Keywords: Integrable Markov models ; Exclusion process ; Twisted SSEP ; set-theoretical Yang–Baxter equation

1 Introduction↩︎

Exclusion Markov processes were first introduced in the context of biology to model protein synthesis in [1] and in statistical physics to describe the dynamics of the Ising model in [2]. The mathematical foundations of these systems were established in [3].

Among these processes, the simplest one is the Symmetric Simple Exclusion Process (SSEP). It describes a one-dimensional lattice gas, which is equivalent to a classical Ising model [4], where particles have equal probabilities of jumping one site to the left or right, with the condition that the target site is empty (hard core interaction). On a periodic lattice, the SSEP reaches thermodynamic equilibrium, contrary to the Asymmetric Simple Exclusion Process (ASEP) where a macroscopic current appears [5]. Generalisations of those models with more than one species of particles have been studied [6]–[8].

The ASEP (respectively the SSEP) can be mapped to the anisotropic (respectively isotropic) Heisenberg quantum chain [9], [10]. More precisely, the Markov matrix containing the transition probabilities for the exclusion process is equivalent to the Hamiltonian of a quantum spin chain. This allows in some cases to use methods originally developed for quantum spin chains to study exclusion processes [11], [12].

Models constructed in this article have the property of being integrable. In the framework of quantum spin chains, the chain is said to be integrable if its Hamiltonian belongs to a set of commuting operators generated by a transfer matrix. A transfer matrix is constructed from an \(R\) matrix which is a solution of the Yang–Baxter equation (YBE). This integrability property allows the exact computation of physical quantities of the spin chain [13]. The definition of an integrable Markov process is obtained by replacing the Hamiltonian with the Markov matrix. Integrability can be used to obtain for instance information on the stationary state and on the dynamical behaviour of the stochastic process.

In order to construct integrable Markov processes, the first step consists in finding solutions to the quantum Yang–Baxter equation [14], [15], which is a notoriously difficult problem in mathematical physics. To address it, Drinfeld in [16] suggested to consider “set-theoretical” solutions of the YBE, that is, solutions of the equation given by a permutation of a set of the form \(X \times X\). These solutions are not produced with the usual quantum group approach [17]. Since Drinfeld’s proposal, set-theoretical solutions have been extensively studied [18]–[30]. One simple class of set-theoretical solutions of the YBE is called Lyubashenko solutions [16].

In this paper, we construct and study new integrable Markov processes defined on a finite one dimensional lattice with periodic boundary conditions. These processes are constructed from set-theoretical solutions of the YBE using the Baxterization method [31]. In particular, we consider involutive Lyubashenko solutions that allow the construction of models with several interesting interpretations.

We prove that those Lyubashenko solutions define Markov processes that are equivalent to some twisted periodic SSEP. The twisted SSEP is a generalisation of the multi-species SSEP on a ring where the exchange rule on one bond of the lattice allows particles to change species. In the context of quantum spin chains, systems with generalised periodic boundary conditions have been studied, such as in [32]–[34], but they are less common in the framework of exclusion Markov processes.

Then we proceed to the study of the evolution of the twisted periodic SSEP for asymptotically long times. The process can be divided into sectors, for which there exists a unique stationary probability distribution. We define the profile and the total charge of a configuration and we prove that they uniquely label a sector. We also give explicit formulas for their number and size. Those formulas illustrate the idea that the twist connects different sectors of the multi-species SSEP.

Further we analyse how a stationary state of the twisted periodic SSEP is affected by a modification of the twist, in the same spirit as a quench procedure in condensed matter physics. Several interesting phenomena can occur such as the spreading and splitting of sectors. We define the branching probability of a stationary state going from one sector to another. Turning on and off the twist several times allows to alternate between different possible stationary states. We study in details some examples.

Finally, we consider more general set-theoretical solutions of the YBE than Lyubashenko solutions. We give an example of such solution and show that the Markov model constructed from it is not equivalent to any twisted SSEP coming from Lyubashenko solutions, thereby leaving room to more general models coming from set-theoretical solutions of the YBE which could be studied in further work.

The plan of the paper goes as follows. Section 2 introduces periodic Markov models constructed from Lyubashenko solutions of the YBE that have the property of being integrable and that have various interpretations.

Section 3 presents the twisted periodic SSEP. In Section 3.4, we prove the equivalence between models constructed from Lyubashenko solutions and some twisted periodic SSEP.

Section 4 focuses on sectors and stationary states of the twisted periodic SSEP. In Section 4.1, we show that the probability distribution of the stationary states is uniform. We classify sector in Section 4.2 and in Sections 4.3 and 4.4, we count the number and find the sizes of sectors.

Section 5 analyses how stationary states of the twisted SSEP are affected by a modification of the twist. In Section 5.2, we describe possible behaviours that can occur when turning on and off the twist.

Section 6 introduces Markov models constructed from more general set-theoretical solutions than Lyubashenko solutions that are not equivalent to some twisted periodic SSEP in general. In Section 6.2, we give an example of such model.

2 Markov models from set-theoretical solutions↩︎

In this section, we introduce Markov models associated to some set-theoretical solutions of the Yang–Baxter equation called Lyubashenko solutions. We show that those models are integrable and provide a non-exhaustive list of interesting interpretations.

2.1 Markov models on a periodic lattice↩︎

We consider a one-dimensional periodic lattice composed of \(L\) sites indexed by the variable \(i\in \llbracket 1,L \rrbracket\). On each site \(i\), the local configuration variable \(\tau_i\) takes values in \(\llbracket 0,N-1 \rrbracket\) with \(N\) a positive integer. The configurations \(\mathfrak{C}\) of the system correspond to \(L\)-tuples \(\vec{\tau}= (\tau_{1},...,\tau_{L})\in \llbracket 0,N-1 \rrbracket^{L}\). The canonical basis vector of a configuration \(\vec{\tau}\) is denoted \(\lvert\vec{\tau} \rangle\). All together they form an orthonormal basis i.e. \(\langle\vec{\tau} \mid\vec{\tau}' \rangle=\delta_{\vec{\tau},\vec{\tau}'}\).

The stochastic dynamics of the model is given by the Markov matrix \[M=\sum_{\vec{\tau},\vec{\tau}'\in \mathfrak{C}}m(\vec{\tau}' \rightarrow \vec{\tau})\,\lvert \vec{\tau}\rangle \langle \vec{\tau}'\rvert\, ,\] where \(m(\vec{\tau}' \rightarrow \vec{\tau})\geq 0\) for \(\vec{\tau}\neq \vec{\tau}'\) is interpreted as a transition rate between configurations \(\vec{\tau}'\) and \(\vec{\tau}\), and by definition, we have \[\label{coeffdiagM}m(\vec{\tau} \rightarrow \vec{\tau})=-\sum_{\vec{\tau}'\in\mathfrak{C},\vec{\tau}'\neq \vec{\tau}}m(\vec{\tau} \rightarrow \vec{\tau}')\;.\tag{1}\] Starting from an initial probability \(P_0\) on the configuration space, the probability of the system to be in configuration \(\vec{\tau}\) at continuous time \(t\) is denoted \(P_{t}(\vec{\tau})\). The probabilities of all configurations at time \(t\) are stored in the vector \(\lvert P_{t}\rangle = \sum_{\vec{\tau} } P_{t}(\vec{\tau}) \lvert \vec{\tau}\rangle\). The evolution of \(\lvert P_{t}\rangle\) is governed by the master equation \[\mathrm{d}\lvert P_{t}\rangle/\mathrm{dt}=M\lvert P_{t}\rangle .\]

