Twisted symmetric exclusion processes and set-theoretical $R$-matrices

T wisted symmetric exclusion pro cesses and set-theoretical R -matrices Mathieu Dabro wski ∗ 1 , Loïc P oulain d’Andecy † 2 , and Eric Ragoucy ‡ 1 1 Lab oratoire d’Annecy de Ph ysique Théorique, 9 Chemin de Bellevue - BP 110 - Annecy-le-Vieux - F-74941 Annecy Cedex - F rance 2 Lab oratoire de Mathématiques de Reims, UMR CNRS 9008, Université Reims Champagne-Ardennes, Moulin de la Housse - BP 1039, 51687 Reims Cedex 2 - F rance Abstract W e inv estigate p erio dic integrable Mark ov mo dels, constructed from set-theoretical solu- tions of the Y ang-Baxter equation. W e first focus on the simplest class of solutions, called Lyubashenk o solutions. W e show that the resulting mo dels are equiv alent to some t wisted Symmetric Simple Exclusion Pro cess (SSEP), whic h are usual perio dic SSEP models where a t wist is added on a bond of the ring. W e also pro vide v arious p ossible in terpretations for these Marko v mo dels. Then, we study the long time dynamics of the twisted SSEP , c haracterising its differen t stationary states and counting them. Allo wing the twist to v ary , we examine the possible transitions betw een the different stationary states. Finally , w e extend our construction of Mark ov models to set-theoretical solutions that are more general than Lyubashenk o solutions and show that suc h mo dels are not equiv alen t to a twisted SSEP in general. Keyw ords: Integrable Mark o v mo dels ; Exclusion pro cess ; T wisted SSEP ; set-theoretical Y ang–Baxter equation 1 In tro duction Exclusion Mark o v pro cesses were first introduced in the context of biology to mo del protein syn thesis in [ 29 ] and in statistical physics to describe the dynamics of the Ising mo del in [ 27 ]. The mathematical foundations of these systems w ere established in [ 38 ]. Among these pro cesses, the simplest one is the Symmetric Simple Exclusion Pro cess (SSEP). It describ es a one-dimensional lattice gas, which is equiv alen t to a classical Ising mo del [ 28 ], where particles hav e equal probabilities of jumping one site to the left or righ t, with the condition that the target site is empty (hard core interaction). On a p eriodic lattice, the SSEP reac hes thermo dynamic equilibrium, con trary to the Asymmetric Simple Exclusion Process (ASEP) where a macroscopic current appears [ 30 ]. Generalisations of those mo dels with more than one sp ecies of particles hav e been studied [ 15 , 26 , 39 ]. The ASEP (respectively the SSEP) can b e mapp ed to the anisotropic (resp ectiv ely isotropic) Heisen b erg quan tum c hain [ 1 , 35 ]. More precisely , the Marko v matrix con taining the transition probabilities for the exclusion pro cess is equiv alent to the Hamiltonian of a quan tum spin chain. This allows in some cases to use methods originally dev elop ed for quan tum spin chains to study exclusion pro cesses [ 20 , 21 ]. ∗ email: mathieudabrowski@outlook.com † email: loic.p oulain-dandecy@univ-reims.fr ‡ email: ragoucy@lapth.cnrs.fr 1 Mo dels constructed in this article ha v e the property of b eing integrable. In the framework of quan tum spin chains, the c hain is said to b e in tegrable if its Hamiltonian b elongs to a set of comm uting op erators generated b y a transfer matrix. A transfer matrix is constructed from an R matrix which is a solution of the Y ang–Baxter equation (YBE). This integrabilit y property allo ws the exact computation of physical quan tities of the spin chain [ 16 ]. The definition of an in tegrable Mark ov pro cess is obtained b y replacing the Hamiltonian with the Marko v matrix. In tegrabilit y can b e used to obtain for instance information on the stationary state and on the dynamical b eha viour of the sto c hastic pro cess. In order to construct integrable Marko v pro cesses, the first step consists in finding solutions to the quan tum Y ang–Baxter equation [ 3 , 40 ], which is a notoriously difficult problem in mathematical ph ysics. T o address it, Drinfeld in [ 13 ] suggested to consider “set-theoretical” solutions of the YBE, that is, solutions of the equation given b y a p erm utation of a set of the form X × X . These solutions are not pro duced with the usual quantum group approac h [ 24 ]. Since Drinfeld’s prop osal, set-theoretical solutions ha v e b een extensiv ely studied [ 2 , 8 – 12 , 14 , 18 , 19 , 22 , 33 , 34 , 36 ]. One simple class of set-theoretical solutions of the YBE is called Lyubashenk o solutions [ 13 ]. In this paper, w e construct and study new integrable Marko v processes defined on a one dimensional lattice with p eriodic b oundary conditions. These pro cesses are constructed from set-theoretical solutions of the YBE using the Baxterization metho d [ 25 ]. In particular, we consider inv olutiv e Lyubashenko solutions that allow the construction of mo dels with sev eral in teresting interpretations. W e pro v e that those Lyubashenko solutions define Marko v pro cesses that are equiv alent to some twisted perio dic SSEP . The twisted SSEP is a generalisation of the SSEP on a ring where the exchange rule on one b ond of the lattice allows particles to change species. In the con text of quan tum spin chains, systems with generalised p eriodic boundary conditions ha ve been studied, suc h as in [ 4 , 5 , 23 ], but they are less common in the framework of exclusion Mark o v pro cesses. Then we proceed to the study the evolution of the twisted perio dic SSEP for asymptotically long times. The pro cess can b e divided in to sectors, for whic h there exists a unique stationary probabilit y distribution. W e define the profile and the total c harge of a configuration and we pro v e that they uniquely lab el a sector. W e also give explicit formulas for their num ber and size. Those formulas illustrate the idea that the twist connects differen t sectors of the SSEP . F urther we analyse how a stationary state of the twisted perio dic SSEP is affected by a mo dification of the t wist, in the same spirit as a quench procedure in condensed matter physics. Sev eral in teresting phenomena can o ccur such as the spreading and splitting of sectors. W e define the branching probabilit y of a stationary state going from one sector to another. T urning on and off the twist sev eral times allows to alternate b et w een different possible stationary states. W e study in details some examples. Finally , w e consider more general set-theoretical solutions of the YBE than Lyubashenk o solutions. W e giv e an example of suc h solution and show that the Marko v mo del constructed from it is not equiv alen t to any twisted SSEP , thereby leaving ro om to more general mo dels coming from set-theoretical solutions of the YBE which could b e studied in further w ork. The plan of the pap er goes as follows. Section 2 introduces p erio dic Mark o v models constructed from Lyubashenk o solutions of the YBE that ha ve the prop ert y of b eing in tegrable and that ha v e v arious in terpretations. Section 3 presen ts the twisted p erio dic SSEP . In Section 3.4 , w e prov e the equiv alence b et w een mo dels constructed from Lyubashenk o solutions and some t wisted p eriodic SSEP . Section 4 focuses on sectors and stationary states of the t wisted p erio dic SSEP . In Section 4.1 , w e show that the probabilit y distribution of the stationary states is uniform. W e classify sector in Section 4.2 and in Sections 4.3 and 4.4 , w e coun t the num ber and find the sizes of sectors. Section 5 analyses ho w stationary states of the twisted SSEP are affected b y a mo dification of the twist. In Section 5.2 , w e describ e p ossible behaviours that can o ccur when turning on and off 2 the twist. Section 6 introduces Marko v models constructed from more general set-theoretical solutions than Lyubashenk o solutions that are not equiv alent to some t wisted p eriodic SSEP in general. In Section 6.2 , we giv e an example of such mo del. 2 Mark o v mo dels from set-theoretical solutions In this section, we in troduce Marko v models associated to some set-theoretical solutions of the Y ang–Baxter equation called Lyubashenko solutions. W e sho w that those mo dels are integrable and provide a non-exhaustiv e list of interesting in terpretations. 