theory with an additional new type of multiple - line connectivity . we have introduced appropriate equivalence classes of graphs and discussed the issue of renormalization . the main building blocks for an algorithmic generation of graphs have been constructed . because of the fast proliferation of graphs already at low orders in the expansion , a computer aided implementation becomes unavoidable , if one is interested in higher orders of the expansion than we have computed so far .    in chapter 6 we have indicated promising applications to spin glass systems . in particular we have identified the dlce graphs contributing to the link - average @xmath371 $ ] of the spin free energy @xmath372 . in the past such quantities often were only accessible by means of the replica trick .    in section 7 we have outlined encouraging results for the transition line of the electroweak phase transition within the su(2)-higgs model . the results have been obtained by dlces applied to gap equations that follow from variational estimates of the free energy . they are in good agreement with corresponding high precision monte carlo results . as we have seen , the main complications of dlces for su(n ) higgs systems come from the internal symmetry structure of the hopping term . in a forthcoming paper we will demonstrate how one has to refine the graphical representation to calculate internal symmetry factors merely within a graphical expansion .    in conclusion , dlces are involved , but practicable , at least with computer aided generation of graphs . they provide an analytical tool to study systems in situations in which it has been impossible so far . we would like to thank reimar khn ( heidelberg ) for pointing out to us ref.s @xcite-@xcite . 9 m.  wortis , `` linked cluster expansion '' , in phase transition and critical phenomena , vol.3 , eds . c.  domb and m.s . green ( academic press , london 1974 ) . c.  itzykson , j .- drouffe , `` statistical field theory '' , vol.2 , cambridge university press , 1989 . guttmann , `` asymptotic analysis of power - series expansions '' , in phase transition and critical phenomena , vol.13 , eds . c.  domb and j.l . lebowitz ( academic press ) . m.  lscher and p.  weisz , nucl .  phys . * b300*[fs22 ] ( 1988 ) 325 . t.  reisz , nucl .  phys . * b450 * ( 1995 ) 569 . t.  reisz , phys .  lett . * 360b * ( 1995 ) 77 . m.  campostrini , a.  pelissetto , p.  rossi and e.  vicari , nucl .  phys . * b459 * ( 1996 ) 207 . s.  zinn , s .- n .  lai and m.  e.  fisher , phys . * e54 * ( 1996 ) 1176 . p.  butera and n.  comi , phys . * e55 * ( 1997 ) 6391 . h.  meyer - ortmanns and t.  reisz , jour .  stat * 87 * ( 1997 ) 755 . a.  pordt and t.  reisz , int . * a12 * ( 1997 ) 3739 - 3757 . d.  sherrington and s.  kirkpatrick , phys . lett * 35 * ( 1975 ) 1972 . v.  dotsenko , s.  franz and m.  mzard , j.  phys . * a27 * ( 1994 ) 2351 . r.  w.  penney , a.  c.  c.  coolen , and d.  sherrington , j.  phys .  a : math . * 26 * ( 1993 ) 3681 . a.  c.  c.  coolen , r.  w.  penney , and d.  sherrington , phys .  rev . * b48 * ( 1993 ) 16116 . z.  fodor , j.  hein , k.  jansen , a.  jaster and i.  montvay , nucl . * b439 * ( 1995)147 . t.  reisz , `` hopping parameter series construction for models with nontrivial vacuum '' , hep - lat/9802023 , submitted to nucl . b. k.  h.  fischer and j.  a.  hertz , `` spin glasses '' , cambridge university press , cambridge 1991 . j.  zinn - justin , `` quantum field theory and critical phenomena '' , 3rd edition , oxford science publication , clarendon press , oxford 1996 . h.  meyer - ortmanns and t.  reisz , in preparation .
dynamical linked cluster expansions are linked cluster expansions with hopping parameter terms endowed with their own dynamics . 
 this amounts to a generalization from 2-point to point - link - point interactions . 
 we develop an associated graph theory with a generalized notion of connectivity and describe an algorithmic generation of the new . 
 we indicate physical applications to spin glasses , partially annealed neural networks and su(n ) gauge higgs systems . 
 in particular the new expansion technique provides the possibility of avoiding the replica - trick in spin glasses . 
 we consider variational estimates for the su(2 ) higgs model of the electroweak phase transition . 
 the results for the transition line , obtained by dynamical linked cluster expansions , agree quite well with corresponding high precision monte carlo results .