The Potts Fully Frustrated model: Thermodynamics, percolation and dynamics in 2 dimensions

We consider a Potts model diluted by fully frustrated Ising spins. The model corresponds to a fully frustrated Potts model with variables having an integer absolute value and a sign. This model presents precursor phenomena of a glass transition in the high-temperature region. We show that the onset of these phenomena can be related to a thermodynamic transition. Furthermore this transition can be mapped onto a percolation transition. We numerically study the phase diagram in 2 dimensions (2D) for this model with frustration and {\em without} disorder and we compare it to the phase diagram of $i)$ the model with frustration {\em and} disorder and of $ii)$ the ferromagnetic model. Introducing a parameter that connects the three models, we generalize the exact expression of the ferromagnetic Potts transition temperature in 2D to the other cases. Finally, we estimate the dynamic critical exponents related to the Potts order parameter and to the energy.

We consider a Potts model diluted by fully frustrated Ising spins. The model corresponds to a fully frustrated Potts model with variables having an integer absolute value and a sign. This model presents precursor phenomena of a glass transition in the high-temperature region. We show that the onset of these phenomena can be related to a thermodynamic transition. Furthermore this transition can be mapped onto a percolation transition. We numerically study the phase diagram in 2 dimensions (2D) for this model with frustration and without disorder and we compare it to the phase diagram of i) the model with frustration and disorder and of ii) the ferromagnetic model. Introducing a parameter that connects the three models, we generalize the exact expression of the ferromagnetic Potts transition temperature in 2D to the other cases. Finally, we estimate the dynamic critical exponents related to the Potts order parameter and to the energy.

I. INTRODUCTION
The glass transition temperature T g for liquids is defined experimentally as the onset of calorimetric anomalies [1]. It is usually understood that T g is not related to a thermodynamic transition [2], but to the slowing down of one or more degrees of freedom. This slowing down prevents the system to reach the equilibrium [3,4]. This question is of interest because almost all liquids can form glasses if cooled at high enough rate. Moreover, many other materials as for example polymers, microemulsions, granular material, vortex glasses, ionic conductors, colloids, plastic glassy crystals and spin glasses (SGs) [5] show glassy properties.
Even well above T g , where glassy systems actually can equilibrate, they show experimentally dynamic anomalies as precursor phenomena of the glass transition [3,[6][7][8]. From a theoretical point of view, one of the open questions is if these precursor phenomena are related to the thermodynamics of the system [2,4,9,10], or if they are not, like the glass transition occurring at lower temperature T g . In particular, for the Ising SG [5] such a relation between precursor phenomena and a thermodynamic free energy essential singularity has been shown [9,11,12]. Indeed, in this model there is a dynamic anomaly at a temperature T * . Above T * the relaxation processes have an exponential behavior, while below T * they have a non-exponential behavior. Theoretical and numerical evidences show that T * coincides with the Griffiths temperature T c [11]. This T c is the transition temperature that the model would have if the frustration due to disorder is removed. A way to remove the frustration is, for example, to substitute every antiferromagnetic interaction with a ferromagnetic interaction. In general, removing the frustration, the model will have ferromagnetic regions and antiferromagnetic regions and T c will be the transition temperature of the unfrustrated model. To be more precise, the free energy of the Ising SG in external field has a singularity which disappears in the limit of zero external field and which occurs at a temperature that goes to T c in the same limit. This singularity is present only in disordered systems. The relation T * = T c has also been shown to be valid numerically in 2 dimensions (2D) for a model with Potts variables that generalizes the Ising SG [13]. This generalized model will be considered in the following and we will refer to it as the Potts SG.
