Ergodicity under twist switching, finite-time twist dynamics, and out-of-equilibrium extensions of integrable set-theoretical Markov processes (twisted SSEP)

Periodic integrable Markov processes built from set-theoretical solutions of the Yang-Baxter equation (YBE) were shown by Dabrowski, Poulain d'Andecy and Ragoucy to coincide with twisted symmetric simple exclusion processes (SSEP): multi-species SSEP on a ring whose exchange across one bond is conjugated by a permutation $f$. That paper classified the sectors and stationary states, computed the branching probabilities under a change of twist, and left open: (i) necessary and sufficient conditions on two permutations such that alternating the corresponding twisted SSEPs reaches any configuration after finitely many switches; (ii) the finite-time response to time-dependent twist perturbations (periodic, transient, etc.), including models where an unstable field on one bond changes particle charges; (iii) genuinely out-of-equilibrium integrable set-theoretical models via parameter-dependent deformations producing asymmetric transition probabilities and via integrable boundaries from reflection equations, and their combination. A same-author follow-up (arXiv:2607.18959) resolved the fourth announced direction (the analysis of the introduced non-Lyubashenko example for all $L>2$) and built integrable open boundaries for the $(p,q)$-SSEP ($p$ singlet species plus $q$ pairs of reactive species that transform into one another) from constant reflection equations, computing non-equilibrium stationary currents and densities for one boundary class; directions (i)-(iii) above remain open. The problem asked here is to settle these three remaining directions.

Background

The symmetric simple exclusion process (SSEP) is a continuous-time Markov process on a ring of $L$ sites in which the particle labels on neighbouring sites are exchanged at rate 1; with labels in $X=\{0,\dots,N-1\}$ (label 0 usually read as a hole) this is the multi-species SSEP, and the bias of jumping left or right is equal (its asymmetric variant, ASEP, has unequal rates). The multi-species SSEP is reducible: its configuration space splits into closed communicating classes, called sectors, and on each sector the dynamics converges to the uniform distribution over that sector. A Markov process is called integrable when its Markov matrix belongs to a commuting family of transfer matrices, the stochastic counterpart of integrability for quantum spin chains; the standard source of such matrices is a solution $R$ of the Yang-Baxter equation (YBE), the algebraic consistency condition that guarantees commuting transfer matrices. These papers use set-theoretical solutions: $R$ is the linear extension of a bijection $\check{r}$ of $X\times X$ satisfying the braid form of the YBE. The simplest class, the Lyubashenko solutions, is indexed by a single bijection $g$ of $X$ through $(x,y)\mapsto(g(y),g^{-1}(x))$; these solutions are involutive ($\check{r}^2=\mathrm{Id}$). From $\check{r}$ one defines a local jump operator $m=\check{r}-\mathrm{Id}$ and, on a ring, a Markov matrix $M=\sum_{i=1}^{L-1}m_{i,i+1}+m_{L,1}$ acting on neighbouring pairs.

The central result of the source paper is that this set-theoretical model is conjugate to a twisted SSEP: with twist $f=g^L$, the local operator on the bond closing the ring becomes $m^f_{L,1}=f\,m_{L,1}\,f^{-1}$, i.e. the ordinary multi-species SSEP whose exchange across that one bond is conjugated by the permutation $f$. The particle species are the cycles of $f$ (a cycle of length $c$ gives a species with $c$ internal charge states), and the sectors of $M_f$ are exactly classified by two invariants: the profile (the count of particles of each species) and the total charge (the sum of internal charges modulo the gcd of the cycle lengths of the species present). Each sector carries a uniform stationary state.

The paper then studies what happens when the twist is allowed to vary. For two bijections $f^{(1)}$ and $f^{(2)}$, a sector of one model is in general cut by the sectors of the other, and the probability of ending in a given sector after relaxation is $prob(C^{(1)}\to C^{(2)}_{\gamma})=|C^{(1)}\cap C^{(2)}_{\gamma}|/|C^{(1)}|$, computable by intersection counting. Alternating the two twists produces spreading, splitting and oscillations of sectors; physically, the switching can be read as an external driving of the boundary bond, or as an unstable field localized on that bond which changes the charges of particles crossing it. Finally, the authors also build Markov models from set-theoretical solutions beyond the Lyubashenko class (such solutions exist already for $|X|=3$) and show by sector counting that one such model is conjugate to no twisted SSEP already for $L=3$.