2.1 Mark o v mo dels on a p erio dic lattice W e consider a one-dimensional p eriodic lattice comp osed of L sites indexed b y the v ariable i ∈ J 1 , L K . On each site i , the local configuration v ariable τ i tak es v alues in J 0 , N − 1 K . The configurations C of the system corresp ond to L -tuples  τ = ( τ 1 , ..., τ L ) ∈ J 0 , N − 1 K L . The canonical basis vector of a configuration  τ is denoted |  τ ⟩ . All together they form an orthonormal basis i.e. ⟨  τ |  τ ′ ⟩ = δ  τ , τ ′ . The sto c hastic dynamics of the mo del is given b y the Mark o v matrix M = X  τ , τ ′ ∈ C m (  τ ′ →  τ ) |  τ ⟩⟨  τ ′ | , (2.1) where m (  τ ′ →  τ ) ≥ 0 for  τ  =  τ ′ is interpreted as a transition rate betw een configurations  τ ′ and  τ , and b y definition, we ha v e m (  τ →  τ ) = − X  τ ′ ∈ C , τ ′  =  τ m (  τ →  τ ′ ) . (2.2) Starting from an initial probabilit y P 0 on the configuration space, the probability of the system to b e in configuration  τ at con tin uous time t is denoted P t (  τ ) . The probabilities of all configurations at time t are stored in the vector | P t ⟩ = P  τ P t (  τ ) |  τ ⟩ . The ev olution of | P t ⟩ is go verned b y the master equation d | P t ⟩ / dt = M | P t ⟩ . (2.3) W e will consider Marko v matrices whic h are lo cal, b y which w e mean of the following form: M = L − 1 X i =1 m i,i +1 + ˜ m L, 1 , (2.4) where m is a local jump op erator acting on tw o sites, and m i,i +1 is its action on sites i and i + 1 . The additional term ˜ m L, 1 is a local jump op erator ˜ m acting on sites L and 1 . When ˜ m = m , the mo del is p eriodic. If ˜ m L, 1 = T L m L, 1 T − 1 L for some N -dimensional matrix T , the mo del is called t wisted p eriodic (b y the twist T ). 2.2 Lyubashenk o solutions of the Y ang-Baxter equation Fix throughout this section a bijectiv e map denoted: g : J 0 , N − 1 K → J 0 , N − 1 K . F rom g we define the follo wing op erator on t wo sites: ˇ r = X i,j ∈ J 0 ,N − 1 K | g ( j ) , g − 1 ( i ) ⟩⟨ i, j | . (2.5) 3 It satisfies the braided Y ang–Baxter equation: ˇ r 12 ˇ r 23 ˇ r 12 = ˇ r 23 ˇ r 12 ˇ r 23 , (2.6) and is called a Lyubashenko solution of the Y ang–Baxter equation. F urther, it is easy to chec k that it is inv olutiv e in the sense that ˇ r 2 = Id . (2.7) Remark 2.1. The solution ˇ r is set-the or etic al b e c ause it c omes fr om a bije ctive map on the set J 0 , N − 1 K 2 , namely the one given by ( x, y ) 7→ ( g ( y ) , g − 1 ( x )) . F rom ˇ r , w e define the lo cal jump op erator m = ˇ r − Id , (2.8) and the Marko v matrix M = L − 1 X i =1 m i,i +1 + m L, 1 . (2.9) It is easy to see that M is indeed a Marko v matrix, namely that its off-diagonal co efficien ts are non-negativ e and that each column sums to 0 . Note that this is a p eriodic mo del. Also ˇ r is symmetric and therefore M to o, whic h is expressed in terms of the transition rates as: m (  τ →  τ ′ ) = m (  τ ′ →  τ ) , ∀  τ ,  τ ′ ∈ C . 2.3 In tegrabilit y of the mo del Here we sho w that the mo del is in tegrable in the sense that the Marko v matrix belongs to a set of comm uting op erators generated b y a transfer matrix. W e follow the approac h dev eloped by the St Petersburg sc hool on quan tum integrable systems, see e.g. [ 16 , 17 ]. The matrix ˇ r can b e Baxterized as follo ws: ˇ R ( z ) = z ˇ r + Id z + 1 . (2.10) Indeed, since ˇ r 2 = Id , it is easy to c hec k that the Y ang–Baxter equation with sp ectral parameters is satisfied: ˇ R 12 ( u ) ˇ R 23 ( u + v ) ˇ R 12 ( v ) = ˇ R 23 ( v ) ˇ R 12 ( u + v ) ˇ R 23 ( u ) . (2.11) The transfer matrix is constructed with the formula t ( z ) = tr 0 R 0 ,L ( z ) R 0 ,L − 1 ( z ) · · · R 0 , 1 ( z ) , (2.12) where R ( z ) = P ˇ R ( z ) with P the p erm utation op erator. The Y ang–Baxter equation ( 2.11 ) ensures that [ t ( z ) , t ( z ′ )] = 0 , so that up on expansion in z , t ( z ) indeed generates a set of commuting op erators. Moreov er, since R (0) = P , the trace in t (0) and t ′ (0) can b e easily computed, and w e reco v er with a straightforw ard calculation the Marko v matrix of ( 2.9 ) as M = t (0) − 1 t ′ (0) = L − 1 X i =1 m i,i +1 + m L, 1 . (2.13) 4 2.4 In terpretations of the mo dels In the next section, the Marko v mo del ab o v e will b e transformed to a t wisted SSEP mo del, closely related to w ell-studied mo dels. Ho w ev er, we suggest here p ossible direct in terpretations of the Mark o v mo del constructed ab o v e. Recall that we are giv en a bijection g of the set J 0 , N − 1 K , in terms of which the transition rates imply lo cal pro cesses on t w o sites of the form ( τ i , τ i +1 ) → ( g ( τ i +1 ) , g − 1 ( τ i ) ) i = 1 , . . . , L, (2.14) where of course L + 1 means 1 . Remark 2.2. If g is the trivial bije ction (the identity map) then the mo del is simply SSEP [ 38 ]. 2.4.1 Symmetric exclusion pro cess with oscillations The first natural interpretation is the usual one, where w e ha v e particles of N differen t sp ecies lab eled b y a v ariable s ∈ J 0 , N − 1 K whic h evolv e on this p eriodic lattice. The particles are sub jected to a hard core interaction, i.e. each site is o ccupied b y at most one particle. Note how ever that starting from a state with tw o sp ecies a, b lo cated in adjacen t sites, ( 2.14 ) sa ys that it may jump to a state with t w o sp ecies g ( b ) , g − 1 ( a ) lo cated in these sites. Roughly sp eaking, the particles are oscillating b et ween differen t sp ecies while mo ving on the lattice. 2.4.2 Sp ecies with in ternal states Here w e make use of the cyclic decomp osition of the bijection g to pro vide different in terpretations. W e denote the decomp osition of the p erm utation g in to disjoint cycles b y g = π 1 . . . π n , where π 1 , . . . , π n are cycles with disjoin t supp ort in J 0 , N − 1 K . W e reindex the set J 0 , N − 1 K to mak e use of such a decomp osition. W e set: J 0 , N − 1 K = { 1 (0) , . . . , 1 ( c 1 − 1) , 2 (0) , . . . , 2 ( c 2 − 1) , . . . . . . , n (0) , . . . , n ( c n − 1) } , so that the p erm utation b ecomes in cyclic notation: g = (1 (0) , . . . , 1 ( c 1 − 1) )(2 (0) , . . . , 2 ( c 2 − 1) ) . . . . . . ( n (0) , . . . , n ( c n − 1) ) . No w the interpretation is that w e hav e n differen t sp ecies and for each sp ecies s , we ha ve an in ternal state lab elled by a “quantum num b er”, that w e will call for brevity a “charge”, which tak es v alues in { 0 , . . . , c s − 1 } . With these notations, the action of g do es not transform sp ecies, it simply increases the v alue of the c harge b y 1 (while, ob viously , g − 1 reduces the charge b y 1 ). The lo cal processes are no w of the form ( s ( e ) , t ( e ′ ) ) → ( t ( e ′ +1) , s ( e − 1) ) . (2.15) Note that for sp ecies s , the v alue of the c harge is understo o d modulo c s . Example 2.3. In this interpr etation, a p articularly simple c ase is when g has only one cycle: g = (1 (0) , . . . , 1 ( N − 1) ) . Then we have only one sp e cies on our lattic e which c arries a quantum numb er, or a char ge, taking values in the inte gers mo dulo N . Remark 2.4. It may happ en that ther e is one cycle (or mor e) in g of length one. The c orr esp onding sp e cies then c arries no char ge and c an b e interpr ete d as a vacuum. 