We will consider Markov matrices which are local, by which we mean of the following form: \[\label{eq:M95model} M=\sum_{i=1}^{L-1}m_{i,i+1}+\tilde{m}_{L,1}\, ,\tag{2}\] where \(m\) is a local jump operator acting on two sites, and \(m_{i,i+1}\) is its action on sites \(i\) and \(i+1\). The additional term \(\tilde{m}_{L,1}\) is a local jump operator \(\tilde{m}\) acting on sites \(L\) and \(1\). When \(\tilde{m}=m\), the model is periodic. If \(\tilde{m}_{L,1}=\mathcal{T}_L\,m_{L,1}\,\mathcal{T}_L^{-1}\) for some \(N\)-dimensional matrix \(\mathcal{T}\), the model is called twisted periodic (by the twist \(\mathcal{T}\)).

2.2 Lyubashenko solutions of the Yang-Baxter equation↩︎

Fix throughout this section a bijective map denoted: \[g\;:\;\llbracket 0,N-1\rrbracket\to\llbracket 0,N-1\rrbracket\;.\] From \(g\) we define the following operator on two sites: \[\label{eq:r95lyub1} \check r=\sum_{i,j\in \llbracket 0,N-1\rrbracket}|g(j),g^{-1}(i)\rangle\langle i,j|\;.\tag{3}\] It satisfies the braided Yang–Baxter equation: \[\label{eq:cYB} \check r_{12}\,\check r_{23}\,\check r_{12}=\check r_{23}\,\check r_{12}\,\check r_{23}\;,\tag{4}\] and is called a Lyubashenko solution of the Yang–Baxter equation. Further, it is easy to check that it is involutive in the sense that \[\label{eq:invo} \check r^{2}=\mathrm{Id}\;.\tag{5}\]

From \(\check r\), we define the local jump operator \[\label{eq:loc95jump} m=\check r-\mathrm{Id}\,,\tag{6}\] and the Markov matrix \[\label{eq:M95Lyub} M=\sum_{i=1}^{L-1}m_{i,i+1}+m_{L,1}\;.\tag{7}\] It is easy to see that \(M\) is indeed a Markov matrix, namely that its off-diagonal coefficients are non-negative and that each column sums to \(0\). Note that this is a periodic model.

Also \(\check r\) is symmetric and therefore \(M\) too, which is expressed in terms of the transition rates as: \[m(\vec{\tau} \rightarrow \vec{\tau}')=m(\vec{\tau}' \rightarrow \vec{\tau})\,,\;\;\;\;\forall \vec{\tau},\vec{\tau}'\in\mathfrak{C}\;.\]

2.3 Integrability of the model↩︎

Here we show that the model is integrable in the sense that the Markov matrix belongs to a set of commuting operators generated by a transfer matrix. We follow the approach developed by the St Petersburg school on quantum integrable systems, see e.g. [13], [35].

The matrix \(\check r\) can be Baxterized as follows: \[\label{eq:R} \check R(z)=\frac{z\, \check{r}+ \mathrm{Id}}{z+1}\;.\tag{8}\] Indeed, since \(\check{r}^{2}=\mathrm{Id}\), it is easy to check that the Yang–Baxter equation with spectral parameters is satisfied: \[\label{eq:YBE-z}\check R_{12}(u)\,\check R_{23}(u+v)\,\check R_{12}(v)=\check R_{23}(v)\,\check R_{12}(u+v)\,\check R_{23}(u)\;.\tag{9}\] The transfer matrix is constructed with the formula \[t(z)=\mathrm{tr}_{0}\, R_{0,L}(z)\,R_{0,L-1}(z)\cdots R_{0,1}(z)\, ,\] where \(R(z)=P\check R(z)\) with \(P\) the permutation operator. The Yang–Baxter equation 9 ensures that \([t(z)\,,\,t(z')]=0\), so that upon expansion in \(z\), \(t(z)\) indeed generates a set of commuting operators. Moreover, since \(R(0)=P\), the trace in \(t(0)\) and \(t'(0)\) can be easily computed, and we recover with a straightforward calculation the Markov matrix of (7 ) as \[\label{eq:M95lyub2}M=t(0)^{-1}t'(0)=\sum_{i=1}^{L-1}m_{i,i+1}+m_{L,1}\;.\tag{10}\]

2.4 Interpretations of the models↩︎

In the next section, the Markov model above will be transformed to a twisted SSEP model, closely related to well-studied models. However, we suggest here possible direct interpretations of the Markov model constructed above.

Recall that we are given a bijection \(g\) of the set \(\llbracket 0,N-1 \rrbracket\), in terms of which the transition rates imply local processes on two sites of the form \[\label{localprocess}(\,\tau_{i}\,,\,\tau_{i+1}\,)\;\;\to\;\;(\,g(\tau_{i+1})\,,\,g^{-1}(\tau_{i}\,)\,)\;\;\;\;i=1,\dots,L,\tag{11}\] where of course \(L+1\) means \(1\).

2.4.1 Symmetric exclusion process with oscillations↩︎

The first natural interpretation is the usual one, where we have particles of \(N\) different species labeled by a variable \(s\in \llbracket 0,N-1 \rrbracket\) which evolve on this periodic lattice. The particles are subjected to a hard core interaction, i.e. each site is occupied by at most one particle.

Note however that starting from a state with two species \(a,b\) located in adjacent sites, (11 ) says that it may jump to a state with two species \(g(b),g^{-1}(a)\) located in these sites. Roughly speaking, the particles are oscillating between different species while moving on the lattice.

2.4.2 Species with internal states↩︎

Here we make use of the cyclic decomposition of the bijection \(g\) to provide different interpretations. We denote the decomposition of the permutation \(g\) into disjoint cycles by \[g=\pi_1\dots\pi_{n}\, ,\] where \(\pi_1,\dots,\pi_{n}\) are cycles with disjoint support in \(\llbracket 0,N-1\rrbracket\). We call \(c_i\) the length of the cycle \(\pi_i\). Clearly, since \(g\) is a permutation of \(\llbracket 0,N-1 \rrbracket\) we must have \(\sum_{i=1}^nc_i=N\). We reindex the set \(\llbracket 0,N-1\rrbracket\) to make use of such a decomposition. We set: \[\llbracket 0,N-1 \rrbracket=\{1^{(0)},\dots,1^{(c_1-1)},2^{(0)},\dots,2^{(c_2-1)},\dots\dots,n^{(0)},\dots,n^{(c_n-1)}\}\, ,\] so that the permutation becomes in cyclic notation: \[g=(1^{(0)},\dots,1^{(c_1-1)})(2^{(0)},\dots,2^{(c_2-1)})\dots\dots(n^{(0)},\dots,n^{(c_n-1)})\, .\] Now the interpretation is that we have \(n\) different species and for each species \(s\), we have an internal state labelled by a “quantum number”, that we will call for brevity a “charge”, which takes values in \(\{0,\dots,c_s-1\}\). With these notations, the action of \(g\) does not transform species, it simply increases the value of the charge by \(1\) (while, obviously, \(g^{-1}\) reduces the charge by \(1\)). The local processes are now of the form \[\label{localprocess2}{(\,s^{(e)}\,,\,t^{(e')}\,)\;\;\to\;\;(\,t^{(e'+1)}\,,\,s^{(e-1)}\,)\;.}\tag{12}\] Note that for species \(s\), the value of the charge is understood modulo \(c_s\).

