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this note considers a stochastic growth model in the _ kpz anisotropic class _ in . this model was introduced in  @xcite and studied in depth for a specific initial condition , the case considered here . this model describes the evolution of particles subject to an _ interlacing property_. the model can also be thought of as a two - dimensional _ stochastically growing interface _ , or as a _ random lozenge tiling model_. another tiling model which shares similar features of the dynamical perspective is _ domino tilings of the aztec diamond _ using _ the shuffling algorithm _ ; see  @xcite . for the model considered in this paper , the evolution of the interface , under hydrodynamic scaling , grows deterministically according to a pde . at the microscopic level in the bulk , the specified boundary conditions of the system are _ forgotten _ : in the bulk of the system one sees an invariant measure which depends only on the normal direction of the macroscopic surface  @xcite . from the lozenge tiling perspective , these limiting measures are determinantal and they are parameterized by the relative proportions of lozenges . these measures match up with the dimer model on the infinite honeycomb graph which are the unique translation invariant stationary measures for any given normal direction  @xcite . stochastic growth models have been studied in many different guises . many of these studies have focussed around the 1 + 1 kpz universality class ; see for example  @xcite for surveys . the @xmath0 anisotropic case has not been as extensively studied  @xcite but there are , however , numerous results in connection with random tiling models  @xcite . one feature of these models which is of particular interest for this note is the _ limit shape _ this is the average profile which the stochastic interface fluctuates around . in particular , we focus on giving an elementary approach to computing the _ speed of growth _ , which is the growth rate of the stochastic growing interface under the prescribed dynamics . this is an important quantity since it determines the limit shape of the system .    in the work by borodin and ferrari  @xcite , the speed of growth was obtained by computing the _ infinitesimal current _ and then taking the bulk scaling limit . the computation was relatively straightforward , but requires the knowledge of the correlations of particles or lozenges at two different times . recently toninelli in  @xcite , considered the same model and a generalized version of the dynamics for the infinite honeycomb graph using a _ bead _ perspective ; beads and the _ bead model _ were introduced in  @xcite . he shows that the model is well - defined for the stationary measure . this requires some effort since the dynamics allow , a - priori , infinitely long - range interactions , but under the stationary measure these interaction probabilities decay exponentially with the distance . the speed of growth was not determined in  @xcite .    by appealing to the underlying combinatorics of the model and the kasteleyn approach for dimer models ( e.g. see  @xcite ) , we are able to determine the speed of growth in the infinite honeycomb case , thus establishing the conjecture in  ( * ? ? ? * eq . ( 3.6 ) ) ; see theorem  [ thminfinitesystem ] . from  @xcite , the speed of growth @xmath1 is given by a specific ( unsigned ) off - diagonal entry of the inverse of the kasteleyn matrix , where the kasteleyn matrix is a type of ( possibly signed ) adjacency matrix  @xcite . the approach used is to first find a recursive formula for this particular entry for the honeycomb graph embedded on a torus which leads to a series , and then take the toroidal exhaustion limit . the resulting limiting series matches with the speed of growth defined through the dynamics under the stationary measure . to prove theorem  [ thminfinitesystem ] , we found a relation for the specific off - diagonal entry of the inverse of the kasteleyn matrix for any _ tileable _ finite honeycomb graph with arbitrary edge weights in terms of single times . as a consequence , we use this relation to motivate an algebraic proof that the expression for the speed of growth of the model in  @xcite and the speed of growth computed from the prescribed dynamics are the same ( the latter is a series with entries given by determintants of increasing size ) . in