5 2.4.3 Rolling p olygons Another p ossible in terpretation is closely related to the previous one. One can see each species s as a regular p olygon with c s sides, each side being labeled b y an in teger whic h corresp onds to the ’c harge’ introduced in the previous interpretation. In this context, the polygons are rolling clo c kwise when mo ving in one direction of the lattice and rolling coun terclo c kwise in the opp osite direction, as illustrated in Figure 1. 1 2 3 3 2 4 1 4 3 1 2 3 1 2 Figure 1: T ransitions for rolling triangles and squares. F or example, if g consists of a single full cycle, g = (0 , 1 , . . . , N − 1) , then the lattice is filled by regular N -gons. 2.4.4 Filling and emptying colored b o xes Finally one can also in terpret a lo cal configuration of the form s ( e ) as a b o x “of color s ” con taining a num b er e of items (particles, balls, etc.). In this interpretation, w e hav e b o xes mo ving on the p erio dic lattice. The b o xes are lab eled or colored by an in teger 1 , . . . , n and a box of t yp e s ma y contain up to c s items. When t wo adjacent b o xes are switched on the lattice, one acquires one more item and the other one loses one item. Again, the n umber of items inside a b o x of color s is considered mo dulo c s . This means in particular that a b o x with c s − 1 items b ecomes empt y when acquiring an additional item (which can b e seen as a discharging of the b o x when full). More delicate to in terpret is when an empt y b o x (with 0 item) “loses” an item and b ecomes a b o x containing c s − 1 items. This interpretation is illustrated in Figures 2 and 3. 4 3 4 3 Figure 2: generic transitions in the colored boxes in terpretation. The num b er on the left of a box indicates its maximal num b er of balls. 4 3 3 4 (3.b) 4 3 3 4 (3.c) Figure 3: Ov erfilling (3.b) / emptying (3.c) boxes. Sp ecial transitions in the colored b o xes in terpretation. 6 Note that if g consists of a single full cycle, g = (0 , 1 , . . . , N − 1) , then we ha ve only one type of b o xes which con tain up to N − 1 items (seen modulo N ). 3 T wisted SSEP In this section, we sho w that the Marko v mo dels from the previous section are mathematically equiv alent to some twisted p eriodic SSEP mo dels. W e first define and in terpret these mo dels, sho w directly their in tegrabilit y , and then pro ceed to the description of the corresp ondence with the previous section. 3.1 Definition of the mo del As in the preceding section, our mo del will dep end on a bijection denoted here f : J 0 , N − 1 K → J 0 , N − 1 K . Remark 3.1. It wil l b e explicitly r elate d (but not e qual) to the bije ction g of the pr e c e ding se ction. F or the Mark o v matrix, we tak e the follo wing: M f = L − 1 X i =1 m i,i +1 + m f L, 1 . (3.1) Here, for i ∈ J 1 , L − 1 K , the local jump op erator m i,i +1 acting on sites i and i + 1 corresp onds to the multi-species SSEP op erator m i,i +1 = X τ i ,τ i +1 ∈ J 0 ,N − 1 K | τ i +1 , τ i ⟩⟨ τ i , τ i +1 | − Id . (3.2) The lo cal jump op erator betw een sites L and 1 is given b y m f L, 1 = X τ L ,τ 1 ∈ J 0 ,N − 1 K | f ( τ 1 ) , f − 1 ( τ L ) ⟩⟨ τ L , τ 1 | − Id . (3.3) Iden tifying the bijection f with the linear map acting on one site: f = X τ ∈ J 0 ,N − 1 K | f ( τ ) ⟩⟨ τ | , the lo cal jump op erator m f L, 1 is of the form m f L, 1 = f L m L, 1 f − 1 L , so that we ha ve a twisted perio dic mo del, where the lo cal SSEP op erator betw een sites L and 1 is twisted b y the bijection f . As in the preceding section, it is easy to see that M f is a symmetric matrix. 3.2 In terpretations The twist f added b et ween sites L and 1 has the same v arious interpretations as those given in the preceding sections. Here note that the dynamics b et ween sites i and i + 1 for i = 1 , . . . , L − 1 is the usual one for SSEP , giv en b y ( τ , τ ′ ) → ( τ ′ , τ ) , 7 while b et ween sites L and 1 , it is giv en b y ( τ , τ ′ ) → ( f ( τ ′ ) , f − 1 ( τ ) ) . Therefore, if w e think ab out J 0 , N − 1 K as representing N differen t sp ecies, the sp ecies are affected b y the dynamics only betw een sites L and 1 . As b efore, reindexing the set J 0 , N − 1 K to make use of the cyclic decomp osition of f , we set f = (1 (0) , . . . , 1 ( c 1 − 1) )(2 (0) , . . . , 2 ( c 2 − 1) ) . . . . . . ( n (0) , . . . , n ( c n − 1) ) , and we use the in terpretation of n differen t sp ecies carrying an in ternal degree of freedom, that w e call a “charge”. Then again the dynamics b et ween sites i and i + 1 for i = 1 , . . . , L − 1 is the usual one for SSEP (simple flip of the t wo lo cal en tries) while b et ween sites L and 1 , the c harges are c hanging according to: ( s ( e ) , t ( e ′ ) ) → ( t ( e ′ +1) , s ( e − 1) ) . (3.4) Th us, a non-trivial function f corresp onds to a purely transmitting impurity (or an “electric” field) lo cated b et ween sites L and 1 and whic h affects the “charge” of the particles when they pass through. As b efore, the v arious interpretations with p olygons rolling or with colored b o xes filling and empt ying are p ossible. The new feature compared to the preceding section is that the rolling of p olygons, or the filling/empt ying of b o xes only happ ens when crossing the b oundary b et ween sites L and 1 . 3.3 In tegrability W e pro ceed to show that the abov e models are also in tegrable by adding the t wist in the construction of the transfer matrix. Again, this property is v ery well-kno wn in the con text of quan tum integrable systems, see e.g. [ 16 , 17 ]. The Baxterized R -matrix this time is given b y R ( z ) = z Id + P z + 1 , (3.5) where P is the flip op erator, and satisfies the Y ang Baxter equation with sp ec tral parameter : R 12 ( u ) R 13 ( u + v ) R 23 ( v ) = R 23 ( v ) R 13 ( u + v ) R 12 ( u ) . In general, an integrable t wist op erator is an in vertible matrix T ( z ) satisfying ˇ R ( u − v ) T ( u ) ⊗ T ( v ) = T ( v ) ⊗ T ( u ) ˇ R ( u − v ) , (3.6) where ˇ R ( z ) = P R ( z ) . The homogeneous t wisted transfer matrix is given b y (see e.g. [ 39 ]) t T ( z ) = tr 0 R 0 ,L ( z ) R 0 ,L − 1 ( z ) · · · R 0 , 1 ( z ) T 0 ( z ) , (3.7) and satisfies the comm utativity relation [ t ( y ) , t ( z )] = 0 . The general expression of the t wisted Mark o v matrix is M T = t T (0) − 1 t ′ T (0) = L − 1 X i =1 m i,i +1 + T − 1 L (0) m L, 1 T L (0) + T L (0) − 1 T ′ L (0) , (3.8) In our case, we simply ha ve to tak e T ( z ) = f − 1 = X τ ∈ J 0 ,N − 1 K | f − 1 ( τ ) ⟩⟨ τ | , 8 where again we ha ve iden tified the bijection f − 1 with the corresp onding linear map. W e hav e that T ( z ) is indeed an integrable t wist since it is in vertible and equation ( 3.6 ) is satisfied because the t wist is indep enden t on z and b ecause of the form of R ( z ) in equation ( 3.5 ) . W e hav e by iden tifying equation ( 3.1 ) and equation ( 3.8 ) that T − 1 L (0) m L, 1 T L (0) = f L m L, 1 f − 1 L = m f L, 1 . (3.9) Therefore, the expression ( 3.8 ) pro duces in this case the Mark ov matrix ( 3.1 ) of our mo del and this shows the in tegrability of the mo del. 3.4 Corresp ondence with set-theoretical mo dels F or this subsection, we denote b y ˜ M g the Marko v matrix ( 2.9 ) from Section 2 constructed from the set-theoretical solution of the Y ang–Baxter equation corresp onding to the bijection g . Here we sho w that ˜ M g is conjugated to a Marko v matrix M f in ( 3.1 ) of the twisted SSEP , and moreov er that this conjugation is simply done by a bijection on the configurations. Recall that the set of configurations is C = {  τ = ( τ 1 , . . . , τ L ) , τ i ∈ J 0 , N − 1 K } . In the following proposition, w e mak e again the identification of a bijection of C with the linear map acting on the v ector space of the mo del. Prop osition 3.2. L et f = g