Until now we talked about glass dynamics in systems with disorder, but numerical simulations show that it is possible to observe glassy behavior with precursor phenomena for spin systems without disorder but with frustration [14,15]. These systems, due to the lack of disorder, are more suitable for a theoretical approach [16,17]. In particular, one can try to answer the question about the relation between the precursor phenomena and the thermodynamics of the system. In Ref. [14] it was considered a simple case: The fully frustrated (FF) Ising model [19] where ferromagnetic and antiferromagnetic interactions are ordered in such a way that any lattice cell has an odd number of antiferromagnetic interactions (i.e. is frustrated). It is shown (by simulations in 2D and 3D) that the onset T * of nonexponential relaxation processes is related to a randombond percolation transition [14]. As for the Ising SG, it is possible to generalize the FF Ising model to a FF model with Potts variables [13]: The Potts FF model that will be described in details in the following. In Ref. [13] the dynamics of the Potts FF model was compared with the dynamics of the Potts SG model and some anticipations on the relation between precursor phenomena and thermodynamics were given. In this work we study in details the thermodynamics of the Potts FF model, showing that T * corresponds to the thermodynamic transition temperature T p of the Potts variables. It is important to note that in any FF model the Griffiths temperature T c cannot be defined for the lack of disorder. Therefore T c cannot play any role in these cases.
The comparison of the results presented here with the analogous study of the Potts SG model with disorder [18] allows to give insight on the role of disorder. To this goal we introduce a formal parameter 0 ≤ X ≤ 1 that connects both models. For X = 0 we have the ferromagnetic Potts model [20] (without frustration and without disorder). For 0 < X < 1 we have the disordered and frustrated Potts model. In particular, for X = 0.5 we have the Potts SG model. For X = 1 we have the Potts FF model (without disorder).
From Ref. [18] and from the present work, it is possible to show that for X = 0.5 and X = 1 there are two thermodynamics transitions. The lower transition is an Ising SG or a FF Ising transition, respectively. The upper transition at T p is in the universality class of a ferromagnetic Potts transition. Furthermore we show that T p corresponds to a percolation temperature. Moreover, we show how it is possible to generalize the exact expression of T p for the model with X = 0 in 2D [20] to the cases X = 0.5 and X = 1.
The organization of the paper is the following. In Sec. II we introduce the model and the known results for X = 0 (ferromagnetic case), for X = 0.5 (disordered and frustrated case) and for X = 1 (ordered and frustrated case). In Sec. III we introduce the cluster formalism used to map the upper thermodynamic transition at T p onto a percolation transition. In Sec. IV we present the phase diagram in 2D for X = 1 as result of Monte Carlo (MC) simulations and we compare it with the cases X = 0 and X = 0.5. In Sec. V we use the spin-flip MC dynamics to study the dynamic critical exponent and the temperature T * , onset of stretched exponentials. In Sec. VI we give the summary and conclusions.

II. THE MODEL
Structural glasses, such as dense molecular glasses, plastic crystal, or orto-therphenyl at low temperature, can be modeled to first approximation as systems with orientational degrees of freedom frustrated by geometrical hindrance between non-spherical molecules. For this reason we will consider the lattice model introduced in Ref. [21], where the orientational degrees of freedom are represented by Potts variables [20] with s states (σ i = 1, . . . , s) and the frustration is modeled by means of ferro/antiferromagnetically interacting Ising spins (S i = ±1), coupled to the Potts variables.