2.4.3 Rolling polygons↩︎

Another possible interpretation is closely related to the previous one. One can see each species \(s\) as a regular polygon with \(c_s\) sides, each side being labeled by an integer which corresponds to the ‘charge’ introduced in the previous interpretation. In this context, the polygons are rolling clockwise when moving in one direction of the lattice and rolling counterclockwise in the opposite direction, as illustrated in Figure 1.

For example, if \(g\) consists of a single full cycle, \(g=(0,1,\dots,N-1)\), then the lattice is filled by regular \(N\)-gons.

2.4.4 Filling and emptying colored boxes↩︎

Finally one can also interpret a local configuration of the form \(s^{(e)}\) as a box “of color \(s\)” containing a number \(e\) of items (particles, balls, etc.).

In this interpretation, we have boxes moving on the periodic lattice. The boxes are labeled or colored by an integer \(1,\dots,n\) and a box of type \(s\) may contain up to \(c_s\) items. When two adjacent boxes are switched on the lattice, one acquires one more item and the other one loses one item.

Again, the number of items inside a box of color \(s\) is considered modulo \(c_s\). This means in particular that a box with \(c_s-1\) items becomes empty when acquiring an additional item (which can be seen as a discharging of the box when full). More delicate to interpret is when an empty box (with \(0\) item) “loses” an item and becomes a box containing \(c_s-1\) items. This interpretation is illustrated in Figures 2 and 3.

Note that if \(g\) consists of a single full cycle, \(g=(0,1,\dots,N-1)\), then we have only one type of boxes which contain up to \(N-1\) items (seen modulo \(N\)).

3 Twisted SSEP↩︎

In this section, we show that the Markov models from the previous section are mathematically equivalent to some twisted periodic SSEP models. We first define and interpret these models, show directly their integrability, and then proceed to the description of the correspondence with the previous section.

3.1 Definition of the model↩︎

As in the preceding section, our model will depend on a bijection denoted here \[f\;:\;\llbracket 0,N-1\rrbracket\to\llbracket 0,N-1\rrbracket\, .\]

For the Markov matrix, we take the following: \[\label{eq:Mf95model} M_f=\sum_{i=1}^{L-1}m_{i,i+1}+m_{L,1}^{f}\;.\tag{13}\] Here, for \(i\in \llbracket 1,L-1 \rrbracket\), the local jump operator \(m_{i,i+1}\) acting on sites \(i\) and \(i+1\) corresponds to the multi-species SSEP operator \[\label{eq:local95m} m_{i,i+1}=\sum_{\tau_{i},\tau_{i+1}\in \llbracket 0,N-1 \rrbracket}\lvert \tau_{i+1},\tau_{i} \rangle \langle \tau_{i},\tau_{i+1} \rvert-\mathrm{Id}\;.\tag{14}\] The local jump operator between sites \(L\) and \(1\) is given by \[\label{eq:local95mtw} m_{L,1}^{f}=\sum_{\tau_{L},\tau_{1}\in \llbracket 0,N-1 \rrbracket}\lvert f(\tau_{1}),f^{-1}(\tau_{L}) \rangle \langle \tau_{L},\tau_{1} \rvert-\mathrm{Id}\;.\tag{15}\] Identifying the bijection \(f\) with the linear operator acting on one site: \[f=\sum_{\tau\in\llbracket 0,N-1\rrbracket}|f(\tau)\rangle\langle \tau|\,,\] the local jump operator \(m_{L,1}^{f}\) is of the form \[m_{L,1}^{f}=f_L\,m_{L,1}\,f_{L}^{-1}\;,\] so that we have a twisted periodic model, where the local SSEP operator between sites \(L\) and \(1\) is twisted by the bijection \(f\).

As in the preceding section, it is easy to see that \(M_f\) is a symmetric matrix.

3.2 Interpretations↩︎

The twist \(f\) added between sites \(L\) and \(1\) has the same various interpretations as those given in the preceding sections. Here note that the dynamics between sites \(i\) and \(i+1\) for \(i=1,\dots,L-1\) is the usual one for SSEP, given by \[(\,\tau\,,\,\tau'\,)\;\;\to\;\;(\,\tau'\,,\,\tau\,)\;\, ,\] while between sites \(L\) and \(1\), it is given by \[(\,\tau\,,\,\tau'\,)\;\;\to\;\;(\,f(\tau')\,,\,f^{-1}(\tau)\,)\;.\] Therefore, if we think about \(\llbracket 0,N-1\rrbracket\) as representing \(N\) different species, the species are affected by the dynamics only between sites \(L\) and \(1\).

As before, reindexing the set \(\llbracket 0,N-1\rrbracket\) to make use of the cyclic decomposition of \(f\), we set \[f=(1^{(0)},\dots,1^{(c_1-1)})(2^{(0)},\dots,2^{(c_2-1)})\dots\dots(n^{(0)},\dots,n^{(c_n-1)})\, ,\] and we use the interpretation of \(n\) different species carrying an internal degree of freedom, that we call a “charge”.

Then again the dynamics between sites \(i\) and \(i+1\) for \(i=1,\dots,L-1\) is the usual one for SSEP (simple flip of the two local entries) while between sites \(L\) and \(1\), the charges are changing according to: \[\label{localprocess2b}(\,s^{(e)}\,,\,t^{(e')}\,)\;\;\to\;\;(\,t^{(e'+1)}\,,\,s^{(e-1)}\,)\;.\tag{16}\] Thus, a non-trivial function \(f\) corresponds to a purely transmitting impurity (or an “electric” field) located between sites \(L\) and 1 and which affects the “charge” of the particles when they pass through.

As before, the various interpretations with polygons rolling or with colored boxes filling and emptying are possible. The new feature compared to the preceding section is that the rolling of polygons, or the filling/emptying of boxes only happens when crossing the boundary between sites \(L\) and \(1\).

3.3 Integrability↩︎

We proceed to show that the above models are also integrable by adding the twist in the construction of the transfer matrix. Again, this property is very well-known in the context of quantum integrable systems, see e.g. [13], [35].