L . W e have: ˜ M g = V − 1 M f V , (3.10) wher e V is the bije ction of C given by V ( τ 1 , τ 2 , . . . , τ L ) = ( τ 1 , g ( τ 2 ) , . . . , g L − 1 ( τ L )) . (3.11) Pr o of. It is well known (see for example [ 11 ] for a more general statement) and easy to chec k that the set-theoretical solution ˇ r = X i,j ∈ J 0 ,N − 1 K | g ( j ) , g − 1 ( i ) ⟩⟨ i, j | , (3.12) is conjugated to the permutation op erator P as follows ˇ r = F − 1 P F = G − 1 P G , (3.13) where the maps F , G are defined by F = X i,j ∈ J 0 ,N − 1 K | g ( i ) , j ⟩⟨ i, j | and G = X i,j ∈ J 0 ,N − 1 K | i, g − 1 ( j ) ⟩⟨ i, j | . (3.14) No w consider U and V the op erators acting on the full lattice with L sites defined by U = L − 1 Y i =1 ( g − 1 i ) L − i and V = L Y i =2 g i − 1 i . (3.15) Recall that g i denotes here the linear map induced by g acting on site i (and similarly for g − 1 i ). Using equation ( 3.13 ) relating ˇ r and P , w e hav e U ˇ r i,i +1 U − 1 = V ˇ r i,i +1 V − 1 = ( P i,i +1 ∀ i ∈ J 1 , L − 1 K , g L L P 1 ,L g − L L for i = L , i + 1 = 1 . (3.16) The formula abov e shows immediately that we ha ve ˜ M g = U − 1 M f U = V − 1 M f V when f = g L , (3.17) and this concludes the proof. 9 An imp ortan t feature of the map V conjugating the tw o Marko v matrices in F ormula ( 3.10 ) is that it is a bijection of C of the form: V = π (1) π (2) . . . π ( L ) , (3.18) where each π ( i ) is a bijection of J 0 , N − 1 K acting on the lo cal configuration v ariable of the site i . Therefore, we conclude that the tw o Mark ov mo dels giv en resp ectiv ely by ˜ M g and M f (with f = g L ) are mathematically equiv alent since an explicit bijection on the configuration space sends one into another. In other words, any mathematical property of one mo del is reflected to an equiv alent prop ert y of the other model; for example, the n um b er of sectors, their cardinalit y , etc. (see the next section). Remark 3.3. One has to b e c ar eful that the bije ction V c orr esp onds to lo c al r elab elings at e ach site i of the lattic e which dep end on the site i . This is not a simple r elab eling of the set J 0 , N − 1 K . Nevertheless, dep ending on the physic al interpr etation of the mo del, one may use the p oint of view c orr esp onding to ˜ M g or the one c orr esp onding to M f . Remark 3.4. Notic e that the existenc e of a p ermutation g satisfying g L = f for a given f is not guar ante e d. A sufficient and ne c essary c ondition for the existenc e of an L -th r o ot of f is given in [ 32 ]. Henc e the set-the or etic al mo dels of the pr evious se ction ar e e quivalent to a sub class (but not al l) of the twiste d SSEP mo dels define d in this se ction. 4 Sectors and stationary states Ha ving sho wn in the preceding section that the mo dels from Section 2 constructed from set- theoretical solutions of the Y ang–Baxter equation are mathematically equiv alen t to some twisted SSEP mo dels from Section 3 , w e pro ceed to the study of these twisted SSEP models. W e will see that there are sev eral stationary states whic h are all giv en b y a uniform probability on sectors (irreducible subsets for M f of the configuration space). Therefore, w e are reduced to the com binatorial study of these sectors. In this section, it will b e con venien t to fix, as before, the bijection f in the form (cycles decomp osition) f = (1 (0) , . . . , 1 ( c 1 − 1) )(2 (0) , . . . , 2 ( c 2 − 1) ) . . . . . . ( n (0) , . . . , n ( c n − 1) ) , and recall that the corresponding Mark ov matrix is denoted M f . W e remind that the particles are denoted s ( e ) where s is the sp ecies and e its charge, with 1 ≤ s ≤ n and 0 ≤ e ≤ c s − 1 . The non-zero transition rates are giv en by ( s ( e ) , t ( e ′ ) ) 1 ← → ( t ( e ′ ) , s ( e ) ) for sites ( i, i + 1) with 1 ≤ i < L , ( s ( e ) , t ( e ′ ) ) 1 ← → ( t ( e ′ +1) , s ( e − 1) ) b et ween sites L and 1 , (4.1) where the charges for the sp ecies s are understo od mo dulo c s . 4.1 Stationary states A stationary state | S ⟩ = P  τ ∈ J 0 ,N − 1 K L S (  τ ) |  τ ⟩ is a probability vector (the co efficien ts sum to 1 ) whic h is inv ariant under the dynamics of the pro cess, that is to sa y , it satisfies M f | S ⟩ = 0 . The num b er of stationary states is equal to the num b er of sectors of the Marko v process, whic h are defined as follows 10 Definition 4.1 (sector) . F or a Markov pr o c ess define d on the c onfigur ation sp ac e C with tr ansitions given by the Markov matrix M , the se ctor of a c onfigur ation  τ ∈ C c orr esp onds to the e quivalenc e class of  τ for the fol lowing e quivalenc e r elation :  τ ∼  τ ′ ⇔ ∃ k , l ∈ N , ⟨  τ | M k |  τ ′ ⟩  = 0 and ⟨  τ ′ | M l |  τ ⟩  = 0 . (4.2) F or all the pro cesses considered in this article, the Mark o v matrices are symmetric so if there exists a k ∈ N suc h that ⟨  τ | M k |  τ ′ ⟩  = 0 , then ⟨  τ ′ | M k |  τ ⟩  = 0 and vice v ersa. Remark 4.2. In the c ase of the twiste d SSEP, it is e asy to se e fr om the definition of M f that the se ctor of a c onfigur ation  τ is e quivalent to the orbit of the gr oup gener ate d by the fol lowing two op er ations : • Permuting the entries of  τ ; • the op er ation f 1 ,L which incr e ases the first char ge by 1 and de cr e ases the last char ge by 1. F or the usual p erio dic SSEP, this gr oup is the symmetric gr oup. F or the twiste d p erio dic SSEP, it includes the symmetric gr oup so the orbits ar e lar ger. Remark 4.3. If a Markov pr o c ess has a single se ctor, the pr o c ess is said to b e irr e ducible. This is not the c ase of the twiste d p erio dic SSEP. W e denote the set of sectors b y S = { sectors } = { C γ , γ ∈ Γ } , (4.3) where Γ is an indexing set for the sectors. F or an y sector C γ , there exists a unique stationary state denoted | S γ ⟩ . This is pro ven in [ 31 , p.111,p118] where the pro cess is assumed to b e recurren t on eac h class, a notion that w e do not discuss further but which is true since the num b er of configurations in a class is finite [ 31 , p.27]. Prop osition 4.4. In e ach se ctor C γ , the pr ob ability distribution of the stationary state is uniform, that is, | S γ ⟩ = 1 | C γ | X  τ ∈ C γ |  τ ⟩ . (4.4) Pr o of. W e already know that the stationary state is unique for a given sector. Since ( 4.4 ) is already normalised correctly (the coefficients sum to 1 ), we only need to show that M f | S γ ⟩ = 0 . Recall that M f is symmetric and denote its matrix co efficien ts b et ween t wo configurations by m (  τ ′ →  τ ) . W e hav e M f X  τ ∈ C γ |  τ ⟩ = X  τ ′ ∈ C γ m (  τ ′ →  τ ) |  τ ′ ⟩ = X  τ ′ ∈ C γ  τ ′  =  τ  m (  τ ′ →  τ ) − m (  τ →  τ ′ )  |  τ ⟩ = 0 , (4.5) where we used F ormula ( 2.2 ) for the diagonal matrix co efficien ts of M f . The case of the usual N -sp ecies SSEP . The usual N -sp ecies SSEP corresp onds to the case f = Id . It is not irreducible and its sectors corresp ond to configurations with a fixed n umber of particles of eac h sp ecies. It is in bijection with the n umber of multisets of the size of the lattice L with elements tak en in the set J 0 , N − 1 K . So the num b er of sectors is |S | =  N − 1 + L N − 1  . 