The model is defined by the Hamiltonian where the sum is extended over all the nearest neighbor sites, J is the strength of interaction, ǫ i,j = ±1 is a quenched variable that represents the sign of the ferro/antiferromagnetic interaction and δ n,m = 0, 1 is a Kronecker delta. To emphasize that the Ising and the Potts variables are interdependent, we can rewrite the Hamiltonian in Eq.(1) as where τ i ≡ S i σ i = ±1, ±2, . . . , ±s is a variables with 2s states and the frustration now is explicitly on the new variable. Any τ i has an absolute value σ i and a sign S i . The model depends on the interaction configuration {ǫ i,j }. Possible choices are the following. (i) If all ǫ i,j = 1 (i.e. all interactions are ferromagnetic), the Eq.(2) is the Hamiltonian of the ferromagnetic Potts model [20] with variables τ i with an even (2s) number of states. It shows a thermodynamic transition at T c (s) whose order depends on s. (ii) If ǫ i,j are quenched random variables, the model corresponds to the Potts SG. This model is a generalization of the Ising SG model that is recovered for s = 1. It shows two thermodynamic transitions [18]. The lower is a SG transition at T SG (s). The upper is a Potts transition at T p (s) > T SG (s). The transition at T p (s) is in the universality class of a ferromagnetic s-states Potts model. Another relevant temperature for this model is the temperature T c (s) defined for the previous case (the ferromagnetic 2s-states Potts model). Indeed, it is possible to show that for finite external field a free energy (Griffiths) singularity arises [11]. In the limit of external field going to zero, the temperature at which this singularity occurs goes to T c (s) and the singularity vanishes. T c (s) is the Griffiths temperature for this model. Furthermore, numerical simulations [13] show that T c (s) > T p (s) corresponds to the onset T * (s) of non-exponential correlation functions for the Ising spins S i . This result generalizes what happens in the Ising SG (s = 1 case), where at the Griffiths temperature T c (1) non-exponential correlation function are seen [12]. It is worth to note that the Ising spins are critical at T SG (s), that is well below T c (s) = T * (s). Moreover, note that the relevant Griffiths temperature for this model is T c (s) of the variables τ i = S i σ i and not the Griffiths temperature of the variables S i alone, that in our notation is T c (1). (iii) If there is an odd number of ǫ i,j = −1 for each elementary cell (as in Fig. 1), the system is fully frustrated (FF). It means that at least one interaction per cell is not satisfied, i.e. the relative energy contribution is 0 instead of −2sJ (as for the edges b, d and f in Fig. 1). The model is called Potts FF and is a generalization of the FF Ising models [19] that is recovered for s = 1. It has frustration but no disorder. For any integer s ≥ 1 the model has in 2D a second-order phase transition at T = 0. Non-exponential correlation functions are reported below a finite temperature T * (s) for s = 2, 1 and 1/2 (the latter case is defined in Sec.III) [14,13]. This dynamic anomaly cannot be related, as in the previous case, to the Griffiths temperature. Indeed, the Griffiths temperature is not defined in FF models for the lack of disorder. The aim of this work is to study the phase diagram of the FF model as function of s and to show that T * (s) corresponds to a Potts transition. Furthermore, using a percolation approach, it is possible to show that the Potts transition corresponds to a percolation transition, defined also for non-integer values of s. The previous three cases can be generalized in a class of Hamiltonians. Consider, as model parameter, the density X of frustrated elementary cells. For X = 0 there are no frustrated cells (like in a ferromagnet), while for X = 1 every cell is frustrated (FF case). For 0 < X < 1 it is possible to partition the lattice in two non-overlapping sub-sets U (X) and F (X) of unfrustrated cells and frustrated cells, respectively. Therefore the Eq.(2) can be written as In this way the ferromagnetic Potts model with an even number of states (case i) is recovered for X = 0, since F (X = 0) is empty; The Potts SG model (case ii) for X = 0.5; The Potts FF model (case iii) for X = 1, since U (X = 1) is empty; Intermediate disordered models are obtained for other values of 0 < X < 1.
To make the comparison between the X = 0.5 and X = 1 model, we will follow the line of Ref. [18] where the phase diagram of the former has been studied. In particular, for X = 1 we consider a square lattice with one ǫ i,j = −1 per cell as shown in Fig.1.