principle it seems to be feasible to obtain theorem  [ thminfinitesystem ] from theorem  [ thmfinitesystem ] by a precise asymptotic analysis and a careful manipulation of sums , but we did not pursue this since we are primarily interested in understanding combinatorial structures behind the identity . * acknowledgments . * the authors are grateful for discussions with f. toninelli about his work and to both icerm and the galileo galileo institute which provided the platform to make such discussions possible . the work is supported by the german research foundation via the sfb 1060b04 project . we first describe the interacting particle system model introduced in  @xcite . it is a model in the @xmath2-dimensional kpz anisotropic class . it is a continuous time markov chain on the state space of interlacing variables @xmath3 @xmath4 is interpreted as the position of the particle with label @xmath5 , but we will also refer to a given particle as @xmath4 . we consider fully - packed initial conditions , namely at time moment @xmath6 we have @xmath7 for all @xmath8 ; see figure  [ figureinitialconditions ] . [ b]@xmath9 [ b]@xmath10 [ b]@xmath11 [ cl]@xmath12 [ cl]@xmath13 [ cl]@xmath14 [ cl]@xmath15 [ cl]@xmath16 [ cl]@xmath17 [ cl]@xmath18 [ cl]@xmath19 [ cl]@xmath20 [ cl]@xmath21 [ cl]@xmath22 [ cl]@xmath23 [ cl]@xmath24 [ cl]@xmath25 [ cl]@xmath26     the particles evolve according to the following dynamics . each particle @xmath4 has an independent exponential clock of rate one , and when the @xmath4-clock rings the particle attempts to jump to the right by one . if at that moment then the jump is blocked . otherwise , we find the largest @xmath27 such that , and all @xmath28 particles in this string jump to the right by one . we illustrate the dynamics using figure  [ figureintro ] , which shows a possible configuration of particles obtained from the fully - packed initial condition . in this state of the system , if the @xmath14-clock rings , then the particle @xmath14 does not move , because it is blocked by particle @xmath13 . if the @xmath17-clock rings , then the particle @xmath17 moves to the right by one unit , but to respect the interlacing property , the particles @xmath21 and @xmath24 also move by one unit to the right at the same time . this aspect of the dynamics is called _ pushing_.    [ b]@xmath9 [ b]@xmath10 [ b]@xmath11 [ cl]@xmath12 [ cl]@xmath13 [ cl]@xmath14 [ cl]@xmath15 [ cl]@xmath16 [ cl]@xmath17 [ cl]@xmath18 [ cl]@xmath19 [ cl]@xmath20 [ cl]@xmath21 [ cl]@xmath22 [ cl]@xmath23 [ cl]@xmath24 [ cl]@xmath25 [ cl]@xmath26     [ remlozenge ] the positions of the particles uniquely determine a lozenge tiling in the region bordered by the thick line in figure  [ figureintro ] .    [ remmeasureunif ] as shown in  @xcite , the measure at time @xmath29 generated by the dynamics starting from the fully - packed initial condition has the property that , conditioned on the measure of the particles at level @xmath30 , the other particles are uniformly distributed on @xmath31 with fixed level @xmath30 configuration . further , it is proved in theorem  1.1 of  @xcite that the correlation function of the particles are determinantal on a subset of space - time  @xcite ( see  @xcite for information on determinantal point processes ) . denote by @xmath32 the random variable that is equal to @xmath33 if there is a particle at @xmath34 at time @xmath29 and @xmath35 otherwise . [ thmdetstructure ] for any given , consider , and @xmath36 . then @xmath37=\det{[{\cal k}(x_i , n_i , t_i;x_j , n_j , t_j)]}_{i , j=1}^m,\ ] ] where @xmath38 } \\ + \frac{1}{(2\pi i)^2}\oint_{\gamma_0}{\mathrm{d}}w\oint_{\gamma_1}{\mathrm{d}}z \frac{e^{t_1/w}}{e^{t_2/z } } \frac{(1-w)^{n_1}}{(1-z)^{n_2 } } \frac{w^{x_1}}{z^{x_2 + 1}}\,\frac{1}{w - z},\end{gathered}\ ] ] the contours @xmath39 , @xmath40 are simple positively oriented closed paths that include the poles @xmath35 and @xmath33 , respectively , and no other poles ( hence , they are disjoint ) . here we used the notation @xmath41    given a configuration of particles , we define the height @xmath42 as @xmath43 in particular , the growth rate of the height at a position @xmath34 is given by the ( infinitesimal ) particle current at @xmath34 , denoted by @xmath44 with @xmath45.\ ] ]    this quantity was computed in the proof of lemma  5.3 of  @xcite with the result @xmath46 where here we use the notation @xmath47 .    