11 The cardinality of the sector C γ corresp onding to p i particles of sp ecies i for i ∈ J 0 , N − 1 K is the m ultinomial co efficien t | C γ | =  L p 0 , ..., p N − 1  . W e are going to generalise these form ulas to the general case (arbitrary f ). Note that it is clear that the presence of the twist will reduce the num b er of sectors (while increasing their sizes), b ecause the t wist induces sp ecies transformations. Therefore, tw o different sectors of the usual SSEP can b elong to the same sector of a twisted SSEP . Remark 4.5. F or any c onfigur ation  τ ,  τ ′ of the twiste d SSEP, the micr osc opic pr ob ability curr ent j  τ →  τ ′ = m (  τ →  τ ′ ) S (  τ ) − m (  τ ′ →  τ ) S (  τ ′ ) vanishes for any stationary state S and the system is at thermo dynamic e quilibrium as in the usu a l N -sp e cies SSEP. 4.2 In v arian ts and sectors 4.2.1 Lab elling sectors Recall that a configuration is denoted by  τ = ( s ( e 1 ) 1 , . . . , s ( e L ) L ) with s i ∈ J 1 , n K and e i ∈ J 0 , c s i − 1 K , where s i corresp onds to the sp ecies and e i to the charge of the particle laying at site i . Definition 4.6 (Profile) . The pr ofile of a c onfigur ation  τ is define d by p (  τ ) = ( p 1 , . . . , p n ) wher e p k = |{ i ∈ { 1 , . . . , L } such that s i = k }| for k = 1 , . . . , n . In wor ds, the value p k r e c or ds how many times the sp e cies k app e ars in  τ . F or a configuration  τ , the sp ecies which app ear at least once are called the  τ -relev ant sp ecies . They dep end only on the profile p (  τ ) and are the ones for which p k  = 0 . Definition 4.7 (T otal charge) . The total char ge of a c onfigur ation  τ is define d by E (  τ ) = L X i =1 e i ! mod D  τ , wher e D  τ = g cd ( c k ∈ K  τ ) , (4.6) with K  τ ⊂ J 1 , n K the set of  τ -r elevant sp e cies. Note that for a giv en p osition i , if in ( 4.6 ) w e replace e i b y , sa y , e i + c s i , this do es not change the v alue of the total c harge, since by construction D  τ divides eac h c s i of  τ -relev ant species s i . In this sense, the total c harge is consistent with the definition of the charge of a sp ecies. Remark 4.8. F or K  τ the set of  τ -r elevant sp e cies, the total char ge of this c onfigur ation is define d mo dulo D  τ = gcd ( c k , k ∈ K  τ ) which dep ends in gener al on  τ . When the bije ction f of the mo del M f has al l its cycles of the same size c , then D  τ = c and it is indep endent on the c onfigur ation. W e are ready to characterise the sectors. Prop osition 4.9. A se ctor is uniquely determine d by a pr ofile and a total char ge. Namely,  τ and  τ ′ ar e in the same se ctor if and only if p (  τ ) = p (  τ ′ ) and E (  τ ) = E (  τ ′ ) . Pr o of. Recalling Remark 4.2 , the sector of a given  τ consists of all elements  τ ′ obtained b y rep eated applications of the p erm utation swapping the en tries of  τ and of the p erm utation f 1 ,L whic h increases the first charge b y 1 and decreases the last c harge b y 1. Combining these t w o op erations, one obtains also for any i, j the op eration f i,j , which adds 1 from the c harge at p osition i and subtracts 1 to the c harge at p osition j . 12 These op erations clearly do not c hange the profile. F or the total c harge, let us consider the op eration f 1 ,L . Then the charge e 1 is changed into e 1 − 1 or e 1 − 1 + c s 1 (if e 1 = 0 ) while the c harge e L b ecomes e L + 1 or e L + 1 − c s L (if e L = c s L ). In an y case, since D  τ divides c s 1 and c s L the sum remains the same. No w assume that p (  τ ) = p (  τ ′ ) and E (  τ ) = E (  τ ′ ) . W e need to sho w that  τ ′ is obtained from  τ b y a succession of the abov e op erations. Denote  τ = ( s ( e 1 ) 1 , . . . , s ( e L ) L ) as b efore. Since the profiles are the same, with permutations w e can bring  τ ′ to b e of the form  τ ′ = ( s ( e ′ 1 ) 1 , . . . , s ( e ′ L ) L ) . Then using op erations of the form f i,L for i = 1 , . . . , L − 1 , w e can bring  τ ′ to b e of the form  τ ′ = ( s ( e 1 ) 1 , . . . , s ( e L − 1 ) L − 1 , s ( e ′ L ) L ) . F rom the assumption that the total c harges are equal, we ha ve that e ′ L = e L mod D  τ . Hence, to conclude the pro of, it is enough to sho w that w e can change the v alue of e ′ L b y D  τ . Let K  τ ⊂ { 1 , . . . , n } b e the set of  τ -relev ant sp ecies. Recall that D  τ = g cd ( c k , k ∈ K  τ ) . Therefore, there exist integers m k ∈ Z (Bezout identit y) such that X k ∈ K  τ m k c k = D  τ . No w for each k ∈ K  τ suc h that k  = s L , we c ho ose a p osition i ∈ { 1 , . . . , L − 1 } suc h that s i = k and we apply the op eration f i,L to the p o wer m s i c s i . W e see that the charge at position i remains the same since we are adding a multiple of c s i . Having done that for every k ∈ K  τ with k  = s L , w e see that the c harge at p osition L has c hanged by X k ∈ K  τ \{ s L } m k c k = D  τ − m s L c s L . This is equiv alent to changing the c harge at p osition L b y D  τ since this charge is considered mo dulo c s L . 4.3 Num b er of sectors In Section 4.1 , w e recalled the n umber of sectors for the usual N -sp ecies SSEP . Using the lab elling of the sectors of the section 4.2.1 , w e can now coun t them and generalise the formula giving their n um b er for an arbitrary p erm utation f . Prop osition 4.10. The numb er of se ctors |S | of the twiste d SSEP on a lattic e with L ≥ 2 sites and with p ermutation f = (1 (0) , . . . , 1 ( c 1 − 1) )(2 (0) , . . . , 2 ( c 2 − 1) ) . . . . . . ( n (0) , . . . , n ( c n − 1) ) is |S | = X ∅ = X ⊂ J 1 ,n K  L − 1 | X | − 1  gcd( c x , x ∈ X ) . (4.7) Pr o of. F rom Proposition 4.9 , the n umber of sectors |S | is equal to the num b er of distinct pairs ( p (  τ ) , E (  τ )) for  τ ∈ J 0 , N − 1 K L . F or a giv en ∅  = X ⊂ J 1 , n K , w e consider the configurations  τ suc h that X is the set of  τ -relev ant species. By definition of the total c harge, such configurations hav e gcd ( c x , x ∈ X ) p ossible v alues for their total c harge. The n umber of p ossible profiles is equal to the num b er of comp osition of L in to | X | parts, which is  L − 1 | X | − 1  . Th us there are  L − 1 | X | − 1  gcd ( c x , x ∈ X ) distinct pairs ( p (  τ ) , E (  τ )) for a given ∅  = X ⊂ J 1 , n K . The total n umber of sectors is obtained by summing ov er all nonempty subsets X ⊂ J 1 , n K . 13 Example 4.11. Her e we apply formula ( 4.7 ) in the c ase wher e f has cycles of the same length, i.e., when al l sp e cies have the same numb er of p ossible char ges. L et f = (1 (0) , . . . , 1 ( N − 1) ) J wher e J ∈ N . The p ower J has the effe ct of splitting the ful l cycle into d cycles, e ach of length N /d wher e d = gcd( N , J ) . In that c ase, formula ( 4.7 ) b e c omes |S | = X ∅ = X ⊂ J 1 ,d K  L − 1 | X | − 1  N d = N d d X k =1  d k   L − 1 k − 1  = N d  L + d − 1 L  , (4.8) wher e we gr oup e d terms c orr esp onding to sets X of size k in the se c ond step and we use d the V andermonde’s identity in the last one. The c ase f = Id c orr esp onds to J = 0 so d = N and we r e c over the numb er of se ctors of the p erio dic SSEP. The c ase f = (1 (0) , . . . , 1 ( N − 1) ) c orr esp onds to J = 1 so d = 1 and ther e ar e N se ctors c orr esp onding to the N p ossible values of the total char ge. Remark 4.12. When f has d cycles of the same length N /d , the numb er of se ctors se en as a function of d , is strictly incr e asing. Inde e d, for S ( d ) = N d  L + d − 1 L  , (4.9) and L ≥ 2 , we have that S ( d +1) S ( d ) = d + L d +1 > 1 . Remark 4.13. The c ases f = Id and f = (1 (0) , . . . , 1 ( N − 1) ) r esp e ctively maximise and minimise the numb er of se ctors of the twiste d SSEP. This is cle ar for f = Id sinc e a se ctor of the SSEP is always include d in a (lar ger) se ctor of any twiste d SSEP. F or f = (1 (0) , . . . , 1 ( c 1 − 1) )(2 (0) , . . . , 2 ( c 2 − 1) ) . . . . . . ( n (0) , . . . , n ( c n − 1) ) we c an r ewrite Equation ( 4.7 ) as |S | = n X k =1  L − 1 1 − 1  c k + X X ⊂ J 1 ,n K | X | > 2  L − 1 | X | − 1  gcd( c x , x ∈ X ) (4.10) = N + X X ⊂ J 1 ,n K | X | > 2  L − 1 | X | − 1  gcd( c x , x ∈ X ) . (4.11) The se c ond term is p ositive so N is a lower b ound for | S | which is r e ache d for the ful l cycle f = (1 (0) , . . . , 1 ( N − 1) ) so N is inde e d the minimum of | S | . 