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when a system phase - separates , it pays for the different domains with a surface energy , which is usually negligible with respect to the bulk energy . as a consequence , any non concave region in the entropy
vs energy curve has to be replaced by a straight line .
this is the result of the usual maxwell construction .
however , the condition of negligible surface energy is violated in presence of long range interactions , as well as for systems with a small number of components . in both cases ,
the possibility of non concave entropies and ensemble inequivalence is well known , and has been demonstrated on numerous models , for instance  @xcite .
the same condition of negligible surface energy is also violated on sparse random graphs : despite the fact that each site has only a small number of neighbors , there will be in general an extensive number of links between two ( extensive ) subsets of the system .
the possibility of ensemble inequivalence in this type of models has been alluded to in some works related to the statistical physics of random graphs and combinatorial optimization  @xcite . however , these authors study the analog of the canonical ensemble , and replace the non concave part of the entropy by a straight line . this phenomenon remains thus to our knowledge unstudied , despite the widespread current interest in complex interaction structures , and networks in general .
the purpose of this work is to present a simple , exactly solvable model on a random regular network , that displays a non concave entropy and ensemble inequivalence .
this is a first step towards the study of more complicated networks , which may also include some local structure , like small world networks .
the paper is organized as follows : in section  [ sec : analytical ] , we present the model , and give its analytical solution ; we then turn in section  [ sec : numerical ] to the comparison with microcanonical simulations using both creutz  @xcite microcanonical dynamics and metropolis  @xcite canonical simulations .
the final section is devoted to conclusions and perspectives .
we study a ferromagnetic system of potts spins with three possible states ( * @xmath1 * , * @xmath2 * and * @xmath3 * ) . the hamiltonian is chosen to be :    @xmath4    where @xmath5 denotes all the bonds in the system
, @xmath6 is the state of spin @xmath7 , and @xmath8 is a kronecker delta . in this form
, the hamiltonian simply counts the number of bonds between spins in different states .
the ground state energy is  @xmath9 .
the spins are located on the nodes of a regular random graph where each node has connectivity  @xmath0 , of order  @xmath10 . a mean field like version of this model , with an
all - to - all coupling , has been studied by ispolatov and cohen  @xcite , and displays ensemble inequivalence .
random regular graphs possess very few loops of size order  @xmath10 , and locally look like trees ; this feature allows us to use standard statistical physics methods , originally developed for bethe lattices .
these calculations are usually done in the canonical ensemble only ; in contrast , we are interested also in the microcanonical solution .
we compute here the free energy and the entropy of the system , by following the formalism of the large deviation cavity method described by o. rivoire in  @xcite .
we consider however only large deviation functions with respect to spin disorder , and not with respect to disorder in the graph structure like in  @xcite .
schematic representation of the iteration ( left ) , link addition ( center ) and site addition ( right).red nodes and solid edges represent the original cavity spins and links , while the green colored nodes and dashed lines identify the additions.,title="fig:",height=170 ] schematic representation of the iteration ( left ) , link addition ( center ) and site addition ( right).red nodes and solid edges represent the original cavity spins and links , while the green colored nodes and dashed lines identify the additions.,title="fig:",height=170 ] schematic representation of the iteration ( left ) , link addition ( center ) and site addition ( right).red nodes and solid edges represent the original cavity spins and links , while the green colored nodes and dashed lines identify the additions.,title="fig:",height=170 ]    we call cavity sites sites which have only @xmath11 neighbors , and one free link . cavity site
@xmath7 sends a field @xmath12 along each link , which tells its state @xmath1 , @xmath2 or @xmath3 .
these field are distributed according to the probability distribution @xmath13 : @xmath14    .[cap : potts - iter ] analysis of the iteration process for @xmath15 : energy shifts and probabilities .
@xmath16 is the field sent by the new cavity site . [
cols="^,^,^,^",options="header " , ]     plugging all the previous results in to eq .
[ eq : f ] , we obtain the expression of the generalized free energy of the system for the general @xmath0 case : @xmath17+\nonumber\\ & & + \frac{k}{2}\ln\left[\frac{1}{3}\left\{\left[p_{a}+\left(p_{b}+p_{c}\right)e^{-\beta}\right]^{k}+\right.\right.\\ & & + \left.\left.\left[p_{b}+\left(p_{a}+p_{c}\right)e^{-\beta}\right]^{k}+\left[p_{c}+\left(p_{a}+p_{b}\right)e^{-\beta}\right]^{k}\right\}\right]\nonumber\end{aligned}\ ] ]    where the three densities @xmath18 , @xmath19 and @xmath20 are solutions of eq .