on the other hand , at any time @xmath29 , the height function @xmath11 at @xmath34 increases by one whenever there is a particle at position @xmath34 which jumps to . by definition of the dynamics , at rate 1 a particle at @xmath34 could jump to the right provided that there is no particle at @xmath48 . however , a particle at @xmath34 can also be pushed if there is a column of particles directly below @xmath34 . more explicitly , for @xmath49 a particle at @xmath34 is pushed to the right by the move of a particle at @xmath50 . this happens at rate @xmath33 provided that @xmath51 is empty and @xmath52 for @xmath53 are occupied . in the case @xmath54 , the constraint that @xmath55 is empty does not exist as there are no particles at level @xmath35 . therefore the growth rate obtained from the dynamics , denoted by @xmath56 , is given by @xmath57+{\mathbbm{e}}\left[\prod_{k=0}^{n-1 } \eta(x , n - k , t)\right].\ ] ]    using theorem  [ thmdetstructure ] and the complementation principle for determinantal point processes ( see appendix of  @xcite ) , the expected value in the right side of ( [ eqspeed ] ) is given by @xmath58_{i , j=0}^{\ell-1 } & \left[-{\cal k}_t(x , n - i;x+1,n-\ell)\right]_{i=0}^{\ell-1}\\ \left[{\cal k}_t(x+1,n-\ell;x , n - j)\right]_{j=0}^{\ell-1 } & 1-{\cal k}_t(x+1,n-\ell;x+1,n-\ell ) \end{array } \right].\ ] ]    in this paper we show directly that indeed @xmath1 and @xmath59 are the same .    [ thmfinitesystem ] it holds @xmath60    this finite - time result can be easily generalized to the case of level - dependent jump rates . this system is still determinantal when starting with fully - packed initial conditions , with correlation kernel obtained in proposition 3.1 of  @xcite ; see corollary  2.26 of  @xcite too . the height function , defined in  , gives a three dimensional surface with cartesian coordinate axis . in a projection in the @xmath61-direction , each unit square of the surface becomes a lozenge , while in the projection of figure  [ figureintro ] there are three types of parallelograms , still referred to as lozenges ; see figure  [ figuretilings ] . [ c]i - @xmath62 [ c]ii - @xmath63 [ c]iii - @xmath28   facets types and their associated lozenges and angles . the gray circle is at position @xmath34 . for dimers , their coordinate is given by the black site . more precisely , in this case we say that dimer of type i is at @xmath34 , of type ii is at @xmath48 , while of type iii at @xmath64.,title="fig:",height=226 ]    another representation useful for the proofs of the theorems in this paper , is through _ perfect matchings _ or _ dimers _ via the dual graph associated to the underlying graph of the lozenge tilings . for a lozenge tiling on a finite graph , the dual graph is a subgraph of the _ honeycomb _ ( or _ hexagonal _ ) graph , with each lozenge representing an edge , called a dimer , on this dual graph ; see figure  [ figuretilings ] . each lozenge tiling represents a _ dimer covering _ of the dual graph ; see figure  [ figuredimers ] for an example . [ r]@xmath34 [ l]@xmath65 [ r]@xmath48 [ l]@xmath66     for the purpose of this paper , we denote @xmath67 to be the infinite ( bipartite ) honeycomb graph ; see the left side of figure  [ figcoordinates ] for a finite snapshot . we set @xmath68 , that is , the restriction of length - size @xmath69 of @xmath67 with periodic boundary conditions . we use the standard terminology for the dimer model ; see for example section 1 in  @xcite for details . now we consider the case where the state space are particles on @xmath70 satisfying interlacing between levels @xmath10 and @xmath71 , for any @xmath72 , i.e. , @xmath73 formally one would like to consider the dynamics on @xmath74 as described for the model in section  [ sectfinitet ] . however , the dynamics are not well - defined for all elements in @xmath75 due to the possibility of pushing from an infinitely long stack of particles . it is shown in  @xcite , that the dynamics are almost surely well - defined starting from the stationary measure because under this measure , the probability of having a stack of particles of length @xmath76 decays exponentially in @xmath76 . the translation invariant stationary measures form a two - parameter family , uniquely determined by the `` two - dimensional slope '' of the height function . in terms of dimers , we give to each type of dimer a weight , say @xmath77 