4.4 Cardinalit y of sectors W e no w determine the size of a sector and show that, for a fixed p erm utation f , it only dep ends on the profile of the configurations of that sector. As in the previous section, we then apply the form ula for a p erm utation with cycles of equal length. Prop osition 4.14. The c ar dinality of a se ctor C γ with pr ofile p = ( p 1 , ..., p n ) and total char ge E is | C γ | =  L ! p 1 ! ...p n !  c p 1 1 ...c p n n gcd( c x , x ∈ X ) , (4.12) wher e X = { k ∈ J 1 , n K with p k  = 0 } is the set of r elevant sp e cies. In p articular, the c ar dinality do es not dep end on the total char ge E . 14 Pr o of. F rom Prop osition 4.9 , a sector C γ is uniquely lab eled b y a pair ( p, E ) where p = ( p 1 , . . . , p n ) is the profile and E is the total c harge taken mo dulo D with D = gcd ( c x , x ∈ X ) . In the following, w e first coun t the num b er of configurations with profile p . Then, among those configurations, we coun t the ones that ha ve total c harge E . Ignoring the charges, the n umber of w ays of distributing p 1 , . . . , p n sp ecies of particles on a lattice with L sites is L ! p 1 ! ...p n ! . No w taking in to account the c harge of particles, once the particle sp ecies is fixed for all sites, there are p s particle of sp ecies s that can take c s p ossible c harges, for eac h s . This implies that the num b er of configurations with profile p is L ! p 1 ! ...p n ! n Y s =1 c p s s . (4.13) No w for a configuration  τ = ( s ( e 1 ) 1 , . . . , s ( e L ) L ) , having the toal c harge E giv es the following constrain ts on the charge: e 1 + · · · + e L = E mo d D . This only restricts the v alue of e L to b e E − ( e 1 + · · · + e L − 1 ) mo d D . Sa y the sp ecies in site L is k then e L is taken modulo c k . Since D divides c k b y construction, there are c k D p ossible v alues for e L instead of c k when the total charge is not fixed. Comparing with ( 4.13 ), this pro v es ( 4.12 ). Example 4.15. W e c an apply formula ( 4.12 ) in the c ase wher e f has d cycles of the same length N /d . W e have | C γ | =  L ! p 1 ! ...p d !   N d  L − 1 . (4.14) The c ase f = Id c orr esp onds to d = N and we r e c over the multinomial c o efficient c orr esp onding to | C γ | for the p erio dic SSEP. The c ase f = (1 (0) , . . . , 1 ( N − 1) ) c orr esp onds to d = 1 and | C γ | = N L − 1 . The p ower L − 1 in the formula tel ls that the char ge for the last site is fixe d by the total char ge. 5 T wist tuning and branc hing probabilities In this part, we analyse the behaviour of a stationary state under a mo dification of the twist. Namely , w e choose a stationary state for some twisted SSEP and w e consider its evolution in a t wisted SSEP with a different t wist. In other w ords, we study a quenc h b et ween t wo t wisted SSEP with differen t t wists. F or example, when one of the models is the usual (non-t wisted) SSEP , this pro cedure can b e seen as turning on/off a twist on a given bond of the p eriodic lattice. 5.1 Ov erlaps and branc hing probabilities W e consider tw o bijections f (1) and f (2) of J 0 , N − 1 K , and w e fix a stationary state | S (1) ⟩ corresp onding to a sector C (1) for the twisted SSEP asso ciated to f (1) . W e recall that | S (1) ⟩ = 1 | C (1) | X  τ ∈ C (1) |  τ ⟩ . (5.1) No w we consider | S (1) ⟩ as the initial condition for the Marko v mo del corresp onding to f (2) , w e let it ev olve using the Mark o v matrix M f (2) and we lo ok at the p ossible stationary states (for M f (2) ) that are reached at t = ∞ . W e denote S (2) the set of sectors for the model associated to f (2) , and by | S (2) γ ⟩ the stationary state asso ciated to γ ∈ S (2) . Definition 5.1. F or γ ∈ S (2) , we denote by pr ob ( C (1) → C (2) γ ) the pr ob ability that the system ends up in the stationary state | S (2) γ ⟩ . W e c al l it the br anching pr ob ability fr om C (1) to C (2) γ . 15 A t t = ∞ , the state will be of the form: X γ ∈S (2) pr ob ( C (1) → C (2) γ ) | S (2) γ ⟩ . (5.2) Note that we ha ve X γ ∈S (2) pr ob ( C (1) → C (2) γ ) = 1 , since the stationary vectors are normalised (in the sense that the sum of their comp onen ts is 1, see ( 5.1 ) ), and the evolution of a probability vector under a Mark ov process remains a probabilit y v ector. The branching probabilities are simply calculated in terms of | S (1) ⟩ and | S (2) γ ⟩ . Indeed, since the stationary states | S (2) γ ⟩ do not evolv e (by definition) under M f (2) , the coefficients in the expansion ( 5.2 ) are the same as the corresp onding coefficients in the initial state | S (1) ⟩ . Therefore w e hav e pr ob ( C (1) → C (2) γ ) = ⟨ S (1) | S (2) γ ⟩ ⟨ S (2) γ | S (2) γ ⟩ . So far, the discussion was v alid in a general setting. No w we use the explicit formula ( 5.1 ) expressing that for our models, the stationary states corresp ond to uniform probabilities on sectors. W e hav e immediately ⟨ S (1) | S (2) γ ⟩ = | C (1) ∩ C (2) γ | | C (1) || C (2) γ | . Therefore, noticing that ⟨ S (2) γ | S (2) γ ⟩ = 1 | C (2) γ | , we conclude with the follo wing result. Prop osition 5.2. The br anching pr ob abilities ar e given by: pr ob ( C (1) → C (2) γ ) = | C (1) ∩ C (2) γ | | C (1) | . 5.2 Spreading, splitting and oscillations of sectors Spreading of a sector. Here w e consider the particular case where the sector C (1) of the first mo del is included in a sector C (2) γ 0 of the second mo del, that is we ha ve C (1) ⊂ C (2) γ 0 , for some sector γ 0 ∈ S (2) . In this case, the branc hing probabilities are trivial, in the sense that, according to Prop osition 5.2 , we ha ve pr ob ( C (1) → C (2) γ ) = δ γ ,γ 0 . The only thing to do is to determine, given C (1) and C (2) γ , whether C (1) ⊂ C (2) γ or C (1) ∩ C (2) γ = ∅ . Examples are detailed in the next subsection. Remark 5.3. Note that although the tr ansition pr ob ability is trivial (0 or 1), the twist affe cts the dynamics of the twiste d SSEP. Inde e d, in the initial state, only the c onfigur ations  τ in C (1) have a non-zer o pr ob ability 1 / | C (1) | while in the final state, mor e c onfigur ations  τ ′ ∈ C (2) γ have a non-zer o (but smal ler) pr ob ability 1 / | C (2) γ | . 16 Splitting of a sector. The opposite situation is when one reverses the role of f (1) and f (2) : w e start with the t wist f (2) and we go on with the twist f (1) . In this case, a sector C (2) for f (2) splits into a disjoin t union of sectors for f (1) : C (2) = C (1) γ 1 ⊔ · · · ⊔ C (1) γ K . No w the branching probabilities are non-trivial and they are giv en, according to Prop osition 5.2 b y pr ob ( C (2) → C (1) γ ) = | C (1) γ | | C (2) | for γ ∈ { γ 1 , . . . , γ K } , and 0 otherwise. Examples are given below. Remark 5.4. The splitting phenomenon, wher e C (2) splits into a disjoint union of classes for f (1) app e ars systematic al ly when f (1) is a p ower of f (2) , namely if f (1) = ( f (2) ) k for some k ≥ 0 . R oughly sp e aking, the p ermutation f (2) p ermutes mor e things than the p ermutation f (1) , ther efor e the se ctors for f (2) ar e lar ger than the ones for f (1) . Oscillations of sectors. Assume w e are in a spreading/splitting situation, namely w e hav e a sector for f (2) whic h is a disjoint union of sector for f (1) : C (2) = C (1) γ 1 ⊔ · · · ⊔ C (1) γ K . Then w e can start from a sector C (1) γ i 1 then w e alternate b et ween turning on the twist f (2) and turning on the twist f (1) . Then sector will alternatively spread and split, that is, schematically: C (2) C (2) C (2) ↗ ↘ ↗ ↘ ↗ ↘ → C (1) γ i 1 C (1) γ i 2 C (1) γ i 3 C (1) γ i 4 ......... where the up-arro w ↗ corresp onds to switching on f (2) (and switc hing off f (1) ) and the do wn arro w to switching on f (1) (and switc hing off f (2) ). W e see that the configuration will oscillate