III. PERCOLATION
We now introduce the percolation map following the Ref. [21]. It is a generalization to X ≥ 0 of Fortuin-Kasteleyn [22] clusters formalism, defined originally only for X = 0. In Ref. [21] it is shown that the partition function Z s,X of the Hamiltonian in Eq.(3) can be written in terms of bond configurations C is the probability to place a bond between two nearest neighbor sites, N (C) is the number of clusters (defined as maximal sets of connected bonds) in the configuration C, |C| is the number of bonds and |C|+|A| is the total number of interactions. A loop of bonds is called frustrated if the product of all the signs ǫ i,j of the interactions along it is equal to −1. Note that for X = 0 there are no frustrated loops and the original Fortuin-Kasteleyn cluster definition is recovered. Since the above cluster definition holds for any s and X, for any model described by the general Hamiltonian in Eq.(3) it is possible to define a percolation temperature for these clusters.
Moreover, the right-hand side expression of Eq.(4) is well defined even for non-integer values of s. In these cases it does not define a Hamiltonian model, but it still defines a percolation model. In particular, the s = 1/2 case for X > 0 is the frustrated percolation model [14].
For X = 0, in the general case in which we consider Potts variables τ i with a generic number q of states instead of an even (2s) number of states in Eqs. (3,5), it is possible to show [20] that the percolation temperature T p (q, X = 0) of the above defined clusters coincides with the ferromagnetic q-state Potts critical temperature T c (q). In particular, in 2D it is possible to prove [20] the relation For X = 0.5 the same kind of relation between the percolation temperature T p (q, X = 0.5) and the Potts transition temperature for q = 2s in 2D has been extended [18]. Furthermore, it was shown numerically that it is possible to generalize the Eq.(6) using a fitting parameter a(X) [18]. The resulting relation is with a(X = 0.5) = 0.800 ± 0.003 [18]. Note that T p (q, X = 0.5) represents a percolation temperature for any q ≥ 0 and also a Potts transition temperature for even integer values q = 2s. Furthermore, for 0 < X < 1 and q = 2s the Eq. (6) for T c (2s) gives by definition the Griffiths temperature of the disordered model.
To study the phase diagram of the model with X = 1 (Potts FF) we have simulated it for s = 2, 7, 20, 50 on square FF lattices with periodic boundary conditions and linear size L from 10 to 80. At low temperature all the Potts variables tend to order ferromagnetically wherever the Ising spins satisfy the ferro/antiferromagnetic interactions (i.e. δ ǫi,j Si,Sj = 1). In particular, at T = 0 the system in 2D has a second-order phase transition of the Ising spins, like in the FF Ising model [19]. We will show that the interplay between Potts variables and Ising spins affects the phase diagram at finite temperature. As in Ref. [18], to study the finite temperature range we can use an efficient cluster dynamics with an annealing procedure. We define a MC step as an update of the whole system. At each temperature we average the data over 10 4 MC steps, discarding the first 5 × 10 3 MC steps.
In our systematic analysis we calculated for each s the Binder's parameter [24] for the energy density E defined as (angular brackets denote the thermal average). This quantity allows to distinguish between first-order and second-order phase transitions. Indeed, in a second-order phase transition for L → ∞ it is V = 2/3 for all temperatures, while in a first-order phase transition V has a well pronounced minimum near the transition temperature.
The thermodynamic of the model is studied by means of the Potts order parameter (where i = 1, . . . s, M i is the density of Potts spins in the i-th state), the susceptibility (where N is the total number of Potts spins) and the specific heat To study the Fortuin-Kasteleyn percolation we calculated the percolation probability per spin P = 1−m 1 and the mean cluster size S = m 2 (where m n = k k n n k is the n-th moment of the distribution of density n k of clusters with size k).
A. The second-order transition for s = 2 For X = 1 and s = 2 the Binder's parameter V goes to the constant value 2/3 for all the temperatures as L increases (see Fig. 2) revealing a second-order phase transition. The transition temperature T s in the thermodynamic limit can be estimated, together with all the critical exponents, using the standard scaling analysis [25] for second-order phase transition, where by definition of critical exponents ν, β, γ and α is ξ ∼ |T − T s | −ν being ξ the correlation length, (and analogous scaling laws for χ and C H ) where f M (x) is an universal function of the dimensionless variable x. The values at which the scaling laws are verified give the estimates of critical exponents and of T s . In Figs. 3 we show the large-sizes data collapses using the set of Ising critical exponents [26] and leaving only T s as free parameter, giving an estimate k B T s /J = 2.73 ± 0.03.