[ eq : pabck ] .
notice that this procedure does not necessarily yield a unique `` free energy '' @xmath21 ; rather , there is one value of @xmath22 for each solution of the consistency equation  ( [ eq : pabck ] ) .
we must then follow all branches of the multi - valued function  @xmath21 to reconstruct the entropy  @xmath23 through a generalized inverse legendre transform ( see for instance  @xcite for a use of this procedure in the context of signal processing ) : @xmath24 where:@xmath25 can easily be calculated numerically using finite differences .
this is the final , implicit , solution for the entropy @xmath23 . in fig .
[ fig : julien - s - y ] , we plot the different solution branches of @xmath22 , and the inverse temperature @xmath26 .
one clearly sees a negative specific heat region , signaled by the presence of multiple function values for the same energy .
left : the three branches of the generalized free energy @xmath27 as a function of the inverse temperature  @xmath28 , for @xmath29 .
right : the corresponding three branches for @xmath26 in the microcanonical ensemble.,title="fig:",width=264]left : the three branches of the generalized free energy @xmath27 as a function of the inverse temperature  @xmath28 , for @xmath29 .
right : the corresponding three branches for @xmath26 in the microcanonical ensemble.,title="fig:",width=264 ]
in this section we compare the analytical solution with the results obtained through numerical simulations .
microcanonical simulations were performed using creutz  @xcite dynamics . during which , a fictitious `` demon '' is introduced , carrying an energy @xmath30 . at each step , a spin flip in the system is attempted , and the corresponding energy change @xmath31 is computed . if @xmath32 , the move is accepted ; if @xmath33 , the move is accepted only if @xmath34 . in both cases
@xmath30 is then updated so that the total energy @xmath35 is kept constant ; the energy of the system  @xmath36 is then constant up to a @xmath37 . for long run times , the demon s energy reaches an exponential distribution @xmath38 , from where one can compute the corresponding microcanonical temperature @xmath39 of our system : @xmath40 ~.\ ] ]    results of the creutz dynamics are plotted on fig .
[ fig : comparison ] and compared with the analytical solution of the previous section .
the agreement between the two is very good , with the @xmath28  vs  energy curve clearly showing a region of negative specific heat .
finally , we performed canonical metropolis@xcite simulations and calculated the average energy in the temperature range where our results predict ensemble inequivalence . as expected , the canonical caloric curve obeys maxwell s construction and clearly `` jumps over '' the region where the specific heat is negative .     between the analytical solution ( solid lines ) , the creutz dynamics results ( stars ) , and the metropolis simulations ( circles ) for @xmath29 .
the creutz simulations were performed on networks with @xmath41 sites , for @xmath42 `` creutz steps '' , and the results were averaged over @xmath43 network realizations .
the metropolis results were obtained using @xmath44 different networks with @xmath45 nodes , by performing @xmath46 monte - carlo steps . in both cases ,
the size of the error bars is comparable to the size of the symbols .
, width=377 ]
we have presented a complete canonical and microcanonical solution of the 3-states potts model on @xmath0-regular random graphs , and shown that this toy model displays ensemble inequivalence .
there is little doubt that this result should generically apply to models on different types of random graphs , such as erds - rnyi ones , among others .
we also expect to observe ensemble inequivalence on small world networks , since in these systems , the presence of random long - range links should prevent the system from separating in two different phases .    beyond the inequivalence between microcanonical and canonical statistical ensemble , non concave large deviation functions
should be expected for some properties on random graphs .
4 of  @xcite gives an example of this .
the present work provides an example where the large deviation cavity method allows to deal with such a situation , and to compute the non concave part of the large deviation function .
we would like to acknowledge useful discussions with stefan boettcher , matthew hastings and zoltn toroczkai , and financial support from grant 0312510 from the division of materials research at the national science foundation .
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we present a complete analytical solution of a system of potts spins on a random @xmath0-regular graph in both the canonical and microcanonical ensembles , using the large deviation cavity method ( ldcm ) . 
 the solution is shown to be composed of three different branches , resulting in an non - concave entropy function.the analytical solution is confirmed with numerical metropolis and creutz simulations and our results clearly demonstrate the presence of a region with negative specific heat and , consequently , ensemble inequivalence between the canonical and microcanonical ensembles .    ,    
 ensemble inequivalence , negative specific heat , random graphs , large deviations .    _ 
 pacs numbers : _ + 05.20.-y classical statistical mechanics . 
 + 05.70.-a thermodynamics . 
 + 89.75.hc networks and genealogical trees .