b et ween the sectors C (1) γ 1 , . . . , C (1) γ K : C (1) γ i 1 prob ( C (2) → C (1) γ i 2 ) − → C (1) γ i 2 prob ( C (2) → C (1) γ i 3 ) − → C (1) γ i 3 prob ( C (2) → C (1) γ i 4 ) − → C (1) γ i 4 → ......... Example 5.5. T aking f (1) = Id , that is the usual SSEP mo del, we se e that turning on and off an arbitr ary twist f (2) , it is p ossible to move b etwe en differ ent stationary states of the usual SSEP mo del. 5.3 Example: usual SSEP and an arbitrary t wist An ob vious case where the spreading/splitting situation happens is when the first mo del is the usual SSEP (the twist f (1) = Id is trivial) and the second mo del can hav e an arbitrary twist f (2) . In this case, we write f (2) with its cycle notation using sp ecies and c harges, that is w e take: f (1) = Id and f (2) = (1 (0) , . . . , 1 ( c 1 − 1) )(2 (0) , . . . , 2 ( c 2 − 1) ) . . . . . . ( n (0) , . . . , n ( c n − 1) ) . A sector C (2) γ for f (2) is indexed b y a profile p (2) = ( p 1 , . . . , p n ) and a total c harge E (2) , as defined in Section 4.2 : C (2) p (2) ,E (2) : p (2) = ( p 1 , . . . , p n ) and E (2) = k mo d D , 17 where D is the greatest common divisor of the relev ant species (see Definition 4.7 ). Recall that the profile p (2) simply coun ts the n umber of o ccurrences of eac h sp ecies while the total c harge E (2) is the sum of the c harges mo dulo D . Now a sector C (1) for the usual SSEP mo del is only indexed by a profile, whic h is of the form: C (1) p (1) : p (1) = ( p 1 (0) , p 1 (1) , . . . , p n ( c n − 1) ) =  p s ( e )  s =1 ,...,n e =0 ,...,c s − 1 . Indeed recall that from the point of view of f (1) = Id , the profile simply coun ts the num b er of o ccurrences of ev ery s ( e ) , for v arious s and e . The inclusion condition is expressed as follo ws: C (1) p (1) ⊂ C (2) p (2) ,E (2) ⇐ ⇒            c s − 1 X e =0 p s ( e ) = p s , ∀ s = 1 , . . . , n , n X s =1 c s − 1 X e =0 ep s ( e ) = E (2) mod D . (5.3) Therefore we ha ve according to Prop osition 5.2 pr ob ( C (1) p (1) → C (2) p (2) ,E (2) ) = ( 1 if ( 5.3 ) is satisfied, 0 otherwise. This was the spreading situation. Now for the splitting situation, w e rev erse the p osition of the sectors, and we get, according to Prop osition 5.2 pr ob ( C (2) p (2) ,E (2) → C (1) p (1) ) =        | C (1) p (1) | | C (2) p (2) ,E (2) | if ( 5.3 ) is satisfied, 0 otherwise. W e recall that the explicit formula for the cardinality of a sector is given in Prop osition 4.14 . 5.4 Example: T wo differen t non-trivial twists Here, for the first mo del, w e take for the t wist the full cycle f (1) = (0 , 1 , . . . , N − 1) . Therefore, in terms of charged sp ecies, for the first mo del, w e ha ve f (1) = (1 (0) , 1 (1) , . . . , 1 ( N − 1) ) , (5.4) and a sector C (1) k is lab eled b y the total c harge E (1) = k mo d N (the profile b eing p (1) = ( L ) ). T o get the c harge k , one can put arbitrary charges in L − 1 sites, the last one b eing used to fix the total charge. Hence, the sector has dimension | C (1) k | = N L − 1 , in accordance with Example 4.15 . The case f (2) = (1 , 3 , . . . N − 1)(2 , 4 , . . . N ) with N ev en. F or the second mo del, we tak e for f (2) the square of the first twist, whic h splits into tw o cycles since N is ev en. In terms of c harged sp ecies, w e hav e f (2) = (1 (0) , 1 (2) , . . . , 1 ( N − 2) )(1 (1) , 1 (3) , . . . , 1 ( N − 1) ) = (2 (0) , 2 (1) , . . . , 2 ( N/ 2 − 1) )(3 (0) , 3 (1) , . . . , 3 ( N/ 2 − 1) ) . (5.5) In the second line, w e hav e written the charged species from the p oin t of view of f (2) . Since f (2) only p erm utes the elements 1 ( e ) for a giv en parit y of e , it is as if w e had tw o different sp ecies. In other words, w e hav e set 2 ( e ) = 1 (2 e ) and 3 ( e ) = 1 (2 e +1) . 18 A sector for f (2) is lab eled by the profile p (2) = ( p 2 , p 3 ) (with p 2 + p 3 = L ) and the total c harge E (2) = ℓ mo d N/ 2 . A particular configuration for this sector is given b y (2 ( ℓ ) , 2 (0) , . . . , 2 (0) | {z } p 2 , 3 (0) , 3 (0) , . . . , 3 (0) | {z } p 3 ) if p 2  = 0 , (3 ( ℓ ) , 3 (0) , . . . , 3 (0) | {z } p 3 = L ) if p 2 = 0 . (5.6) In b oth cases, the charge E (1) for this configuration is E (1) = 2 ℓ + p 3 mo d N . Since p 2 + p 3 = L is alw a ys satisfied, pr ob ( C (1) k → C (2) p (2) ,ℓ ) is non-v anishing if and only if k = 2 ℓ + p 3 mo d N . Denoting b y C (2) γ the sectors which obey this equalit y , the probabilit y is given b y pr ob ( C (1) k → C (2) γ ) = | C (2) γ | | C (1) k | = L ! p 2 ! p 3 !  1 2  L − 1 . (5.7) Remark 5.6. Using this expr ession, one c an che ck that P γ pr ob ( C (1) k → C (2) γ ) = 1 , as it should. Note that the sum should b e taken over l and ( p 2 , p 3 ) with the c onstr aints that p 2 + p 3 = L and p 3 = k − 2 l mo d N . Dep ending on k , only the even or only the o dd values of p 3 ar e al lowe d. Then one uses the e quality X p 2 + p 3 = L p 3 = a mo d 2 L ! p 2 ! p 3 ! = 2 L − 1 , (5.8) valid for a = 0 , 1 (this is che cke d by exp anding (1 + 1) L and (1 − 1) L ). The case f (2) = ( f (1) ) n − 1 with N = ( n − 1) D . More generally , keeping f (1) as ab o v e, w e assume now that N = ( n − 1) D (the previous case w as n = 3 ). W e take f (2) = (2 (0) , ..., 2 ( D − 1) )(3 (0) , ..., 3 ( D − 1) ) · · · ( n (0) , ..., n ( D − 1) ) , (5.9) with the reindexing done similarly as b efore. The sectors are no w characterised by the profile p (2) = ( p 2 , p 3 , . . . , p n ) and a total charge E (2) = ℓ mo d D . A particular configuration for this sector is giv en by (2 ( ℓ ) , 2 (0) , . . . , 2 (0) | {z } p 2 , 3 (0) , 3 (0) , . . . , 3 (0) | {z } p 3 , . . . , n (0) , n (0) , . . . , n (0) | {z } p n ) , (5.10) where we hav e supp osed that p 2  = 0 without loss of generality . The condition of non-empty in tersection b et ween C (1) k and C (2) p (2) ,ℓ is expressed as ( n − 1) ℓ + n X j =2 ( j − 2) p j = k mo d N , (5.11) and, when this condition is satisfied, pr ob ( C (1) k → C (2) γ ) = L ! p 2 ! p 3 ! . . . p n !  1 n − 1  L − 1 . (5.12) Remark 5.7. As in R emark 5.6 , the pr op erty P γ pr ob ( C (1) k → C (2) γ ) = 1 c an b e che cke d dir e ctly and is ensur e d by the identity X p 2 + ... + p n = L p 3 +2 p 4 + .. +( n − 2) p n = a mo d n − 1 L ! p 2 ! p 3 ! . . . p n ! = ( n − 1) L − 1 , (5.13) which is valid for al l values a = 0 , 1 , ..., n − 2 . This is che cke d by exp anding (1 + ξ + · · · + ξ n − 2 ) L for al l ( n − 1) -st r o ots of unity ξ . 19 6 More general solutions of set-theoretical Y ang-Baxter equation In this section, w e introduce Marko v mo dels constructed from more general set-theoretical maps than the ones previously considered. W e then giv e a non trivial example of such map. W e sho w that its corresp onding Marko v mo del for a small lattice ( L = 3 ) is not equiv alent to any t wisted p eriodic SSEP as defined in this pap er. 6.1 Mark ov mo dels W e no w consider set-theoretical maps that are more general than the ones in Section 2.2 . F or i ∈ J 0 , N − 1 K , we consider the family of bijective maps g i , f i : J 0 , N − 1 K → J 0 , N − 1 K , that are chosen in suc h a w a y that the follo wing operator on t w o sites ˇ r = X i,j ∈ J 0 ,N − 1 K | g i ( j ) , f j ( i ) ⟩⟨ i, j | . (6.1) satisfies the YBE ( 2.6 ) and is inv olutiv e as in equation ( 2.7 ). Remark 6.1. A Lyub ashenko solution has the same form as Equation ( 6.1 ) , but the bije ctions g i and f i ar e indep endent of i , so that they ar e r esp e ctively al l e qual to some g and f . The maps g i and f i suc h that ˇ r ob eys the YBE, satisfy for all i, j, k ∈ J 0 , N − 1 K g i ( g j )( k ) = g g i ( j ) ( g f j ( i ) ( k )) , f k ( f j ( i )) = f f k ( j ) ( f g j ( k ) ( i )) , f g f j ( i ) ( k ) ( g i ( j )) = g f g j ( k ) ( i ) ( f