The Baxterized \(R\)-matrix this time is given by \[\label{eq:R95twist} R(z)=\frac{z\mathrm{Id}+P}{z+1}\, ,\tag{17}\] where \(P\) is the flip operator, and satisfies the Yang Baxter equation with spectral 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 twist operator is an invertible matrix \(\mathcal{T}(z)\) satisfying \[\label{eq:int95twist} \check{R}(u-v)\mathcal{T}(u) \otimes \mathcal{T}(v)= \mathcal{T}(v) \otimes \mathcal{T}(u) \check{R}(u-v)\;,\tag{18}\] where \(\check{R}(z)=PR(z)\). The homogeneous twisted transfer matrix is given by (see e.g. [8]) \[t_{\mathcal{T}}(z)=\mathrm{tr}_{0} R_{0,L}(z)R_{0,L-1}(z)\cdots R_{0,1}(z)\mathcal{T}_{0}(z)\, ,\] and satisfies the commutativity relation \([t(y),t(z)]=0\). The general expression of the twisted Markov matrix is \[\label{eq:M95twist} M_{\mathcal{T}}=t_{\mathcal{T}}(0)^{-1}t'_{\mathcal{T}}(0)=\sum_{i=1}^{L-1}m_{i,i+1}+\mathcal{T}_{L}^{-1}(0)m_{L,1}\mathcal{T}_{L}(0)+\mathcal{T}_{L}(0)^{-1}\mathcal{T}'_{L}(0)\, ,\tag{19}\] In our case, we simply have to take \[\mathcal{T}(z)=f^{-1}=\sum_{\tau\in\llbracket 0,N-1\rrbracket}|f^{-1}(\tau)\rangle\langle \tau|\,,\] where again we have identified the bijection \(f^{-1}\) with the corresponding linear operator. We have that \(\mathcal{T}(z)\) is indeed an integrable twist since it is invertible and equation 18 is satisfied because the twist is independent on \(z\) and because of the form of \(R(z)\) in equation 17 . We have by identifying equation 13 and equation 19 that \[\label{eq:m95L1} \mathcal{T}_{L}^{-1}(0)m_{L,1}\mathcal{T}_{L}(0)=f_Lm_{L,1}f_L^{-1}=m_{L,1}^{f}\, .\tag{20}\] Therefore, the expression (19 ) produces in this case the Markov matrix 13 of our model and this shows the integrability of the model.

3.4 Correspondence with set-theoretical models↩︎

For this subsection, we denote by \(\tilde{M}_g\) the Markov matrix 7 from Section 2 constructed from the set-theoretical solution of the Yang–Baxter equation corresponding to the bijection \(g\).

Here we show that \(\tilde{M}_g\) is conjugated to a Markov matrix \(M_f\) in 13 of the twisted SSEP, and moreover that this conjugation is simply done by a bijection on the configurations.

Recall that the set of configurations is \[\mathfrak{C}=\{\vec{\tau}=(\tau_1,\dots,\tau_L)\;,\;\;\tau_i\in\llbracket 0,N-1\rrbracket\}\, .\] In the following proposition, we make again the identification of a bijection of \(\mathfrak{C}\) with the linear operator acting on the vector space of the model.

An important feature of the operator \(V\) conjugating the two Markov matrices in Formula ?? is that it is induced by a bijection of \(\mathfrak{C}\) of the form \(\pi^{(1)}\pi^{(2)}\dots \pi^{(L)}\,,\) where each \(\pi^{(i)}\) permutes the particles species at site \(i\). Therefore, we conclude that the two Markov models given respectively by \(\tilde{M}_g\) and \(M_f\) (with \(f=g^L\)) are mathematically equivalent since an explicit bijection on the configuration space sends one into another. In other words, any mathematical property of one model is reflected to an equivalent property of the other model; for example, the number of sectors, their cardinality, etc. (see the next section).

4 Sectors and stationary states↩︎

Having shown in the preceding section that the models from Section 2 constructed from set-theoretical solutions of the Yang–Baxter equation are mathematically equivalent to some twisted SSEP models from Section 3, we proceed to the study of these twisted SSEP models. We will see that there are several stationary states which are all given by a uniform probability on sectors (irreducible subsets for \(M_f\) of the configuration space). Therefore, we are reduced to the combinatorial study of these sectors.

In this section, it will be convenient to fix, as before, the bijection \(f\) in the form (cycles decomposition) \[f=(1^{(0)},\dots,1^{(c_1-1)})(2^{(0)},\dots,2^{(c_2-1)})\dots\dots(n^{(0)},\dots,n^{(c_n-1)})\;,\] and recall that the corresponding Markov matrix is denoted \(M_f\).

We remind that the particles are denoted \(s^{(e)}\) where \(s\) is the species and \(e\) its charge, with \(1\leq s\leq n\) and \(0\leq e\leq c_s-1\). The non-zero transition rates are given by \[\begin{align} & (s^{(e)}\,,\,t^{(e')}) \;\substack{1\\ \longleftrightarrow}\;(t^{(e')}\,,\,s^{(e)})\quad for sites (i,i+1) with 1\leq i<L\,,\\ & (s^{(e)}\,,\,t^{(e')}) \;\substack{1\\ \longleftrightarrow}\;(t^{(e'+1)}\,,\,s^{(e-1)})\quad between sites L and 1\,, \end{align}\] where the charges for the species \(s\) are understood modulo \(c_s\).

4.1 Stationary states↩︎

A stationary state \(\lvert S \rangle=\sum_{\vec{\tau}\in \llbracket 0,N-1 \rrbracket^{L}} S(\vec{\tau})\lvert \vec{\tau}\rangle\) is a probability vector (the coefficients sum to \(1\)) which is invariant under the dynamics of the process, that is to say, it satisfies \[M_f\lvert S \rangle=0\;.\] The number of stationary states is equal to the number of sectors of the Markov process, which are defined as follows

For all the processes considered in this article, the Markov matrices are symmetric so if there exists a \(k\in \mathbb{N}\) such that \(\langle \vec{\tau} \rvert M^{k} \lvert \vec{\tau}' \rangle\neq 0\), then \(\langle \vec{\tau}' \rvert M^{k} \lvert \vec{\tau} \rangle\neq 0\) and vice versa.

We denote the set of sectors by \[\mathcal{S}=\{\text{sectors}\}=\{C_{\gamma}\,,\;\gamma\in\Gamma\}\, ,\] where \(\Gamma\) is an indexing set for the sectors.

For any sector \(C_{\gamma}\), there exists a unique stationary state denoted \(\lvert S_{\gamma} \rangle\). This is proven in [37] where the process is assumed to be positive recurrent on each class, a notion that we do not discuss further but which is true since the number of configurations in a class is finite [37].

4.1.0.1 The case of the usual \(N\)-species SSEP.

The usual \(N\)-species SSEP corresponds to the case \(f=\text{Id}\). It is not irreducible and its sectors correspond to configurations with a fixed number of particles of each species. It is in bijection with the number of multisets of the size of the lattice \(L\) with elements taken in the set \(\llbracket 0,N-1\rrbracket\). So the number of sectors is \[|\mathcal{S}|=\begin{pmatrix} N-1+L\\ N-1 \end{pmatrix}\;.\] The cardinality of the sector \(C_\gamma\) corresponding to \(p_{i}\) particles of species \(i\) for \(i\in \llbracket 0,N-1 \rrbracket\) is the multinomial coefficient \[|C_{\gamma}|=\begin{pmatrix} L\\ p_{0},...,p_{N-1} \end{pmatrix}\;.\] We are going to generalise these formulas to the general case (arbitrary \(f\)). Note that it is clear that the presence of the twist will reduce the number of sectors (while increasing their sizes), because the twist induces species transformations. Therefore, two different sectors of the usual multi-species SSEP can belong to the same sector of a twisted SSEP.