as indicated in figure  [ figuretilings ] . the probability of a dimer configuration is then proportional to the product of the weights of each dimer configuration , thus we effectively have only two free parameters . one nice property of stationary measures on @xmath70 is that their correlation functions are determinantal with a correlation kernel given only in terms of the slope ( which is determined by @xmath77 ) . correlations functions , at different times for the model on @xmath70 with stationary initial conditions , are not explicitly known ( and for the asymmetric version studied by toninelli  @xcite they are unlikely to be determinantal ) and therefore one can not use ( [ eq2.6 ] ) to determine the speed of growth . however , the speed of growth is defined by the dynamics via ( [ eqspeed ] ) with @xmath10 replaced by @xmath78 . sheffield showed that indeed the translation invariant measures are uniquely determined by the slope of the height function . [ thmsheffield ] for each @xmath79 with @xmath80 with @xmath81 there is a unique translation - invariant ergodic gibbs measure @xmath82 on the set of dimer coverings of @xmath83 , for which the height function has average normal @xmath84 . this measure can be obtained as the limit as @xmath85 of the uniform measure on the set of those dimer coverings of @xmath86 , whose proportion of dimers in the three orientations is @xmath87 , up to errors tending to zero as @xmath85 . moreover every ergodic gibbs measure on @xmath83 is of the above type for some @xmath84 . the correlation functions of the stationary measure @xmath82 are determinantal and they are given in terms of the so - called inverse kasteleyn matrix , denoted by @xmath88 ( see more details in section  [ sect3.2 ] and the lecture notes  @xcite for a complete treatment of the subject ) . it is given as follows ( it is obtained from eq . ( 4 ) in  @xcite with appropriate change of coordinates ) @xmath89 with @xmath90 as shown by kenyon , okounkov and sheffield in  @xcite , the latter is the limiting inverse kasteleyn matrix obtained in the toroidal exhaustion limit where the edge weights @xmath91 and @xmath28 are depicted in figure  [ figuretilings ] . kenyon introduced a very useful mapping from @xmath92 to the upper - half complex plane @xmath93 illustrated in figure  [ figgeometry ] . [ l]@xmath94 [ r]@xmath95 [ c]@xmath96 [ c]@xmath35 [ c]@xmath33   associated with @xmath92 is given by constructing the triangle with basis @xmath97 and angles @xmath98 , @xmath99 as indicated . the length of the segment @xmath100 is @xmath95 and the length of the segment of @xmath101 is @xmath94.,title="fig:",height=151 ]    by change of variables @xmath102 and @xmath103 in eq . one obtains @xmath104 a simple computation ( see appendix ) leads to the following representation of the kernel @xmath105 where for @xmath106 the integration contour crosses @xmath107 , while for @xmath108 the contour crosses @xmath109 . this is consistent with theorem  5.1 and proposition  3.2 of  @xcite , where the correlation kernel was obtained by taking the finite system considered in theorem  [ thmfinitesystem ] and followed by taking the large time / space limit in such a way that the local normal direction of the surface is given by @xmath92 : this introduced a conjugation factor and a shift in the @xmath9 by @xmath10 . finally , the point in the complex plane @xmath110 in  @xcite equals @xmath111 here . ] . [ thminfinitesystem ] consider the particle model on @xmath70 distributed according to @xmath82 , that is , with determinantal correlation functions given by the correlation kernel ( [ eqapp1 ] ) . then the speed of growth is given by @xmath112    the speed of growth of the model defined in  @xcite is then given by , where @xmath113 are the two parameters in the model of  @xcite . the totally asymmetric dynamics is the one given by @xmath114 and @xmath115 . the algebraic proof presented here is strongly inspired by the combinatorial proof for finite graphs obtained beforehand that will be used to prove theorem  [ thminfinitesystem ] . in this proof we will suppress all the @xmath29 indices and have reasonably sized matrices , we use the notation @xmath116    we use the coordinate system illustrated in figure  [ figcoordinates ] . we derive a series expansion of @xmath117 by expanding it step - by - step . the idea behind the expansion is that @xmath117 is ( intuitively ) suggestive of an `` edge '' between @xmath118 and @xmath119 . if this `` edge '' is covered by a dimer , then either there is a dimer covering @xmath120 or not . in the latter case , then there are dimers covering @xmath121 and @xmath122 . this can be repeated for @xmath123 and for @xmath124 . the latter case will not occur and therefore the series naturally ends . this idea is exploited in depth for the finite graph lozenge tiling ; see proposition  [ proprecursiongraph ] . we start with two algebraic identities satisfied by the kernel ( [ startingkernel ] ) . [ lemidentity ] it holds @xmath125    it is quite trivial . one uses linearity of the integrals . [ lemidentityb ] it holds @xmath126    it is quite trivial . one uses linearity of the integrals . we start with a proposition that will be used recursively . [ proprecursionfinitet ] consider any set of @xmath127 @xmath128 black / white couples of ( disjoint ) vertices that do not include black vertices at @xmath129 and white at @xmath130 . then @xmath131=\det\left[\begin{array}{lll } { \cal k}_{x_i , m_i;x_j',m_j ' } & { \cal k}_{x_i , m_i;x , m } & { \cal k}_{x_i , m_i;x+1,m}\\ { \cal k}_{x , m;x_j',m_j ' } & { \cal k}_{x , m;x , m } & { \cal k}_{x , m;x+1,m } \\ { \cal k}_{x , m-1;x_j',m_j ' } & { \cal k}_{x , m-1;x , m } & { \cal k}_{x , m-1;x+1,m } \end{array}\right]\\ & + \det\left[\begin{array}{llll } { \cal k}_{x_i , m_i;x_j',m_j ' } & { \cal k}_{x_i , m_i;x , m } & { \cal k}_{x_i , m_i;x+1,m-1 } & { \cal k}_{x_i , m_i;x+1,m}\\ { \cal k}_{x , m;x_j',m_j ' } & { \cal k}_{x , m;x , m } & { \cal k}_{x , m;x+1,m-1 } & { \cal k}_{x , m;x+1,m}\\ { \cal k}_{x , m-1;x_j',m_j ' } & { \cal k}_{x , m-1;x , m } & { \cal k}_{x , m-1;x+1,m-1 } & { \cal k}_{x , m-1;x+1,m}\\ { \cal k}_{x+1,m-1;x_j',m_j ' } & { \cal k}_{x+1,m-1;x , m } & { \cal k}_{x+1,m-1;x+1,m-1 } & { \cal k}_{x+1,m-1;x+1,m } \end{array}\right ] , \end{aligned}\ ] ] where , whenever there is a @xmath132 or @xmath59 index , this means a block - matrix with @xmath132 and/or @xmath59 from @xmath33 to @xmath127 . schematically this is represented in figure  [ figdimerrec ] , where ( [ eq3.3 ] ) equals ( a)=(c1)+(c2 ) . [ c]@xmath133 [ c]@xmath134 [ c]@xmath135 [ c](a ) [ c](b1 ) [ c](b2 ) [ c](c1 ) [ c](c2 ) . the link in ( a ) is between @xmath136.,title="fig:",height=113 ]    the left side of ( [ eq3.3 ] ) is represented in figure  [ figdimerrec](a ) . the scheme of figure  [ figdimerrec](b1 ) is given by @xmath137.\ ] ] the scheme of figure  [ figdimerrec](b2 ) is given by @xmath138,\ ] ] we subtract from the last column the previous two . using the identity in lemma  [ lemidentity ] we then obtain @xmath139\\ & = \det\left[\begin{array}{lll } { \cal k}_{x_i , m_i;x_j',m_j ' } & { \cal k}_{x_i , m_i;x+1,m } & { \cal k}_{x_i , m_i;x , m}\\ { \cal k}_{x , m;x_j',m_j ' } & { \cal k}_{x , m;x+1,m } & { \cal k}_{x , m;x , m } \\ { \cal k}_{x , m-1;x_j',m_j ' } & { \cal k}_{x , m-1;x+1,m } & { \cal k}_{x , m-1;x , m } \end{array}\right ] . \end{aligned}\ ] ] the determinants in ( [ eq3.5 ] ) and ( [ eq3.8 ] ) differs only by the last row . thus summing them up and using the identity in lemma  [ lemidentityb ] one immediately sees that the last row and the second - last row are identical except for an extra @xmath140 term in the last matrix entry . therefore we have recovered left side of ( [ eq3.3 ] ) . finally one has to verify the equality between the schemes of figure  [ figdimerrec](b1)/(b2 ) and figure  [ figdimerrec](c1)/(c2 ) . this is trivial since it corresponds to permuting the position of one column and take care of the signature of the permutation . now we are ready to finish the proof of theorem  [ thmfinitesystem ] . the schematic representation of the proof is in figure  [ figprooffinitet ] . [ c]@xmath133 [ c]@xmath134 [ cb]@xmath141 .,title="fig:",height=113 ]    recall that @xmath142 is the random variable of a particle located at @xmath34 . similarly , denote by @xmath143 the random variable of having a white lozenge ( type iii in figure  [ figuretilings ] ) at @xmath34 , where the position is given by the one of the black triangle . using proposition  [ proprecursionfinitet ] repeatedly by starting with the case @xmath144 and @xmath145 and recalling the correspondence between dimers and lozenges ( see figure  [ figuretilings ] ) we get @xmath146.