k ( j )) . (6.2) The maps g i and f i suc h that ˇ r 2 = Id satisfy for all i, j ∈ J 0 , N − 1 K g g i ( j ) ( f j ( i )) = i, f f j ( i ) ( g i ( j )) = j . (6.3) In v olutiv e solutions of the YBE where g i , f i are bijections are equiv alent to algebraic structures called braces [ 34 ]. The lo cal jump op erator and the Mark o v matrix constructed from this op erator are defined iden tically as previously in Equations ( 2.8 ) and ( 2.9 ) . It can b e easily chec ked that the resulting mo del is a Mark ovian perio dic mo del. It also has the prop ert y of b eing in tegrable. This comes from the fact that the op erator ( 6.1 ) is in volutiv e, which allows to Baxterize it and to construct a set of commuting operators as in Section 2.3 . In terpretation. The form of the map ( 6.1 ) implies that the transition rates b et ween tw o adjacen t sites hav e the form ( τ i , τ i +1 ) → ( g τ i ( τ i +1 ) , f τ i +1 ( τ i ) ) i = 1 , . . . , L . (6.4) The in terpretations that w e can give for models constructed from those more general set-theoretical solutions are in general harder to find than the ones giv en in section 2.4 . Separable and non-separable bijections. In Section 3.4 , Prop osition 3.2 , w e prov ed the equiv alence b et ween Mark ov mo dels constructed from Lyubashenko solutions and the t wisted p eriodic SSEP . In that case, the bijection on the configurations had the form V sep = π (1) π (2) ...π ( L ) , (6.5) 20 and w e call it separable. F or set-theoretical solutions ( 6.1 ) , there exist non separable bijections of C given b y U ( τ 1 , τ 2 , . . . , τ L ) = ( f τ L f τ L − 1 . . . f τ 2 ( τ 1 ) , . . . , f τ L ( τ L − 1 ) , τ L ) , (6.6) and V ( τ 1 , τ 2 , . . . , τ L ) = ( τ 1 , g τ 1 ( τ 2 ) , . . . , g τ 1 g τ 2 . . . g τ L − 1 ( τ L )) , (6.7) satisfying ∀ i ∈ J 1 , L − 1 K , ˇ r i,i +1 = U − 1 P i,i +1 U = V − 1 P i,i +1 V . (6.8) The proof of equation ( 6.8 ) for the map U is done by induction on L in [ 14 , prop1.7,p7] and is similar for V . Ho wev er, in addition to the fact that U and V are not separable, the conjugation of the mo del constructed from ˇ r b y the maps U or V do es not giv e in general a twisted perio dic SSEP . This is b ecause the map U ˇ r L, 1 U − 1 (or V ˇ r L, 1 V − 1 ) acts on the whole lattice in general and not only on sited L and 1 . Remark 6.2. If a set-the or etic al solution ( 6.1 ) is c onjugate d to the p ermutation op er ator via a sep ar able bije ction of c onfigur ations, then it is a Lyub ashenko solution. Inde e d, for two sites supp ose we have ˇ r 1 , 2 = V − 1 sep P 1 , 2 V sep , (6.9) with V sep = π (1) π (2) . On one side, we have for u, v ∈ J 0 , N − 1 K ˇ r 1 , 2 ( u, v ) = ( g u ( v ) , f v ( u )) . (6.10) On the other side, we have V − 1 sep P 1 , 2 V sep ( u, v ) = (( π (1) ) − 1 π (2) ( v ) , ( π (2) ) − 1 π (1) ( u )) . (6.11) Sinc e u and v ar e arbitr ary, by e quating e ach c omp onent, this implies that g u = g for al l u and f v = f for al l v . Using the involutivity of ˇ r , this implies that f = g − 1 so it has the form of Equation ( 2.5 ) . 6.2 A set-theoretical mo del not equiv alent to t wisted SSEP W e no w giv e an example of a set-theoretical solution of the form of Equation ( 6.1 ) that is not a Lyubashenk o solution. W e then show that the corresp onding Mark ov mo del is not equiv alen t to an y twisted perio dic SSEP for L = 3 . This example is for N = 3 . The maps f i and g j of Equation ( 6.1 ) are giv en b y g 0 (0) = f 0 (0) = 2 , g 0 (1) = f 0 (1) = 1 , g 0 (2) = f 0 (2) = 0 g 1 (0) = f 1 (0) = 0 , g 1 (1) = f 1 (1) = 1 , g 1 (2) = f 1 (2) = 2 g 2 (0) = f 2 (0) = 2 , g 2 (1) = f 2 (1) = 1 , g 2 (2) = f 2 (2) = 0 . They are indeed bijective and satisfy equations ( 6.2 ) and ( 6.3 ). In the basis B = {| 00 ⟩ , | 01 ⟩ , | 02 ⟩ , | 10 ⟩ , | 11 ⟩ , | 12 ⟩ , | 20 ⟩ , | 21 ⟩ , | 22 ⟩} , the matrix ˇ r reads ˇ r =               · · · · · · · · 1 · · · 1 · · · · · · · 1 · · · · · · · 1 · · · · · · · · · · · 1 · · · · · · · · · · · 1 · · · · · · · 1 · · · · · · · 1 · · · 1 · · · · · · · ·               , (6.12) 21 where the dots stand 0 . The Marko v matrix of the mo del constructed from ( 6.12 ) is noted M ˇ r and we note M f the Marko v matrix of the t wisted SSEP with the twist defined b y f . W e are going to show that the matrix M ˇ r is not conjugated to an y matrix M f for L = 3 (and in particular, there is no bijection of the configuration space, separable or not, which sends M ˇ r to a matrix M f ). T o do that, we sho w that the n umber of sectors for M ˇ r is different from the n umber of sectors for M f for an y p erm utation f . This will show that those mo dels are not conjugated since the num b er of sectors of a mo del is equal to the dimension of the kernel of the corresp onding Mark ov matrix, and this is inv ariant b y conjugation. F or N = 3 , there are 3! = 6 different p erm utations f . As we hav e seen in Section 4 , the n um b er of sectors only dep ends on the cycle structure of f . Using the notation in tro duced in Section 2.4 , it suffices to consider f 1 = Id , f 2 = (1 (0) , 1 (1) )(2 (0) ) and f 3 = (1 (0) , 1 (1) , 1 (2) ) . Using Equation ( 4.10 ) , the n umber of sectors in the twisted SSEP corresponding to f 1 , f 2 and f 3 for L = 3 are resp ectiv ely 10, 5 and 3. On the other hand, by considering the action of ( 6.12 ) on the 3 3 configurations, we find that the num b er of sectors in the corresponding set-theoretical mo del constructed for L = 3 is 7. This pro v es that this set-theoretical mo del is not conjugated to any t wisted SSEP for L = 3 . Remark 6.3. This mo del wil l b e studie d in mor e details in a futur e work. In p articular, we wil l show that, for al l L > 2 , it is not e quivalent to any twiste d SSEP mo del with 3 sp e cies. 7 Outlo ok The main fo cus of this pap er w as the construction and study of Mark o v mo dels from the simple Lyubashenk o solutions of the YBE. W e also introduced Marko v mo dels constructed from more general set-theoretical solutions of the YBE that are not alw a ys equiv alen t to some twisted SSEP and we gav e an example of such solution. In a future w ork, w e will study in more details the Mark o v mo del constructed from this solution. A dditionally , the oscillation phenomenon describ ed in Section 5.2 could b e further inv estigated. On the mathematical side, given t w o twisted SSEP , what are the conditions on the p erm utations defining the t wists in order to ha v e a non-zero probability of o ccup ying any configuration after a certain n umber of switch betw een the twists of b oth models? On the condensed matter physics side, w e can imagine that the mo difications of the twist are caused b y an unstable field on a b ond of the lattice and affecting the c harges of particles. W e could study the finite time effects of this field on the system for differen t types of instabilities (p eriodic, transient, etc.). A similar analysis can b e carried in the original set-theoretical model where the charge of particles can v ary on eac h b ond of the lattice when exchanging (because of unstable chemical bonds for instance). W e fo cused throughout this article on mo dels at thermo dynamic equilibrium defined on a ring. As a natural contin uation of this w ork, we plan to extend our approach to broader classes of mo dels, with particular emphasis on those leading to out-of-equilibrium systems. A first direction consists in adding integrable b oundaries to the system. The framework for suc h construction has b een developed in [ 7 , 12 , 37 ]. It led to the notion of set-theoretical reflection equation [ 6 , 7 ], whose solutions pro vide the p ossible in tegrable b oundaries. 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