The study of the percolation quantities shows a smooth behavior of P and a cusp in S increasing with L. Therefore one can make the ansatz that the percolation transition is of second-order. Let us define a Fortuin-Kasteleyn percolation temperature T p and a set of percolation critical exponents ν p , β p and γ p by means of the relations ξ p ∼ |T − T p | −νp where ξ p is the connectedness length of the clusters (i.e. the typical linear cluster size) and Applying the standard scaling analysis for percolation [27], one obtains in a consistent way the results summarized in Fig. 4 using the corresponding thermodynamic critical exponents for the 2D Ising model. The numerical estimate for the percolation temperature is k B T p /J = 2.73 ± 0.03 coincident with the estimates of T s . Therefore the percolation transition and the Potts transition occur at the same temperature and with the same set of critical exponents, i.e. they coincide. For X = 1 and s = 7, 20 and 50 we have considered systems with L from 10 to 50 lattice steps with periodic boundary conditions. On the base of the mean field results [17] for the Potts FF model and the knowledge of the Potts model [20], we expect that the order of transition changes for s > 4. In fact, the thermodynamic order parameter M becomes more and more discontinuous as s increases. At the same time the percolation order parameter P develops a better and better pronounced discontinuity.
In particular, the study of the Binder's parameter V reveals that the model for s = 7, 20 and 50 has a firstorder phase transition, since, for each considered lattice size L between 10 and 50, V has a non vanishing minimum, as shown in Fig. 5 for s = 7. In this cases the estimates of infinite size transition temperatures T s (s) and T p (s), for the thermodynamic and the percolation transition, respectively, can be done through the relation [25] T max (L) − T max (∞) ∼ L −D (13) where D = 2 is the Euclidean dimension, T max (L) is the finite-size temperature of the maximum of C H and S, respectively, and T max (∞) is the transition temperature in the thermodynamic limit. Therefore T s (s) and

C. Phase diagram
The numerical results in Secs. IV.A and IV.B give rise to the phase diagram for X = 1. It is qualitatively similar to the phase diagram for X = 0.5 [18]. In Fig.6 we compare both of them with the phase diagrams for X = 0.
For X = 0 (Potts model with 2s-states variables) for any integer s there is a paramagnetic/ferromagnetic phase transition at finite temperature T c (s). The transition is of second-order for 2s ≤ 4 and of first-order for 2s > 4 [20]. T c (s) is given by Eq.(6) with q = 2s. It coincides with the Fortuin-Kasteleyn percolation temperature. The percolation temperature is defined even for non-integer values of s.
For X = 0.5 (Potts SG model with frustration induced by disorder and with 2s-states variables) there are two phase transitions for integer values of s. The lower transition is a SG transition. It is supposed at T SG = 0 for any s in 2D and at T SG (s) > 0 in higher dimensions as in the Ising SG [28]. The high-temperature transition is a s-states Potts ferromagnetic transition occurring at T p (s, X) given by Eq. (7) with q = 2s and a(X = 0.5) = 0.800 ± 0.003 [18]. It is a second-order transition for s ≤ 4 and a first-order transition for s > 4. T p (s, X) corresponds also to the Fortuin-Kasteleyn percolation temperature, that is defined even for non-integer s [18]. This model is disordered and its Griffiths temperature is by definition the transition temperature of the 2s-states Potts model T c (s), given by the Eq.(6) with q = 2s. It is possible to see that T c (s) is numerically consistent with the onset of non-exponential relaxations at T = T * (s) for s = 2 in 2D [13] and for s = 1 (Ising SG) in 2D and 3D [12,29].