\ ] ]    [ l]forced [ l]only two choices ) and ( [ eqspeed]).,title="fig:",height=113 ]    the conditions of theorem  [ thmfinitesystem ] stipulate that the system is bounded from below by level @xmath33 and this is the reason why the above series is finite . finally we have to see that ( [ eq3.11 ] ) and ( [ eqspeed ] ) match . first consider @xmath147 . as illustrated in figure  [ figprooffinitetb ] , by observing that there are particles at @xmath148 , it implies that the square lozenges at occur with probability one and thus we remove them from the right side of ( [ eq3.11 ] ) . further , if we have a particle at @xmath50 , then either there is a particle at @xmath51 or there is a white lozenge at @xmath51 since the other type of lozenge do not fit . therefore we can replace in ( [ eq3.11 ] ) @xmath149 with @xmath150 . finally , for @xmath54 , @xmath55 is forced to be white and therefore we can remove it from the product too . this finishes the proof of theorem  [ thmfinitesystem ] . the strategy is first to obtain a recursion relation analogue of proposition  [ proprecursionfinitet ] for a finite honeycomb graph with generic weights , then use this result to extend the recursion relation to @xmath86 with @xmath77 weights . using theorem  [ thmsheffield ] we take the toroidal exhaustion limit and have the same recursion relation for the infinite honeycomb graph @xmath151 . the proof of theorem  [ thminfinitesystem ] ends applying recursively the recursion relation . let @xmath152 be a finite subgraph of the honeycomb graph which is tileable by dimers . @xmath153 is bipartite and has the same number of white and black vertices . the geometry of the honeycomb graph is as illustrated in figure  [ figcoordinates ] so that the black vertices are in ( a subset of ) @xmath70 and the white vertices are in ( a subset of ) @xmath154 . to avoid using half - integer coordinates , we adopt a notation so that the white vertices are also on @xmath70 , more precisely , let @xmath155 , @xmath156 , and @xmath157 , then @xmath158 edges of the graphs are of the form @xmath159 to @xmath160 , @xmath161 , and . we denote by @xmath162 and @xmath163 the set of the black and white vertices respectively with the above coordinates . assign @xmath164 to be the edge weights and denote the kasteleyn matrix , the matrix whose rows are indexed by all the white vertices and whose columns are indexed by black vertices , by @xmath165 for all @xmath166 and @xmath167 . the above formulation defines a valid kasteleyn orientation of the graph , that is , the number of counterclockwise edges in any face is odd ; see figure  [ figkast1 ] . [ c](a ) [ c](b )     the kasteleyn matrix was originally introduced by kasteleyn  @xcite to count the number of dimer covers of a graph as the latter is given by @xmath168 , but its use goes beyond the uniform weight case . the probability @xmath169 on dimer configurations is defined as follows . for a given dimer configuration , we associate a weight to be the product of all the weights of the dimers present in that configuration . the partition function @xmath170 is then the sum of all weighted dimer configurations of @xmath153 . then , the probability of a dimer configuration is given by its weight divided by @xmath170 . in particular , given any disjoint set of edges @xmath171 , the probability of seeing dimers on the edges @xmath171 is given by @xmath172 kenyon in  @xcite showed the following . [ thmkenyon97 ] consider a set of @xmath173 disjoints edges of @xmath153 , , @xmath174 . then , @xmath175=\det\left[{\mathbb{k}}_g^{-1}(b_i , w_j)\right]_{i , j=1}^m \prod_{i=1}^m { \mathbb{k}}_g(w_i , b_i ) , \ ] ] where @xmath176 represents the inverse of @xmath177 .    in other words , the dimers form a determinantal point process with correlation kernel @xmath178 given by @xmath179 for @xmath180 and @xmath181 . the kasteleyn matrix approach has been used with some success for computing combinatorial and asymptotics of random tiling models ; see  @xcite for domino tiling models and  @xcite for the honeycomb case . the first result is the one that inspired proposition  [ proprecursionfinitet ] . to state it , we introduce some notations . for @xmath182 set @xmath183 and @xmath184 . [ proprecursiongraph ] assume that the set of vertices @xmath185 belong to the graph @xmath153 for @xmath186 and let the edges @xmath187 for @xmath188 . then @xmath189+r_g(n),\ ] ] with @xmath190 here @xmath170 denotes the partition function of @xmath153 and @xmath191 denotes the partition function of the graph obtained from removing @xmath192 from @xmath153 . we add an _ auxiliary edge _ @xmath193 , which is an edge not present in the graph but helpful for computations ; a similar idea to that used in  @xcite . we assign a weight @xmath33 to the auxiliary edge @xmath193 . to preserve the kasteleyn orientation of the new graph , this edge is directed from @xmath159 to @xmath194 . since @xmath159 and @xmath194 are on the same face , removing them from the graph preserves the kasteleyn orientation ( see figure  [ figkast1 ] ) . therefore , each matching of @xmath195 has the same sign . cramer s rule gives @xmath196}{\det[{\mathbb{k}}_g]}= \frac{z_{g\setminus \sigma_0}}{z_g}.\ ] ]    for @xmath197 , @xmath160 is either matched to @xmath198 or @xmath199 . if @xmath160 is matched to @xmath198 , then the edge @xmath200 is forced to be matched too . notice that @xmath201 the edge weight of is equal to and the edge weights of @xmath202 and @xmath200 are equal to @xmath203 and @xmath204 respectively . notice that the product of the latter two matrix entries is exactly @xmath205 . therefore , @xmath206 now we proceed by induction . consider the graph @xmath207 , which can be thought as the graph where the vertices in @xmath208 are matched as in figure  [ figkast2](a ) . [ r]@xmath34 [ r]@xmath209 [ c]@xmath210 [ c]@xmath211 [ c]@xmath212 [ c]@xmath213 . ( a ) for @xmath208 , ( b ) for @xmath214 , ( c ) for @xmath215 , and ( d ) for @xmath216.,title="fig:",height=188 ]    the vertex @xmath217 is either matched to or to . for the former , then the vertex is incident to only one vertex , which means that also is matched . since @xmath218 then we have @xmath219 define @xmath220 the vertex set @xmath216 is equal to @xmath214 . therefore ( [ eq3.21 ] ) reads @xmath221 we iterate ( [ eq3.22 ] ) and find that @xmath222 now , from  , we have @xmath223}{c_2(m+1)},\ ] ] because @xmath224 is the product of the edge weights of the edges @xmath225 . therefore , dividing ( [ eq3.23 ] ) by @xmath170 , we find the right side of ( [ eq3.15 ] ) , while the left side follows from  .    in order to prove theorem  [ thminfinitesystem ] we start by considering a finite honeycomb graphs with @xmath226 weights ( see figure  [ figuretilings ] ) . as above , define @xmath227 to be the random variable of having a lozenge of type i at @xmath34 . that is @xmath228 then proposition  [ proprecursiongraph ] gives the following . [ corfiniteabc ] for any @xmath226-weighted honeycomb graph @xmath153 ( satisfying the assumptions of proposition  [ proprecursiongraph ] ) , @xmath229 + \widetilde r_g(n)\ ] ] with @xmath230 . the @xmath226-weighting means that @xmath231 see figure  [ figuretilings ] . then @xmath232 as well as @xmath233 so that proposition  [ proprecursiongraph ] gives immediately @xmath234 + r_g(n)\ ] ] with @xmath235 . next notice that having a chain of particles @xmath236 which corresponds to the edges @xmath237 and no particle at @xmath238 corresponding to the edge @xmath239 , then the dimer configurations incident to the edge @xmath240 for @xmath241 are forced since each vertex becomes incident to one edge by increasing @xmath132 . hence the dimers for @xmath242 are forced . this implies @xmath243={\mathbbm{p}}[e_0 ^ 0,\ldots , e_m^0,e_m^1]={\mathbbm{e}}\bigg[(1-\eta(x+1,n - m))\prod_{i=0}^m \eta(x , n - i)\bigg].\ ] ] this ends the proof of the corollary . we focus on the @xmath226-weighting of the honeycomb graph . recall that @xmath151 denotes the infinite honeycomb graph and @xmath86 denotes the honeycomb graph embedded on a torus of size @xmath69 . the first result is the extension of the recursion relation to the infinite honeycomb graph . [ proprecursioninfinite ] on the infinite honeycomb graph with @xmath226-weights it holds @xmath244+r_n\ ] ] with @xmath245\ ] ] for some finite constant @xmath246 . on @xmath86 there are four parity classes corresponding to vertical / horizontal height changes when crossing the torus ( see  @xcite ) : @xmath247 where @xmath248 is obtained from the kasteleyn matrix which has even horizontal and vertical height change and so on . we apply corollary  [ corfiniteabc ] to each parity class : let @xmath249 be the kasteleyn matrix associated with these four parity classes , i.e. , @xmath250=z_{{\mathcal{h}}_l}^{ij}$ ] . then , we have @xmath251\end{gathered}\ ] ] where @xmath252 and @xmath253 represents the particle probability measure for the parity class @xmath254 . we sum ( [ eq3.32 ] ) over the parity classes with the correct sign . after taking the thermodynamic limit @xmath85 ( which does exist because in  @xcite is was shown that it converges to the infinite gibbs measure characterized by the inverse kasteleyn matrix @xmath255 ) we obtain ( [ eq3.40 ] ) .    