4.2 Invariants and sectors↩︎

4.2.1 Labelling sectors↩︎

Recall that a configuration is denoted by \[\vec{\tau}=(s_1^{(e_1)},\dots,s_L^{(e_L)})\;\;\;\;\text{with}\;s_i\in\llbracket 1,n \rrbracket\;\text{and}\;e_i\in\llbracket 0,c_{s_i}-1 \rrbracket\, ,\] where \(s_i\) corresponds to the species and \(e_i\) to the charge of the particle laying at site \(i\).

For a configuration \(\vec{\tau}\), the species which appear at least once are called the \(\vec{\tau}\)-relevant species. They depend only on the profile \(p(\vec{\tau})\) and are the ones for which \(p_k\neq 0\).

Note that for a given position \(i\), if in (?? ) we replace \(e_i\) by, say, \(e_i+c_{s_i}\), this does not change the value of the total charge, since by construction \(D_{\vec{\tau}}\) divides each \(c_{s_i}\) of \(\vec{\tau}\)-relevant species \(s_i\). In this sense, the total charge is consistent with the definition of the charge of a species.

We are ready to characterise the sectors.

4.3 Number of sectors↩︎

In Section 4.1, we recalled the number of sectors for the usual \(N\)-species SSEP. Using the labelling of the sectors of the section 4.2.1, we can now count them and generalise the formula giving their number for an arbitrary permutation \(f\).

4.4 Cardinality of sectors↩︎

We now determine the size of a sector and show that, for a fixed permutation \(f\), it only depends on the profile of the configurations of that sector. As in the previous section, we then apply the formula for a permutation with cycles of equal length.

5 Twist tuning and branching probabilities↩︎

In this part, we analyse the behaviour of a stationary state under a modification of the twist. Namely, we choose a stationary state for some twisted SSEP and we consider its evolution in a twisted SSEP with a different twist. In other words, we study a quench between two twisted SSEP with different twists. For example, when one of the models is the usual (non-twisted) SSEP, this procedure can be seen as turning on/off a twist on a given bond of the periodic lattice.

5.1 Overlaps and branching probabilities↩︎

We consider two bijections \(f^{(1)}\) and \(f^{(2)}\) of \(\llbracket 0,N-1\rrbracket\), and we fix a stationary state \(|S^{(1)}\rangle\) corresponding to a sector \(C^{(1)}\) for the twisted SSEP associated to \(f^{(1)}\). We recall that \[\label{probvector}|S^{(1)}\rangle=\frac{1}{|C^{(1)}|}\sum_{\vec{\tau}\in C^{(1)}}|\vec{\tau}\rangle\;.\tag{27}\] Now we consider \(|S^{(1)}\rangle\) as the initial condition for the Markov model corresponding to \(f^{(2)}\), we let it evolve using the Markov matrix \(M_{f^{(2)}}\) and we look at the possible stationary states (for \(M_{f^{(2)}}\)) that are reached at \(t=\infty\).

We denote \(\mathcal{S}^{(2)}\) the set of sectors for the model associated to \(f^{(2)}\), and by \(|S^{(2)}_\gamma\rangle\) the stationary state associated to \(\gamma\in\mathcal{S}^{(2)}\).

At \(t=\infty\), the state will be of the form: \[\label{dec95alphagamma}\sum_{\gamma\in\mathcal{S}^{(2)}}prob(C^{(1)}\to C_{\gamma}^{(2)})|S^{(2)}_\gamma\rangle\;.\tag{28}\] Note that we have \[\sum_{\gamma\in\mathcal{S}^{(2)}}prob(C^{(1)}\to C_{\gamma}^{(2)})=1\,,\] since the stationary vectors are normalised (in the sense that the sum of their components is 1, see 27 ), and the evolution of a probability vector under a Markov process remains a probability vector.

The branching probabilities are simply calculated in terms of \(|S^{(1)}\rangle\) and \(|S_{\gamma}^{(2)}\rangle\). Indeed, since the stationary states \(|S_{\gamma}^{(2)}\rangle\) do not evolve (by definition) under \(M_{f^{(2)}}\), the coefficients in the expansion (28 ) are the same as the corresponding coefficients in the initial state \(|S^{(1)}\rangle\). Therefore we have \[prob(C^{(1)}\to C_{\gamma}^{(2)})=\frac{\langle S^{(1)}|S_{\gamma}^{(2)}\rangle}{\langle S_{\gamma}^{(2)}|S_{\gamma}^{(2)}\rangle}\;.\] So far, the discussion was valid in a general setting. Now we use the explicit formula (27 ) expressing that for our models, the stationary states correspond to uniform probabilities on sectors. We have immediately \[\langle S^{(1)}|S_{\gamma}^{(2)}\rangle=\frac{|C^{(1)}\cap C_\gamma^{(2)}|}{|C^{(1)}||C_\gamma^{(2)}|}\;.\] Therefore, noticing that \(\langle S_{\gamma}^{(2)}|S_{\gamma}^{(2)}\rangle=\frac{1}{|C^{(2)}_\gamma|}\), we conclude with the following result.

5.2 Spreading, splitting and oscillations of sectors↩︎

5.2.0.1 Spreading of a sector.

Here we consider the particular case where the sector \(C^{(1)}\) of the first model is included in a sector \(C_{\gamma_0}^{(2)}\) of the second model, that is we have \[C^{(1)}\subset C_{\gamma_0}^{(2)}\,,\;\;\;\;\text{for some sector \gamma_0\in\mathcal{S}^{(2)}.}\] In this case, the branching probabilities are trivial, in the sense that, according to Proposition 18, we have \[prob(C^{(1)}\to C_{\gamma}^{(2)})=\delta_{\gamma,\gamma_0}\;.\] The only thing to do is to determine, given \(C^{(1)}\) and \(C_{\gamma}^{(2)}\), whether \(C^{(1)}\subset C_{\gamma}^{(2)}\) or \(C^{(1)}\cap C_{\gamma}^{(2)}=\emptyset\). Examples are detailed in the next subsection.

5.2.0.2 Splitting of a sector.

The opposite situation is when one reverses the role of \(f^{(1)}\) and \(f^{(2)}\): we start with the twist \(f^{(2)}\) and we go on with the twist \(f^{(1)}\). In this case, a sector \(C^{(2)}\) for \(f^{(2)}\) splits into a disjoint union of sectors for \(f^{(1)}\): \[C^{(2)}=C_{\gamma_1}^{(1)}\sqcup \dots \sqcup C_{\gamma_K}^{(1)}\;.\] Now the branching probabilities are non-trivial and they are given, according to Proposition 18 by \[prob(C^{(2)}\to C_{\gamma}^{(1)})=\frac{|C_{\gamma}^{(1)}|}{|C^{(2)}|}\;\;\;\;\text{for \gamma\in\{\gamma_1,\dots,\gamma_K\}}\,,\] and \(0\) otherwise. Examples are given below.