For X = 1 (Potts FF model with frustration and no disorder and with 2s-states variables), considered here, there are two transitions for integer values of s, as well as for X = 0.5. The lower transition is at T F F = 0 in 2D and at T F F (s) > 0 in higher dimensions, as for the FF Ising model (s = 1) [19,32]. As seen in this section, in this case the upper transition is a s-states Potts ferromagnetic transition at T p (s, X = 1). As shown in Fig. 6, Eq.(7) describes well T p (s, X = 1), using a fit parameter a(X = 1) = 0.690 ± 0.003. Furthermore we have shown that it coincides with a Fortuin-Kasteleyn percolation transition. As consequence the Fortuin-Kasteleyn clusters represent the regions of correlated Potts variables. This means that the cluster's characteristic linear size is equal to the correlation length of the Potts variables. Analysis for s = 1/2 (frustrated percolation) and s = 1 (FF Ising model) in 2D and 3D are given in Ref. [14]. The main difference between X = 1 and X = 0.5 cases is that the Griffiths temperature is defined in the latter (disordered model), but not in the former (for the lack of disorder). The important consequence of this fact is that for X = 1 the dynamic anomalies are present only below T p (s, X = 1) [30], while for X = 0.5 they are present also above T p (s, X = 0.5) and below T c (s) [31].
Note that all the Potts-percolation transition temperatures for X = 0, 0.5, 1 can be described by the same form in Eq.(7) as function of s with a X-dependent parameter a(X). This parameter has a regular behavior as function of X (see the inset in Fig.6).

V. DYNAMIC CRITICAL EXPONENT
As shown in Sec. IV.A, at T p (s = 2, X = 1) the Potts variables have a second-order phase transition. Therefore their correlation length diverges and their dynamics slows down. A measure of the slowing down for any observables A is given by the dynamic critical exponent z defined by τ A (T p (L), L) ∼ L z . Here, omitting for sake of simplicity the dependence on s and X, T p (L) is the transition temperature for the system with finite size L [33] and τ A (T, L) is the correlation time at temperature T and size L associated to the correlation function for A where t is the time. More then one definition of τ A is possible for any A, but all of them, even if numerically different, have the same qualitatively behavior [12]. In particular, facing the difficulty that the greater size the greater τ A (T p (L), L), we define τ M and τ E for Potts order parameter M and for energy density E as the time To study the behavior of f M and f E in the thermodynamic limit we have extrapolated the data in the infinite size limit following the procedure suggested in Ref. [36]. It consists in plotting at any t the generic f A (t, T, L) for finite size L versus 1/L and in extrapolating for 1/L → 0. We consider the temperature range 2.65 < k B T /J < 3.25 [35]. The results are shown in Figs that is a combination of a form b) with a power law. In the previous functions, f 0 , τ , x and β are T -dependent parameters. Note that for the form c) it is possible to estimate f 0 and x separately from β and τ , since the first two describe the short-time behavior while the second two the long-time regime. The Ogielski form turns out to describe very well f M and f E as shown in Figs. 8 and 9. The fitting parameters are presented in Fig.9 and 10. The forms b) and c) give always compatible estimates of β exponent, consistently with the very low values of x (that is approximately the slope of the function at t = 0 in Figs.8 and 9).
From Fig. 10 for the parameter β of f M , we see that it is β = 1 above the Potts transition at k B T p /J = 2.73 ± 0.03 and β < 1 below T p . Therefore there is a dynamic transition between a high-temperature exponential behavior to a low-temperature stretched exponential behavior of f M . This result shows the presence of a complex dynamics below T p , consistently with the analysis of non-linear susceptibility correlation function of the Ising spins [13]. It is important to note that, while the Potts variables have a transition at T p , the Ising spins have no transition at T p and, in principle, no dynamical anomalies are expected for them. On the other hand, the data for f E show that the longtime behavior is well described by an exponential function. In particular, the form c) with β = 1 and a non-zero x fits very well also the data for the lowest temperature considered here. Note that, in principle, one can expect a dynamical anomaly even for the energy density E, since it depends explicitly on the critical Potts variables.