For out-of-equilibrium physics, two standard devices exist in integrable systems. Open boundaries preserving integrability are built, as in quantum spin chains, from solutions of the reflection equation (a consistency condition for boundary matrices analogous to the YBE); its set-theoretical version, the set-theoretical reflection equation, has solutions known as reflection maps, classified so far only for restricted classes of bulk solutions. Baxterisation, i.e. introducing a spectral parameter linearly as in $\check{r}(u)=(u\check{r}+\mathrm{Id})/(u+1)$, supplies the parameter dependence needed for both the reflection-equation framework and for deformed local jump operators; if the deformation also breaks the left-right symmetry of the transition rates, one obtains set-theoretical analogues of the passage from SSEP to biased (ASEP-type) dynamics. A same-author follow-up paper used exactly these devices to define the $(p,q)$-SSEP ($p$ singlet species plus $q$ pairs of reactive species that can transform into one another, generalizing the non-Lyubashenko example above) and to construct its integrable open boundaries. The boundary matrices it obtains from the constant reflection equation form a general boundary class, in which the boundary may treat some reactive pairs symmetrically and distinguish the two species of other pairs; its symmetry-breaking part consists of those boundaries that distinguish the two species of at least one reactive pair through unequal rates, so that the exchange symmetry $a^-\leftrightarrow a^+$ and the projection onto a multi-species SSEP are lost. The follow-up determined the stationary currents and densities only for the fully symmetry-preserving case, in which the boundary treats every reactive pair symmetrically; the questions below concern what still remains open after it.

Problem Statement

The common objective is to extend the exactly-solved description of integrable set-theoretical Markov processes (twisted SSEP and its non-Lyubashenko generalizations) from the fixed-twist, periodic, equilibrium setting of the source paper to the time-dependent and genuinely out-of-equilibrium settings, retaining integrability throughout. Three linked directions are asked; direction (1) supplies the mathematical foundation for direction (2), and direction (3) is the structural-construction counterpart. (The fourth direction announced in the source, the detailed study of its non-Lyubashenko example for all $L>2$, has been resolved by the authors' follow-up arXiv:2607.18959 and is not part of this problem.)

(1) Twist-switching ergodicity classification. Determine necessary and sufficient conditions on two permutations $f^{(1)}$ and $f^{(2)}$ of $\{0,\dots,N-1\}$ such that, starting from any configuration of the ring, alternating the dynamics of the twisted SSEP $M_{f^{(1)}}$ and $M_{f^{(2)}}$ (turning one twist on and the other off, each for a positive time interval, with relaxation within sectors) gives a non-zero probability of occupying any given configuration after finitely many switches.

(2) Finite-time response to time-dependent twists. Quantify the finite-time effects of time-dependent twist perturbations (periodic, transient, and other prescribed protocols) on the evolution and on observables, in particular how the stationary-sector weights redistribute before relaxation, including the physically motivated case where the modification of the twist is caused by an unstable field on a bond of the lattice affecting the charges of the particles, and the analogous set-theoretical models where particle charges can change at every bond upon exchange. A complete answer is a general, protocol-parametrised finite-time result covering an arbitrary prescribed time-dependent twist, with the periodic and transient (single-switch) protocols required at least, and quantifying the pre-relaxation redistribution of the stationary-sector weights together with its effect on observables such as the mean charge transport; no other protocol needs separate treatment.

(3) Out-of-equilibrium integrable set-theoretical models. Construct and analyse integrable set-theoretical Markov models with non-equilibrium steady states, specifically: (a) parameter-dependent deformations of set-theoretical YBE solutions whose local jump operators have asymmetric transition probabilities (the set-theoretical analogue of the passage from SSEP to ASEP-type bias); (b) integrable boundaries obtained from solutions of the set-theoretical reflection equation, together with a classification of such solutions in this framework, going beyond the constant reflection equation solutions used in the follow-up; and (c) combinations of parameter-dependent bulk deformations with integrable boundaries. For the resulting open models, determine the stationary distributions and physical quantities (mean currents, densities, and reaction-type rates for reactive species) for general and symmetry-breaking boundary classes, for which no Matrix Ansatz (matrix-product stationary-state method) is currently available. All three construction targets (a)-(c) are jointly required, with (c) the general combination that subsumes (a) and (b) as special cases; a solution of (c) therefore settles (a) and (b) as well, whereas a solution of only (a) or only (b) does not by itself settle the direction. The closing requirement on stationary distributions and physical quantities applies to every model constructed and must cover the whole general boundary class of the Background, including its symmetry-breaking part.

References

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