5.2.0.3 Oscillations of sectors.

Assume we are in a spreading/splitting situation, namely we have a sector for \(f^{(2)}\) which is a disjoint union of sector for \(f^{(1)}\): \[C^{(2)}=C_{\gamma_1}^{(1)}\sqcup \dots \sqcup C_{\gamma_K}^{(1)}\;.\] Then we can start from a sector \(C_{\gamma_{i_1}}^{(1)}\) then we alternate between turning on the twist \(f^{(2)}\) and turning on the twist \(f^{(1)}\). Then sector will alternatively spread and split, that is, schematically: \[\begin{array}{ccccccccccccccc} &&C^{(2)}& && &C^{(2)}& && &C^{(2)}& && &\\ &\nearrow &&\searrow &&\nearrow&&\searrow &&\nearrow &&\searrow &&\to &\\ C_{\gamma_{i_1}}^{(1)}& && &C_{\gamma_{i_2}}^{(1)}& && &C_{\gamma_{i_3}}^{(1)}& && &C_{\gamma_{i_4}}^{(1)}& &.........\\ \end{array}\] where the up-arrow \(\nearrow\) corresponds to switching on \(f^{(2)}\) (and switching off \(f^{(1)}\)) and the down arrow to switching on \(f^{(1)}\) (and switching off \(f^{(2)}\)). We see that the configuration will oscillate between the sectors \(C_{\gamma_1}^{(1)},\dots, C_{\gamma_K}^{(1)}\): \[C_{\gamma_{i_1}}^{(1)}\quad\raisebox{1ex}{\substack{\scriptscriptstyle{prob(C^{(2)}\to C_{\gamma_{i_2}}^{(1)})}\\ \longrightarrow}}\quad C_{\gamma_{i_2}}^{(1)} \quad\raisebox{1ex}{\substack{\scriptscriptstyle{prob(C^{(2)}\to C_{\gamma_{i_3}}^{(1)})}\\ \longrightarrow}}\quad C_{\gamma_{i_3}}^{(1)} \quad\raisebox{1ex}{\substack{\scriptscriptstyle{prob(C^{(2)}\to C_{\gamma_{i_4}}^{(1)})}\\ \longrightarrow}}\quad C_{\gamma_{i_4}}^{(1)}\to .........\]

5.3 Example: usual multi-species SSEP and an arbitrary twist↩︎

An obvious case where the spreading/splitting situation happens is when the first model is the usual multi-species SSEP (the twist \(f^{(1)}=\mathrm{Id}\) is trivial) and the second model can have an arbitrary twist \(f^{(2)}\).

In this case, we write \(f^{(2)}\) with its cycle notation using species and charges, that is we take: \[f^{(1)}=\mathrm{Id}\qquad\text{and}\qquad f^{(2)}=(1^{(0)},\dots,1^{(c_1-1)})(2^{(0)},\dots,2^{(c_2-1)})\dots\dots(n^{(0)},\dots,n^{(c_n-1)})\;.\] A sector \(C_{\gamma}^{(2)}\) for \(f^{(2)}\) is indexed by a profile \(p^{(2)}=(p_1,\dots,p_n)\) and a total charge \(E^{(2)}\), as defined in Section 4.2: \[C_{p^{(2)},E^{(2)}}^{(2)}\quad :\qquad\quad p^{(2)}=(p_1,\dots,p_n)\;\;\;\text{and}\;\;\;E^{(2)}= k\;\mathrm{mod}\,D\,,\] where \(D\) is the greatest common divisor of the relevant species (see Definition 3). Recall that the profile \(p^{(2)}\) simply counts the number of occurrences of each species while the total charge \(E^{(2)}\) is the sum of the charges modulo \(D\). Now a sector \(C^{(1)}\) for the usual multi-species SSEP model is only indexed by a profile, which is of the form: \[C_{p^{(1)}}^{(1)}\quad:\qquad\quad p^{(1)}=(p_{1^{(0)}},p_{1^{(1)}},\dots,p_{n^{(c_n-1)}})=\Bigl(p_{s^{(e)}}\Bigr)_{\substack{s=1,\dots,n\\ e=0,\dots,c_s-1}}\;.\] Indeed recall that from the point of view of \(f^{(1)}=\mathrm{Id}\), the profile simply counts the number of occurrences of every \(s^{(e)}\), for various \(s\) and \(e\).

The inclusion condition is expressed as follows: \[\label{inclusion1} C_{p^{(1)}}^{(1)}\subset C_{p^{(2)},E^{(2)}}^{(2)}\;\;\;\;\Longleftrightarrow\;\;\;\;\left\{\begin{array}{l} \displaystyle\sum_{e=0}^{c_s-1}p_{s^{(e)}}=p_s\,,\;\;\forall s=1,\dots,n\,,\\[0.5em] \displaystyle\sum_{s=1}^n\sum_{e=0}^{c_s-1}e p_{s^{(e)}}=E^{(2)}\;mod\,D\,. \end{array}\right.\tag{29}\] Therefore we have according to Proposition 18 \[prob(C_{p^{(1)}}^{(1)}\to C_{p^{(2)},E^{(2)}}^{(2)})=\left\{\begin{array}{ll} 1 & \text{if (\ref{inclusion1}) is satisfied,}\\[0.4em] 0 & \text{otherwise.} \end{array}\right.\] This was the spreading situation. Now for the splitting situation, we reverse the position of the sectors, and we get, according to Proposition 18 \[prob(C_{p^{(2)},E^{(2)}}^{(2)}\to C_{p^{(1)}}^{(1)})=\left\{\begin{array}{ll} \displaystyle\frac{|C_{p^{(1)}}^{(1)}|}{|C_{p^{(2)},E^{(2)}}^{(2)}|} & \text{if (\ref{inclusion1}) is satisfied,}\\[0.5em] 0 & \text{otherwise.} \end{array}\right.\] We recall that the explicit formula for the cardinality of a sector is given in Proposition 17.

5.4 Example: Two different non-trivial twists↩︎

Here, for the first model, we take for the twist the full cycle \(f^{(1)}=(0,1,\dots,N-1)\). Therefore, in terms of charged species, for the first model, we have \[f^{(1)}=(1^{(0)},1^{(1)},\dots,1^{(N-1)})\, ,\] and a sector \(C^{(1)}_k\) is labeled by the total charge \(E^{(1)}= k mod\;N\) (the profile being \(p^{(1)}=(L)\)). To get the charge \(k\), one can put arbitrary charges in \(L-1\) sites, the last one being used to fix the total charge. Hence, the sector has dimension \(|C^{(1)}_k|=N^{L-1}\), in accordance with Example 4.

5.4.0.1 The case \(f^{(2)}=(1,3,\dots N-1)(2,4,\dots N)\) with \(N\) even.

For the second model, we take for \(f^{(2)}\) the square of the first twist, which splits into two cycles since \(N\) is even. In terms of charged species, we have \[\begin{align} f^{(2)}=\;&(1^{(0)},1^{(2)},\dots,1^{(N-2)})(1^{(1)},1^{(3)},\dots,1^{(N-1)})\\ =\;&(2^{(0)},2^{(1)},\dots,2^{(N/2-1)})(3^{(0)},3^{(1)},\dots,3^{(N/2-1)})\,. \end{align}\] In the second line, we have written the charged species from the point of view of \(f^{(2)}\). Since \(f^{(2)}\) only permutes the elements \(1^{(e)}\) for a given parity of \(e\), it is as if we had two different species. In other words, we have set \(2^{(e)}=1^{(2e)}\) and \(3^{(e)}=1^{(2e+1)}\).