to prove ( [ eq3.32b ] ) , notice that by the extra restrictions imposed on the number of dimer covers by reducing the number of vertices to be covered and by comparing allowed dimer configurations on the graphs @xmath256 , @xmath257 and @xmath258 , we have @xmath259 for some constant @xmath260 . we divide the above equation by @xmath261 which gives @xmath262 this means that by using   we have @xmath263 \leq \tilde c bc \,\frac{(bc)^n z^{ij}_{{\mathcal{h}}_l\setminus\sigma_n}}{z^{ij}_{{\mathcal{h}}_l}}. \end{aligned}\ ] ]    using ( [ eq3.30 ] ) , setting @xmath264 and summing over the parity classes gives the desired bound .      with the result of proposition  [ proprecursioninfinite ] we can now easily end the proof of theorem  [ thminfinitesystem ] .    notice that the left side of ( [ eq3.40 ] ) does not depend on @xmath30 and is finite , while the right side of ( [ eq3.40 ] ) is a sum of positive numbers . therefore the last terms in the sum with @xmath265 goes to zero faster than @xmath266 , which in turns implies that @xmath267 as @xmath268 ( in reality , the decay is exponential as shown in  @xcite ) . therefore , by taking @xmath268 in ( [ eq3.40 ] ) we obtain @xmath269.\ ] ] finally notice that the prefactor in ( [ eq2.12 ] ) for @xmath270 and @xmath271 is exactly @xmath272 , i.e. , @xmath273 the other equalities are easy to compute and were already obtained in  @xcite . it seems plausible that theorem  [ thminfinitesystem ] could also be verified using proposition  [ proprecursionfinitet ] recursively for the kernel @xmath274 giving an algebraic proof . however , a bound for the remainder , analogous to ( [ eq3.32b ] ) , would still be required . this bound seems mysterious from the algebraic expression and even the positivity of the remainder is not obvious from the determinant expression . here we derive a single integral representation for the inverse kasteleyn matrix @xmath274 given in ( [ eq2.14 ] ) . let us do the change of variables @xmath275 so that @xmath276 _ case @xmath106_. in this case when @xmath277 is not in the arc of the circle of radius @xmath95 ( anticlockwise oriented ) from @xmath278 to @xmath96 , then no poles of @xmath279 lies inside its integration contour and the contribution is @xmath35 . if @xmath277 is in the arc of circle from @xmath278 to @xmath96 , then there is a simple pole at @xmath280 , which leads to @xmath281 here the integration path can be any path crossing the real axis on @xmath107 . _ case @xmath108_. in this case when @xmath277 is in the arc of the circle of radius @xmath95 ( anticlockwise oriented ) from @xmath278 to @xmath96 , then outside the integration contour of @xmath279 there are no poles and the integral over @xmath279 gives @xmath35 . if @xmath277 is in the arc of circle from @xmath96 to @xmath278 , then there is a simple pole at @xmath280 , which leads to @xmath282 this is equal to the expression ( [ eqapp1 ] ) if the paths are chosen to cross real axis on @xmath109 . a.  borodin , _ determinantal point processes _ , the oxford handbook of random matrix theory ( g.  akemann , j.  baik , and ph .  di francesco , eds . ) , oxford university press , 2011 , pp .  233252 [ arxiv:0911.1153 ] .                                  k.  johansson , _ random matrices and determinantal processes _ , mathematical statistical physics , session lxxxiii : lecture notes of the les houches summer school 2005 ( a.  bovier , f.  dunlop , a.  van enter , f.  den hollander , and j.  dalibard , eds . ) , elsevier science , 2006 , pp .
the speed of growth for a particular stochastic growth model introduced by borodin and ferrari in  @xcite , which belongs to the kpz anisotropic universality class , was computed using multi - time correlations . 
 the model was recently generalized by toninelli in  @xcite and for this generalization the stationary measure is known but the time correlations are unknown . in this note , we obtain algebraic and combinatorial proofs for the expression of the speed of growth from the prescribed dynamics .