In the fitting forms used, the correlation time τ is defined as a fitting parameter. Another possible definition is the following (integral correlation time): Due to the divergence of the correlation time, the definition in Eq. (16) does not converge near the transition temperature. Therefore it is not shown in Figs. 9 and 10 for the temperature with the largest τ . However, where it converges, all the different estimates of the correlation time are numerically consistent as show in Fig.9 and 10. Note that, even if the data are extrapolated to the thermodynamic limit, finite size effects are still present. Indeed, the critical temperature estimated from τ measurements, within our temperature mesh, is between 2.75J/k B and 2.85J/k B , that is at variance with the estimate of T p .

VI. SUMMARY AND CONCLUSIONS
We have studied a schematic model for glasses where frustrated orientational degrees of freedom, associated to 2s-states Potts variables τ i , induce complex dynamics. In real systems the frustration can be associated, for example, to steric hindrance of non spherical molecules in structural glasses. Each τ i has an absolute value (σ i = 1, . . . , s) and a sign (S i = ±1). The frustration is over the signs and can be with disorder or without disorder, depending on a model parameter X. For 0 < X < 1 there is disorder, while for X = 1 the model is (fully) frustrated without disorder. For X = 0 there is no frustration, and the model recovers the 2s-states ferromagnetic Potts model.
For 0 < X ≤ 1 the model has two thermodynamic transitions. The high-temperature transition at T p (s, X) for the model with 2s states is in the universality class of the s-states ferromagnetic Potts model. Therefore the fluctuations (χ and C H ) of the orientational degrees of freedom diverge at T p (s, X) (for s ≤ 4), as well as the correlation times of quantities depending on them (like the Potts order parameter M or the energy density E). For them we estimated the dynamic critical exponents z M and z E . The low-temperature transition is a spin glass transition for X = 0.5, or a fully frustrated transition for X = 1, and marks the ordering transition of the signs S i (Ising variables).
The diverging fluctuations at upper and lower transition temperatures are expected to be experimentally observable only using specific probes that couple with them. Examples of such probes could be those associated to dielectric measurements in supercooled-liquids and plastic glassy crystals or to electron spin resonance spectroscopy measurements [6,7].
For X = 1 and s = 2 the model shows a complex dynamics in correspondence of T p . In particular, the correlation function for M and the correlation function for the signs S i change their behaviors. They have an exponential behavior above T p and a non-exponential behavior below T p . This behavior is expected at least for any s ≤ 4, because it is associated to the free energy singularity occurring at the second-order transition of the Potts variables. Therefore the complex dynamics corresponds to a real thermodynamic transition.
For X = 0.5 and s = 2 the onset of this dynamic anomaly is shifted to a higher temperature, above T p . It occurs in correspondence of the Griffiths essential singularity of the free energy. This singularity is not defined in the X = 1 case. For zero external field it goes to the transition temperature of the ferromagnetic 2s-states Potts model T c (2s) and it vanishes. Therefore the onset of complex behavior does not correspond to a real thermodynamic transition.
We have shown that the Potts transition at T p (s, X) for any s and X considered here, coincides with a percolation transition. It is not worthless to note that the dynamic transition at T p (s, X = 1) persists also for s = 1 and s = 1/2 [14]. In these cases T p (s, X = 1) does not correspond to a thermodynamic transition, but only to a percolation transition. Therefore for s < 2 the dynamic anomaly is no more related to a thermodynamic transition, but to a percolation transition in the real space [14]. This result could be in some relation with experimental results on microemulsions [8].
Finally, we have shown that it is possible to generalize the exact relation for the ferromagnetic Potts transition temperature T c (s) in 2D to a transition temperature T p (s, X) for any 0 ≤ X ≤ 1, using a fitting parameter a(X). In particular, a(X) acts like a renomarlization factor for the number of states of the model and it is a(X = 0) = 1 (ferromagnetic case) and a(X) decreasing regularly with increasing X.