A sector for \(f^{(2)}\) is labeled by the profile \(p^{(2)}=(p_2,p_3)\) (with \(p_2+p_3=L\)) and the total charge \(E^{(2)}= \ell mod\;N/2\). A particular configuration for this sector is given by \[\begin{align} & (\underbrace{2^{(\ell)},2^{(0)},\dots,2^{(0)}}_{p_2},\underbrace{3^{(0)},3^{(0)},\dots,3^{(0)}}_{p_3})\quad ifp_2\neq0\,,\quad (\underbrace{3^{(\ell)},3^{(0)},\dots,3^{(0)}}_{p_3=L})\quad ifp_2=0\,. \end{align}\] In both cases, the charge \(E^{(1)}\) for this configuration is \(E^{(1)}=2\ell+p_3\) mod \(N\). Since \(p_2+p_3=L\) is always satisfied, \(prob(C^{(1)}_k\to C^{(2)}_{p^{(2)},\ell})\) is non-vanishing if and only if \(k=2\ell+p_3\) mod \(N\). Denoting by \(C^{(2)}_\gamma\) the sectors which obey this equality, the probability is given by \[prob(C^{(1)}_k\to C^{(2)}_{\gamma})=\frac{|C^{(2)}_{\gamma}|}{|C^{(1)}_k|}=\frac{L!}{p_2!\,p_3!}\left(\frac{1}{2}\right)^{L-1}\,.\]

5.4.0.2 The case \(f^{(2)}=(f^{(1)})^{n-1}\) with \(N=(n-1)D\).

More generally, keeping \(f^{(1)}\) as above, we assume now that \(N=(n-1)D\) (the previous case was \(n=3\)). We take \[f^{(2)}=(2^{(0)},...,2^{(D-1)}) (3^{(0)},...,3^{(D-1)}) \cdots (n^{(0)},...,n^{(D-1)})\,,\] with the reindexing done similarly as before.

The sectors are now characterised by the profile \(p^{(2)}=(p_2,p_3,\dots,p_{n})\) and a total charge \(E^{(2)}= \ell mod\;D\). A particular configuration for this sector is given by \[(\underbrace{2^{(\ell)},2^{(0)},\dots,2^{(0)}}_{p_2},\underbrace{3^{(0)},3^{(0)},\dots,3^{(0)}}_{p_3},\dots,\underbrace{n^{(0)},n^{(0)},\dots,n^{(0)}}_{p_n})\,,\] where we have supposed that \(p_2\neq0\) without loss of generality. The condition of non-empty intersection between \(C^{(1)}_k\) and \(C^{(2)}_{p^{(2)},\ell}\) is expressed as \[(n-1)\,\ell+\sum_{j=2}^n (j-2)p_j = k mod N\, ,\] and, when this condition is satisfied, \[prob(C^{(1)}_k\to C^{(2)}_{\gamma})= \frac{L!}{p_2! p_3!\dots p_n!}\,\left(\frac{1}{n-1}\right)^{L-1}\,.\]

6 More general solutions of set-theoretical Yang-Baxter equation↩︎

In this section, we introduce Markov models constructed from more general set-theoretical maps than the ones previously considered. We then give a non trivial example of such map. We show that its corresponding Markov model for a small lattice (\(L=3\)) is not equivalent to any twisted periodic SSEP coming from Lyubashenko solutions.

6.1 Markov models↩︎

We now consider set-theoretical maps that are more general than the ones in Section 2.2. For \(i\in \llbracket 0,N-1 \rrbracket\), we consider the family of bijective maps \[g_{i},\;f_{i}:\;\llbracket 0,N-1\rrbracket\to\llbracket 0,N-1\rrbracket \, ,\] that are chosen in such a way that the following operator on two sites \[\label{eq:r95gene} \check r=\sum_{i,j\in \llbracket 0,N-1\rrbracket}|g_{i}(j),f_{j}(i)\rangle\langle i,j|\;.\tag{30}\] satisfies the YBE 4 and is involutive as in equation 5 .

The maps \(g_{i}\) and \(f_{i}\) such that \(\check{r}\) obeys the YBE, satisfy for all \(i,j,k\in \llbracket 0,N-1 \rrbracket\) \[\label{eq:yb95rel} g_{i}(g_{j})(k)=g_{g_{i}(j)}(g_{f_{j}(i)}(k)),\quad f_{k}(f_{j}(i))=f_{f_{k}(j)}(f_{g_{j}(k)}(i)),\quad f_{g_{f_{j}(i)}(k)}(g_{i}(j))=g_{f_{g_{j}(k)}(i)}(f_{k}(j))\,.\tag{31}\] The maps \(g_{i}\) and \(f_{i}\) such that \(\check{r}^{2}=\mathrm{Id}\) satisfy for all \(i,j\in \llbracket 0,N-1 \rrbracket\) \[\label{eq:inv95rel} g_{g_{i}(j)}(f_{j}(i))=i,\quad f_{f_{j}(i)}(g_{i}(j))=j\,.\tag{32}\] Involutive solutions of the YBE where \(g_{i}\), \(f_{i}\) are bijections are equivalent to algebraic structures called braces [24].

The local jump operator and the Markov matrix constructed from this operator are defined identically as previously in Equations 6 and 7 . It can be easily checked that the resulting model is a Markovian periodic model. It also has the property of being integrable. This comes from the fact that the operator 30 is involutive, which allows to Baxterize it and to construct a set of commuting operators as in Section 2.3.

6.1.0.1 Interpretation.

The form of the operator 30 implies that the transition rates between two adjacent sites have the form \[\label{localprocess95gene} (\,\tau_{i}\,,\,\tau_{i+1}\,)\;\;\to\;\;(\,g_{\tau_{i}}(\tau_{i+1})\,,\,f_{\tau_{i+1}}(\tau_{i})\,)\; \; \;\;i=1,\dots,L\, .\tag{33}\] The interpretations that we can give for models constructed from those more general set-theoretical solutions are in general harder to find than the ones given in section 2.4.

6.1.0.2 Separable and non-separable bijections.

In Section 3.4, Proposition 5, we proved the equivalence between Markov models constructed from Lyubashenko solutions and the twisted periodic SSEP. In that case, the bijection on the configurations had the form \[\label{eq:V95sep} V_{\mathrm{sep}}=\pi^{(1)} \pi^{(2)}... \pi^{(L)}\, ,\tag{34}\] and we call it separable. For set-theoretical solutions 30 , there exist non separable bijections of \(\mathfrak{C}\) given by \[\label{Ufactor95gene} U(\tau_1,\tau_2,\dots,\tau_L)=(f_{\tau_{L}}f_{\tau_{L-1}}\dots f_{\tau_{2}}(\tau_{1}),\dots,f_{\tau_{L}}(\tau_{L-1}),\tau_{L})\, ,\tag{35}\] and \[\label{Vfactor95gene} V(\tau_1,\tau_2,\dots,\tau_L)=(\tau_1,g_{\tau_{1}}(\tau_2),\dots,g_{\tau_{1}}g_{\tau_{2}}\dots g_{\tau_{L-1}}(\tau_L))\,,\tag{36}\] satisfying

\[\label{eq:loc95corr95UV} \forall i\in \llbracket 1,L-1 \rrbracket,\;\check{r}_{i,i+1}=U^{-1}P_{i,i+1} U=V^{-1}P_{i,i+1} V\, .\tag{37}\]

The proof of equation 37 for \(U\) is done by induction on \(L\) in [18] and is similar for \(V\). However, in addition to the fact that \(U\) and \(V\) are not separable, the conjugation of the model constructed from \(\check{r}\) by \(U\) or \(V\) does not give in general a twisted periodic SSEP. This is because \(U\check{r}_{L,1}U^{-1}\) (or \(V\check{r}_{L,1}V^{-1}\)) acts on the whole lattice in general and not only on sited \(L\) and \(1\).

6.2 Example↩︎

We now give an example of a set-theoretical solution of the form of Equation 30 that is not a Lyubashenko solution. We then show that the corresponding Markov model is not equivalent to any twisted SSEP coming from Lyubashenko solutions for \(L=3\).

This example is for \(N=3\). The maps \(f_{i}\) and \(g_{j}\) of Equation 30 are given by \[\begin{align} g_{0}(0)=f_{0}(0)=2,\quad g_{0}(1)=f_{0}(1)=1,\quad g_{0}(2)=f_{0}(2)=0\\ g_{1}(0)=f_{1}(0)=0,\quad g_{1}(1)=f_{1}(1)=1,\quad g_{1}(2)=f_{1}(2)=2\\ g_{2}(0)=f_{2}(0)=2,\quad g_{2}(1)=f_{2}(1)=1,\quad g_{2}(2)=f_{2}(2)=0\, . \end{align}\] They are indeed bijective and satisfy equations 31 and 32 .

In the basis \(\mathcal{B}=\{ \lvert 00 \rangle, \lvert 01 \rangle, \lvert 02 \rangle, \lvert 10 \rangle, \lvert 11 \rangle, \lvert 12 \rangle, \lvert 20 \rangle, \lvert 21 \rangle, \lvert 22 \rangle\}\), the matrix \(\check{r}\) reads \[\label{eq:r95brace} \check{r}= \begin{pmatrix} \cdot &\cdot&\cdot &\cdot&\cdot &\cdot&\cdot &\cdot&1\\ \cdot &\cdot&\cdot &1&\cdot &\cdot&\cdot &\cdot&\cdot\\ \cdot &\cdot&1 &\cdot&\cdot &\cdot&\cdot &\cdot&\cdot\\ \cdot &1&\cdot &\cdot&\cdot &\cdot&\cdot &\cdot&\cdot\\ \cdot &\cdot&\cdot &\cdot&1 &\cdot&\cdot &\cdot&\cdot\\ \cdot &\cdot&\cdot &\cdot&\cdot &\cdot&\cdot &1&\cdot\\ \cdot &\cdot&\cdot &\cdot&\cdot &\cdot&1 &\cdot&\cdot\\ \cdot &\cdot&\cdot &\cdot&\cdot &1&\cdot &\cdot&\cdot\\ 1 &\cdot&\cdot &\cdot&\cdot &\cdot&\cdot &\cdot&\cdot\\ \end{pmatrix}\, ,\tag{38}\] where the dots stand \(0\). The Markov matrix of the model constructed from 38 is denoted \(M_{\check{r}}\) and we denote \(M_{f}\) the Markov matrix of the twisted SSEP with the twist defined by \(f\).

We are going to show that the matrix \(M_{\check{r}}\) is not conjugated to any matrix \(M_{f}\) for \(L=3\) (and in particular, there is no bijection of the configuration space, separable or not, which sends \(M_{\check{r}}\) to a matrix \(M_{f}\)). To do that, we show that the number of sectors for \(M_{\check r}\) is different from the number of sectors for \(M_f\) for any permutation \(f\). This will show that those models are not conjugated since the number of sectors of a model is equal to the dimension of the kernel of the corresponding Markov matrix, and this is invariant by conjugation.

For \(N=3\), there are \(3!=6\) different permutations \(f\). As we have seen in Section 4, the number of sectors only depends on the cycle structure of \(f\). Using the notation introduced in Section 2.4, it suffices to consider \(f_{1}=\mathrm{Id}\), \(f_{2}=(1^{(0)},1^{(1)})(2^{(0)})\) and \(f_{3}=(1^{(0)},1^{(1)},1^{(2)})\). Using Equation [prop:comm95nb95gene], the number of sectors in the twisted SSEP corresponding to \(f_{1}\), \(f_{2}\) and \(f_{3}\) for \(L=3\) are respectively 10, 5 and 3.

On the other hand, by considering the action of 38 on the \(3^3\) configurations, we find that the number of sectors in the corresponding set-theoretical model constructed for \(L=3\) is 7. This proves that this set-theoretical model is not conjugated to any twisted SSEP coming from Lyubashenko solutions for \(L=3\).

7 Outlook↩︎

The main focus of this paper was the construction and study of Markov models from the simple Lyubashenko solutions of the YBE. We also introduced Markov models constructed from more general set-theoretical solutions of the YBE that are not always equivalent to some twisted SSEP and we gave an example of such solution. In a future work, we will study in more details the Markov model constructed from this solution.

Additionally, the oscillation phenomenon described in Section 5.2 could be further investigated. On the mathematical side, given two twisted SSEP, what are the conditions on the permutations defining the twists in order to have a non-zero probability of occupying any configuration after a certain number of switch between the twists of both models? On the condensed matter physics side, we can imagine that the modifications of the twist are caused by an unstable field on a bond of the lattice and affecting the charges of particles. We could study the finite time effects of this field on the system for different types of instabilities (periodic, transient, etc.). A similar analysis can be carried in the original set-theoretical model where the charge of particles can vary on each bond of the lattice when exchanging (because of unstable chemical bonds for instance).

We focused throughout this article on models at thermodynamic equilibrium defined on a ring. As a natural continuation of this work, we plan to extend our approach to broader classes of models, with particular emphasis on those leading to out-of-equilibrium systems.

A first direction consists in adding integrable boundaries to the system. The framework for such construction has been developed in [28], [38], [39]. It led to the notion of set-theoretical reflection equation [38], [40], whose solutions provide the possible integrable boundaries.

A second approach preserves periodic boundary conditions but the set-theoretical solution is deformed so that the local jump operator constructed from it has a parameter dependence leading to an asymmetry in the transition probabilities.

Clearly, it will be highly desirable to combine these two approaches, namely by studying solutions of the set-theoretic reflection equation in conjunction with parameter-dependent set-theoretic-like solutions of the YBE.

7.0.0.1 Acknowledgements.

We acknowledge support from the grant IRP AAPT. We warmly thank Nicolas Crampé for helpful discussions. M. D. also thanks the Laboratoire de Mathématiques de Reims for its hospitality during a research visit related to this project.

References↩︎

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1. email: mathieudabrowski@outlook.com↩︎

2. Author to whom any correspondence should be addressed.↩︎

3. email: loic.poulain-dandecy@univ-reims.fr↩︎

4. email: ragoucy@lapth.cnrs.fr↩︎